Working in ZF+GCH where GCH is expressed in the algebraic form "Never $\mathfrak{n} < \mathfrak{m} < 2^\mathfrak{n} $ for infinite cardinals" is there a simple and direct proof, preferably of an algebraic nature and bypassing AC, that $\mathfrak{m}+\mathfrak{n}=\mathfrak{m}\cdot\mathfrak{n}=\max (\mathfrak{m},\mathfrak{n})$.
Usually given ZF the arithmetic of cardinals is developed and it turns out to have an ordered ring structure with exponentiation. Then AC is introduced and cardinal arithmetic is rendered trivial, except for exponentiation. I want to go straight from ZF+GCH to cardinal arithmetic being trivial, without going from ZF+GCH to AC then ZF+AC to cardinal arithmetic being trivial.
By an algebraic style of proof I mean similar in style to this:-
... but then $2\mathfrak{m}=\mathfrak{m}$, because $\mathfrak{m}\le2\mathfrak{m}=\mathfrak{m}+\mathfrak{m}\le2^{\mathfrak{m}}+2^{\mathfrak{m}}=2\cdot2^{\mathfrak{m}}=2^{\mathfrak{m}+1}=2^{\mathfrak{m}}$ and no intermediates by GCH so $\mathfrak{m}=2\mathfrak{m}$ or $2\mathfrak{m}=2^{\mathfrak{m}}$, but $2\mathfrak{m}<2^{\mathfrak{m}}$ ...