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Theorem dfrhm2 18483
Description: The property of a ring homomorphism can be decomposed into separate homomorphic conditions for addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Assertion
Ref Expression
dfrhm2 RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))
Distinct variable group:   𝑠,𝑟

Proof of Theorem dfrhm2
Dummy variables 𝑣 𝑤 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rnghom 18481 . 2 RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))))})
2 ringgrp 18318 . . . . . . . 8 (𝑟 ∈ Ring → 𝑟 ∈ Grp)
3 ringgrp 18318 . . . . . . . 8 (𝑠 ∈ Ring → 𝑠 ∈ Grp)
4 eqid 2606 . . . . . . . . 9 (Base‘𝑟) = (Base‘𝑟)
5 eqid 2606 . . . . . . . . 9 (Base‘𝑠) = (Base‘𝑠)
6 eqid 2606 . . . . . . . . 9 (+g𝑟) = (+g𝑟)
7 eqid 2606 . . . . . . . . 9 (+g𝑠) = (+g𝑠)
84, 5, 6, 7isghm3 17427 . . . . . . . 8 ((𝑟 ∈ Grp ∧ 𝑠 ∈ Grp) → (𝑓 ∈ (𝑟 GrpHom 𝑠) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)))))
92, 3, 8syl2an 492 . . . . . . 7 ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → (𝑓 ∈ (𝑟 GrpHom 𝑠) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)))))
109abbi2dv 2725 . . . . . 6 ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → (𝑟 GrpHom 𝑠) = {𝑓 ∣ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)))})
11 df-rab 2901 . . . . . . 7 {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦))} = {𝑓 ∣ (𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)))}
12 fvex 6095 . . . . . . . . . 10 (Base‘𝑠) ∈ V
13 fvex 6095 . . . . . . . . . 10 (Base‘𝑟) ∈ V
1412, 13elmap 7746 . . . . . . . . 9 (𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ↔ 𝑓:(Base‘𝑟)⟶(Base‘𝑠))
1514anbi1i 726 . . . . . . . 8 ((𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦))) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦))))
1615abbii 2722 . . . . . . 7 {𝑓 ∣ (𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)))} = {𝑓 ∣ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)))}
1711, 16eqtri 2628 . . . . . 6 {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦))} = {𝑓 ∣ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)))}
1810, 17syl6eqr 2658 . . . . 5 ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → (𝑟 GrpHom 𝑠) = {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦))})
19 eqid 2606 . . . . . . . . 9 (mulGrp‘𝑟) = (mulGrp‘𝑟)
2019ringmgp 18319 . . . . . . . 8 (𝑟 ∈ Ring → (mulGrp‘𝑟) ∈ Mnd)
21 eqid 2606 . . . . . . . . 9 (mulGrp‘𝑠) = (mulGrp‘𝑠)
2221ringmgp 18319 . . . . . . . 8 (𝑠 ∈ Ring → (mulGrp‘𝑠) ∈ Mnd)
2319, 4mgpbas 18261 . . . . . . . . . 10 (Base‘𝑟) = (Base‘(mulGrp‘𝑟))
2421, 5mgpbas 18261 . . . . . . . . . 10 (Base‘𝑠) = (Base‘(mulGrp‘𝑠))
25 eqid 2606 . . . . . . . . . . 11 (.r𝑟) = (.r𝑟)
2619, 25mgpplusg 18259 . . . . . . . . . 10 (.r𝑟) = (+g‘(mulGrp‘𝑟))
27 eqid 2606 . . . . . . . . . . 11 (.r𝑠) = (.r𝑠)
2821, 27mgpplusg 18259 . . . . . . . . . 10 (.r𝑠) = (+g‘(mulGrp‘𝑠))
29 eqid 2606 . . . . . . . . . . 11 (1r𝑟) = (1r𝑟)
3019, 29ringidval 18269 . . . . . . . . . 10 (1r𝑟) = (0g‘(mulGrp‘𝑟))
31 eqid 2606 . . . . . . . . . . 11 (1r𝑠) = (1r𝑠)
3221, 31ringidval 18269 . . . . . . . . . 10 (1r𝑠) = (0g‘(mulGrp‘𝑠))
3323, 24, 26, 28, 30, 32ismhm 17103 . . . . . . . . 9 (𝑓 ∈ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)) ↔ (((mulGrp‘𝑟) ∈ Mnd ∧ (mulGrp‘𝑠) ∈ Mnd) ∧ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))))
3433baib 941 . . . . . . . 8 (((mulGrp‘𝑟) ∈ Mnd ∧ (mulGrp‘𝑠) ∈ Mnd) → (𝑓 ∈ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))))
3520, 22, 34syl2an 492 . . . . . . 7 ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → (𝑓 ∈ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))))
3635abbi2dv 2725 . . . . . 6 ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)) = {𝑓 ∣ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))})
37 df-rab 2901 . . . . . . 7 {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))} = {𝑓 ∣ (𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠)))}
3814anbi1i 726 . . . . . . . . 9 ((𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))))
39 3anass 1034 . . . . . . . . 9 ((𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠)) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))))
4038, 39bitr4i 265 . . . . . . . 8 ((𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))) ↔ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠)))
4140abbii 2722 . . . . . . 7 {𝑓 ∣ (𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠)))} = {𝑓 ∣ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))}
4237, 41eqtri 2628 . . . . . 6 {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))} = {𝑓 ∣ (𝑓:(Base‘𝑟)⟶(Base‘𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))}
4336, 42syl6eqr 2658 . . . . 5 ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)) = {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))})
4418, 43ineq12d 3773 . . . 4 ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))) = ({𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦))} ∩ {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))}))
45 ancom 464 . . . . . . . 8 (((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))) ↔ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ∧ (𝑓‘(1r𝑟)) = (1r𝑠)))
46 r19.26-2 3043 . . . . . . . . 9 (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))))
4746anbi1i 726 . . . . . . . 8 ((∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ∧ (𝑓‘(1r𝑟)) = (1r𝑠)) ↔ ((∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ∧ (𝑓‘(1r𝑟)) = (1r𝑠)))
48 anass 678 . . . . . . . 8 (((∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ∧ (𝑓‘(1r𝑟)) = (1r𝑠)) ↔ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))))
4945, 47, 483bitri 284 . . . . . . 7 (((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))) ↔ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))))
5049a1i 11 . . . . . 6 (𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) → (((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))) ↔ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠)))))
5150rabbiia 3157 . . . . 5 {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ ((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))))} = {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠)))}
52 oveq12 6533 . . . . . . . 8 ((𝑤 = (Base‘𝑠) ∧ 𝑣 = (Base‘𝑟)) → (𝑤𝑚 𝑣) = ((Base‘𝑠) ↑𝑚 (Base‘𝑟)))
5352ancoms 467 . . . . . . 7 ((𝑣 = (Base‘𝑟) ∧ 𝑤 = (Base‘𝑠)) → (𝑤𝑚 𝑣) = ((Base‘𝑠) ↑𝑚 (Base‘𝑟)))
54 raleq 3111 . . . . . . . . . 10 (𝑣 = (Base‘𝑟) → (∀𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))))
5554raleqbi1dv 3119 . . . . . . . . 9 (𝑣 = (Base‘𝑟) → (∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))))
5655adantr 479 . . . . . . . 8 ((𝑣 = (Base‘𝑟) ∧ 𝑤 = (Base‘𝑠)) → (∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))))
5756anbi2d 735 . . . . . . 7 ((𝑣 = (Base‘𝑟) ∧ 𝑤 = (Base‘𝑠)) → (((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))) ↔ ((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))))))
5853, 57rabeqbidv 3164 . . . . . 6 ((𝑣 = (Base‘𝑟) ∧ 𝑤 = (Base‘𝑠)) → {𝑓 ∈ (𝑤𝑚 𝑣) ∣ ((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))))} = {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ ((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))))})
5913, 12, 58csbie2 3525 . . . . 5 (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))))} = {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ ((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))))}
60 inrab 3854 . . . . 5 ({𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦))} ∩ {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))}) = {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠)))}
6151, 59, 603eqtr4i 2638 . . . 4 (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))))} = ({𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦))} ∩ {𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ∣ (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)(𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ∧ (𝑓‘(1r𝑟)) = (1r𝑠))})
6244, 61syl6reqr 2659 . . 3 ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) → (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))))} = ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))
6362mpt2eq3ia 6593 . 2 (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))))}) = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))
641, 63eqtri 2628 1 RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  {cab 2592  wral 2892  {crab 2896  csb 3495  cin 3535  wf 5783  cfv 5787  (class class class)co 6524  cmpt2 6526  𝑚 cmap 7718  Basecbs 15638  +gcplusg 15711  .rcmulr 15712  Mndcmnd 17060   MndHom cmhm 17099  Grpcgrp 17188   GrpHom cghm 17423  mulGrpcmgp 18255  1rcur 18267  Ringcrg 18313   RingHom crh 18478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-er 7603  df-map 7720  df-en 7816  df-dom 7817  df-sdom 7818  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-2 10923  df-ndx 15641  df-slot 15642  df-base 15643  df-sets 15644  df-plusg 15724  df-0g 15868  df-mhm 17101  df-ghm 17424  df-mgp 18256  df-ur 18268  df-ring 18315  df-rnghom 18481
This theorem is referenced by:  rhmrcl1  18485  rhmrcl2  18486  isrhm  18487  zrhval  19617  rhmfn  41707  rhmval  41708  rhmsubclem1  41877  rhmsubcALTVlem1  41896
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