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Theorem dfrhm2 18698
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 18696 . 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 18533 . . . . . . . 8 (𝑟 ∈ Ring → 𝑟 ∈ Grp)
3 ringgrp 18533 . . . . . . . 8 (𝑠 ∈ Ring → 𝑠 ∈ Grp)
4 eqid 2620 . . . . . . . . 9 (Base‘𝑟) = (Base‘𝑟)
5 eqid 2620 . . . . . . . . 9 (Base‘𝑠) = (Base‘𝑠)
6 eqid 2620 . . . . . . . . 9 (+g𝑟) = (+g𝑟)
7 eqid 2620 . . . . . . . . 9 (+g𝑠) = (+g𝑠)
84, 5, 6, 7isghm3 17642 . . . . . . . 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 494 . . . . . . 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 2740 . . . . . 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 2918 . . . . . . 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 6188 . . . . . . . . . 10 (Base‘𝑠) ∈ V
13 fvex 6188 . . . . . . . . . 10 (Base‘𝑟) ∈ V
1412, 13elmap 7871 . . . . . . . . 9 (𝑓 ∈ ((Base‘𝑠) ↑𝑚 (Base‘𝑟)) ↔ 𝑓:(Base‘𝑟)⟶(Base‘𝑠))
1514anbi1i 730 . . . . . . . 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 2737 . . . . . . 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 2642 . . . . . 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 2672 . . . . 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 2620 . . . . . . . . 9 (mulGrp‘𝑟) = (mulGrp‘𝑟)
2019ringmgp 18534 . . . . . . . 8 (𝑟 ∈ Ring → (mulGrp‘𝑟) ∈ Mnd)
21 eqid 2620 . . . . . . . . 9 (mulGrp‘𝑠) = (mulGrp‘𝑠)
2221ringmgp 18534 . . . . . . . 8 (𝑠 ∈ Ring → (mulGrp‘𝑠) ∈ Mnd)
2319, 4mgpbas 18476 . . . . . . . . . 10 (Base‘𝑟) = (Base‘(mulGrp‘𝑟))
2421, 5mgpbas 18476 . . . . . . . . . 10 (Base‘𝑠) = (Base‘(mulGrp‘𝑠))
25 eqid 2620 . . . . . . . . . . 11 (.r𝑟) = (.r𝑟)
2619, 25mgpplusg 18474 . . . . . . . . . 10 (.r𝑟) = (+g‘(mulGrp‘𝑟))
27 eqid 2620 . . . . . . . . . . 11 (.r𝑠) = (.r𝑠)
2821, 27mgpplusg 18474 . . . . . . . . . 10 (.r𝑠) = (+g‘(mulGrp‘𝑠))
29 eqid 2620 . . . . . . . . . . 11 (1r𝑟) = (1r𝑟)
3019, 29ringidval 18484 . . . . . . . . . 10 (1r𝑟) = (0g‘(mulGrp‘𝑟))
31 eqid 2620 . . . . . . . . . . 11 (1r𝑠) = (1r𝑠)
3221, 31ringidval 18484 . . . . . . . . . 10 (1r𝑠) = (0g‘(mulGrp‘𝑠))
3323, 24, 26, 28, 30, 32ismhm 17318 . . . . . . . . 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 943 . . . . . . . 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 494 . . . . . . 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 2740 . . . . . 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 2918 . . . . . . 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 730 . . . . . . . . 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 1040 . . . . . . . . 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 267 . . . . . . . 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 2737 . . . . . . 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 2642 . . . . . 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 2672 . . . . 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 3807 . . . 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 466 . . . . . . . 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 3061 . . . . . . . . 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 730 . . . . . . . 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 680 . . . . . . . 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 286 . . . . . . 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 3180 . . . . 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 6644 . . . . . . . 8 ((𝑤 = (Base‘𝑠) ∧ 𝑣 = (Base‘𝑟)) → (𝑤𝑚 𝑣) = ((Base‘𝑠) ↑𝑚 (Base‘𝑟)))
5352ancoms 469 . . . . . . 7 ((𝑣 = (Base‘𝑟) ∧ 𝑤 = (Base‘𝑠)) → (𝑤𝑚 𝑣) = ((Base‘𝑠) ↑𝑚 (Base‘𝑟)))
54 raleq 3133 . . . . . . . . . 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 3141 . . . . . . . . 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 481 . . . . . . . 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 739 . . . . . . 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 3190 . . . . . 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 3556 . . . . 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 3891 . . . . 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 2652 . . . 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 2673 . . 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 6705 . 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 2642 1 RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  {cab 2606  wral 2909  {crab 2913  csb 3526  cin 3566  wf 5872  cfv 5876  (class class class)co 6635  cmpt2 6637  𝑚 cmap 7842  Basecbs 15838  +gcplusg 15922  .rcmulr 15923  Mndcmnd 17275   MndHom cmhm 17314  Grpcgrp 17403   GrpHom cghm 17638  mulGrpcmgp 18470  1rcur 18482  Ringcrg 18528   RingHom crh 18693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-plusg 15935  df-0g 16083  df-mhm 17316  df-ghm 17639  df-mgp 18471  df-ur 18483  df-ring 18530  df-rnghom 18696
This theorem is referenced by:  rhmrcl1  18700  rhmrcl2  18701  isrhm  18702  zrhval  19837  rhmfn  41683  rhmval  41684  rhmsubclem1  41851  rhmsubcALTVlem1  41869
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