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Theorem dfrhm2 19463
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 19461 . 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 19293 . . . . . . . 8 (𝑟 ∈ Ring → 𝑟 ∈ Grp)
3 ringgrp 19293 . . . . . . . 8 (𝑠 ∈ Ring → 𝑠 ∈ Grp)
4 eqid 2822 . . . . . . . . 9 (Base‘𝑟) = (Base‘𝑟)
5 eqid 2822 . . . . . . . . 9 (Base‘𝑠) = (Base‘𝑠)
6 eqid 2822 . . . . . . . . 9 (+g𝑟) = (+g𝑟)
7 eqid 2822 . . . . . . . . 9 (+g𝑠) = (+g𝑠)
84, 5, 6, 7isghm3 18350 . . . . . . . 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 598 . . . . . . 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 2951 . . . . . 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 3139 . . . . . . 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 6665 . . . . . . . . . 10 (Base‘𝑠) ∈ V
13 fvex 6665 . . . . . . . . . 10 (Base‘𝑟) ∈ V
1412, 13elmap 8422 . . . . . . . . 9 (𝑓 ∈ ((Base‘𝑠) ↑m (Base‘𝑟)) ↔ 𝑓:(Base‘𝑟)⟶(Base‘𝑠))
1514anbi1i 626 . . . . . . . 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 2887 . . . . . . 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 2845 . . . . . 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, 17eqtr4di 2875 . . . . 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 2822 . . . . . . . . 9 (mulGrp‘𝑟) = (mulGrp‘𝑟)
2019ringmgp 19294 . . . . . . . 8 (𝑟 ∈ Ring → (mulGrp‘𝑟) ∈ Mnd)
21 eqid 2822 . . . . . . . . 9 (mulGrp‘𝑠) = (mulGrp‘𝑠)
2221ringmgp 19294 . . . . . . . 8 (𝑠 ∈ Ring → (mulGrp‘𝑠) ∈ Mnd)
2319, 4mgpbas 19236 . . . . . . . . . 10 (Base‘𝑟) = (Base‘(mulGrp‘𝑟))
2421, 5mgpbas 19236 . . . . . . . . . 10 (Base‘𝑠) = (Base‘(mulGrp‘𝑠))
25 eqid 2822 . . . . . . . . . . 11 (.r𝑟) = (.r𝑟)
2619, 25mgpplusg 19234 . . . . . . . . . 10 (.r𝑟) = (+g‘(mulGrp‘𝑟))
27 eqid 2822 . . . . . . . . . . 11 (.r𝑠) = (.r𝑠)
2821, 27mgpplusg 19234 . . . . . . . . . 10 (.r𝑠) = (+g‘(mulGrp‘𝑠))
29 eqid 2822 . . . . . . . . . . 11 (1r𝑟) = (1r𝑟)
3019, 29ringidval 19244 . . . . . . . . . 10 (1r𝑟) = (0g‘(mulGrp‘𝑟))
31 eqid 2822 . . . . . . . . . . 11 (1r𝑠) = (1r𝑠)
3221, 31ringidval 19244 . . . . . . . . . 10 (1r𝑠) = (0g‘(mulGrp‘𝑠))
3323, 24, 26, 28, 30, 32ismhm 17949 . . . . . . . . 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 539 . . . . . . . 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 598 . . . . . . 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 2951 . . . . . 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 3139 . . . . . . 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 626 . . . . . . . . 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 1092 . . . . . . . . 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 281 . . . . . . . 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 2887 . . . . . . 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 2845 . . . . . 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, 42eqtr4di 2875 . . . . 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 4164 . . . 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 . . . . . . 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)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ∧ (𝑓‘(1r𝑟)) = (1r𝑠)))
46 r19.26-2 3163 . . . . . . . 8 (∀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 626 . . . . . . 7 ((∀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 472 . . . . . . 7 (((∀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 300 . . . . . 6 (((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))))
5049rabbii 3448 . . . . 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)))}
51 oveq12 7149 . . . . . . . 8 ((𝑤 = (Base‘𝑠) ∧ 𝑣 = (Base‘𝑟)) → (𝑤m 𝑣) = ((Base‘𝑠) ↑m (Base‘𝑟)))
5251ancoms 462 . . . . . . 7 ((𝑣 = (Base‘𝑟) ∧ 𝑤 = (Base‘𝑠)) → (𝑤m 𝑣) = ((Base‘𝑠) ↑m (Base‘𝑟)))
53 raleq 3386 . . . . . . . . . 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)))))
5453raleqbi1dv 3384 . . . . . . . . 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)))))
5554adantr 484 . . . . . . . 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)))))
5655anbi2d 631 . . . . . . 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))))))
5752, 56rabeqbidv 3461 . . . . . 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))))})
5813, 12, 57csbie2 3894 . . . . 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))))}
59 inrab 4249 . . . . 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)))}
6050, 58, 593eqtr4i 2855 . . . 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))})
6144, 60syl6reqr 2876 . . 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))))
6261mpoeq3ia 7216 . 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))))
631, 62eqtri 2845 1 RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  {cab 2800  wral 3130  {crab 3134  csb 3855  cin 3907  wf 6330  cfv 6334  (class class class)co 7140  cmpo 7142  m cmap 8393  Basecbs 16474  +gcplusg 16556  .rcmulr 16557  Mndcmnd 17902   MndHom cmhm 17945  Grpcgrp 18094   GrpHom cghm 18346  mulGrpcmgp 19230  1rcur 19242  Ringcrg 19288   RingHom crh 19458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-plusg 16569  df-0g 16706  df-mhm 17947  df-ghm 18347  df-mgp 19231  df-ur 19243  df-ring 19290  df-rnghom 19461
This theorem is referenced by:  rhmrcl1  19465  rhmrcl2  19466  isrhm  19467  zrhval  20199  rhmfn  44482  rhmval  44483  rhmsubclem1  44650  rhmsubcALTVlem1  44668
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