rngohom1 - Mathbox for Jeff Madsen

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Theorem rngohom1 37997

Description: A ring homomorphism preserves 1. (Contributed by Jeff Madsen, 24-Jun-2011.)

**Hypotheses**
Ref	Expression
rnghom1.1	⊢ 𝐻 = (2^nd ‘𝑅)
rnghom1.2	⊢ 𝑈 = (GId‘𝐻)
rnghom1.3	⊢ 𝐾 = (2^nd ‘𝑆)
rnghom1.4	⊢ 𝑉 = (GId‘𝐾)

**Assertion**
Ref	Expression
rngohom1	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑈) = 𝑉)

Proof of Theorem rngohom1 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . . 5 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		rnghom1.1	. . . . 5 ⊢ 𝐻 = (2^nd ‘𝑅)
3		eqid 2736	. . . . 5 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4		rnghom1.2	. . . . 5 ⊢ 𝑈 = (GId‘𝐻)
5		eqid 2736	. . . . 5 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
6		rnghom1.3	. . . . 5 ⊢ 𝐾 = (2^nd ‘𝑆)
7		eqid 2736	. . . . 5 ⊢ ran (1^st ‘𝑆) = ran (1^st ‘𝑆)
8		rnghom1.4	. . . . 5 ⊢ 𝑉 = (GId‘𝐾)
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 37994	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ↔ (𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆) ∧ (𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))))
10	9	biimpa 476	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆) ∧ (𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))))
11	10	simp2d 1143	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑈) = 𝑉)
12	11	3impa 1109	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑈) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3052 ran crn 5660 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 GIdcgi 30476 RingOpscrngo 37923 RingOpsHom crngohom 37989

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-map 8847 df-rngohom 37992

This theorem is referenced by: rngohomco 38003 rngoisocnv 38010

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