rngohom1 - Mathbox for Jeff Madsen

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Theorem rngohom1 34391

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 RngHom 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 2778	. . . . 5 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		rnghom1.1	. . . . 5 ⊢ 𝐻 = (2^nd ‘𝑅)
3		eqid 2778	. . . . 5 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4		rnghom1.2	. . . . 5 ⊢ 𝑈 = (GId‘𝐻)
5		eqid 2778	. . . . 5 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
6		rnghom1.3	. . . . 5 ⊢ 𝐾 = (2^nd ‘𝑆)
7		eqid 2778	. . . . 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 34388	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 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 470	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 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 1134	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉)
12	11	3impa 1097	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ran crn 5356 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 1^st c1st 7443 2^nd c2nd 7444 GIdcgi 27917 RingOpscrngo 34317 RngHom crnghom 34383

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-map 8142 df-rngohom 34386

This theorem is referenced by: rngohomco 34397 rngoisocnv 34404

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