rngohom1 - Mathbox for Jeff Madsen

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Theorem rngohom1 38721

Description: Obsolete theorem, use rhm1 20638 instead. A ring homomorphism preserves 1. (Contributed by Jeff Madsen, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

**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 2760	. . . . 5 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		rnghom1.1	. . . . 5 ⊢ 𝐻 = (2^nd ‘𝑅)
3		eqid 2760	. . . . 5 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4		rnghom1.2	. . . . 5 ⊢ 𝑈 = (GId‘𝐻)
5		eqid 2760	. . . . 5 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
6		rnghom1.3	. . . . 5 ⊢ 𝐾 = (2^nd ‘𝑆)
7		eqid 2760	. . . . 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 38718	. . . 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 482	. . 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 1161	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑈) = 𝑉)
12	11	3impa 1127	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑈) = 𝑉)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ran crn 5656 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 1^st c1st 7985 2^nd c2nd 7986 GIdcgi 30974 RingOpscrngo 38647 RingOpsHom crngohom 38713

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8831 df-rngohom 38716

This theorem is used by: rngohomco 38727 rngoisocnv 38734

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