rngohom1 - Mathbox for Jeff Madsen

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Theorem rngohom1 38306

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 2737	. . . . 5 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		rnghom1.1	. . . . 5 ⊢ 𝐻 = (2^nd ‘𝑅)
3		eqid 2737	. . . . 5 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4		rnghom1.2	. . . . 5 ⊢ 𝑈 = (GId‘𝐻)
5		eqid 2737	. . . . 5 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
6		rnghom1.3	. . . . 5 ⊢ 𝐾 = (2^nd ‘𝑆)
7		eqid 2737	. . . . 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 38303	. . . 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 1144	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑈) = 𝑉)
12	11	3impa 1110	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 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ran crn 5626 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 1^st c1st 7934 2^nd c2nd 7935 GIdcgi 30579 RingOpscrngo 38232 RingOpsHom crngohom 38298

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7364 df-oprab 7365 df-mpo 7366 df-map 8769 df-rngohom 38301

This theorem is referenced by: rngohomco 38312 rngoisocnv 38319

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