rngohomf - Mathbox for Jeff Madsen

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Theorem rngohomf 37953

Description: A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.)

**Hypotheses**
Ref	Expression
rnghomf.1	⊢ 𝐺 = (1^st ‘𝑅)
rnghomf.2	⊢ 𝑋 = ran 𝐺
rnghomf.3	⊢ 𝐽 = (1^st ‘𝑆)
rnghomf.4	⊢ 𝑌 = ran 𝐽

**Assertion**
Ref	Expression
rngohomf	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:𝑋⟶𝑌)

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

Step	Hyp	Ref	Expression
1		rnghomf.1	. . . . 5 ⊢ 𝐺 = (1^st ‘𝑅)
2		eqid 2735	. . . . 5 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		rnghomf.2	. . . . 5 ⊢ 𝑋 = ran 𝐺
4		eqid 2735	. . . . 5 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
5		rnghomf.3	. . . . 5 ⊢ 𝐽 = (1^st ‘𝑆)
6		eqid 2735	. . . . 5 ⊢ (2^nd ‘𝑆) = (2^nd ‘𝑆)
7		rnghomf.4	. . . . 5 ⊢ 𝑌 = ran 𝐽
8		eqid 2735	. . . . 5 ⊢ (GId‘(2^nd ‘𝑆)) = (GId‘(2^nd ‘𝑆))
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 37952	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑆)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥(2^nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦))))))
10	9	biimpa 476	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹:𝑋⟶𝑌 ∧ (𝐹‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑆)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥(2^nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦)))))
11	10	simp1d 1141	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:𝑋⟶𝑌)
12	11	3impa 1109	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:𝑋⟶𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1^st c1st 8011 2^nd c2nd 8012 GIdcgi 30519 RingOpscrngo 37881 RingOpsHom crngohom 37947

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-rngohom 37950

This theorem is referenced by: rngohomcl 37954 rngogrphom 37958 rngohomco 37961 keridl 38019

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