rngohomf - Mathbox for Jeff Madsen

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Theorem rngohomf 37480

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 2728	. . . . 5 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		rnghomf.2	. . . . 5 ⊢ 𝑋 = ran 𝐺
4		eqid 2728	. . . . 5 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
5		rnghomf.3	. . . . 5 ⊢ 𝐽 = (1^st ‘𝑆)
6		eqid 2728	. . . . 5 ⊢ (2^nd ‘𝑆) = (2^nd ‘𝑆)
7		rnghomf.4	. . . . 5 ⊢ 𝑌 = ran 𝐽
8		eqid 2728	. . . . 5 ⊢ (GId‘(2^nd ‘𝑆)) = (GId‘(2^nd ‘𝑆))
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 37479	. . . 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 475	. . 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 1139	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:𝑋⟶𝑌)
12	11	3impa 1107	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:𝑋⟶𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ran crn 5683 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 1^st c1st 7999 2^nd c2nd 8000 GIdcgi 30328 RingOpscrngo 37408 RingOpsHom crngohom 37474

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-map 8855 df-rngohom 37477

This theorem is referenced by: rngohomcl 37481 rngogrphom 37485 rngohomco 37488 keridl 37546

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