rngohomf - Mathbox for Jeff Madsen

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Theorem rngohomf 38016

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 2731	. . . . 5 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		rnghomf.2	. . . . 5 ⊢ 𝑋 = ran 𝐺
4		eqid 2731	. . . . 5 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
5		rnghomf.3	. . . . 5 ⊢ 𝐽 = (1^st ‘𝑆)
6		eqid 2731	. . . . 5 ⊢ (2^nd ‘𝑆) = (2^nd ‘𝑆)
7		rnghomf.4	. . . . 5 ⊢ 𝑌 = ran 𝐽
8		eqid 2731	. . . . 5 ⊢ (GId‘(2^nd ‘𝑆)) = (GId‘(2^nd ‘𝑆))
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 38015	. . . 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 1142	. 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 1541 ∈ wcel 2111 ∀wral 3047 ran crn 5615 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 2^nd c2nd 7920 GIdcgi 30470 RingOpscrngo 37944 RingOpsHom crngohom 38010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-map 8752 df-rngohom 38013

This theorem is referenced by: rngohomcl 38017 rngogrphom 38021 rngohomco 38024 keridl 38082

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