rngohomf - Mathbox for Jeff Madsen

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Theorem rngohomf 37960

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 2729	. . . . 5 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		rnghomf.2	. . . . 5 ⊢ 𝑋 = ran 𝐺
4		eqid 2729	. . . . 5 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
5		rnghomf.3	. . . . 5 ⊢ 𝐽 = (1^st ‘𝑆)
6		eqid 2729	. . . . 5 ⊢ (2^nd ‘𝑆) = (2^nd ‘𝑆)
7		rnghomf.4	. . . . 5 ⊢ 𝑌 = ran 𝐽
8		eqid 2729	. . . . 5 ⊢ (GId‘(2^nd ‘𝑆)) = (GId‘(2^nd ‘𝑆))
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 37959	. . . 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 1540 ∈ wcel 2109 ∀wral 3044 ran crn 5639 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 GIdcgi 30419 RingOpscrngo 37888 RingOpsHom crngohom 37954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-map 8801 df-rngohom 37957

This theorem is referenced by: rngohomcl 37961 rngogrphom 37965 rngohomco 37968 keridl 38026

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