rngohommul - Mathbox for Jeff Madsen

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Theorem rngohommul 36826

Description: Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.)

**Hypotheses**
Ref	Expression
rnghommul.1	⊢ 𝐺 = (1^st ‘𝑅)
rnghommul.2	⊢ 𝑋 = ran 𝐺
rnghommul.3	⊢ 𝐻 = (2^nd ‘𝑅)
rnghommul.4	⊢ 𝐾 = (2^nd ‘𝑆)

**Assertion**
Ref	Expression
rngohommul	⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))

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

Step	Hyp	Ref	Expression
1		rnghommul.1	. . . . . . 7 ⊢ 𝐺 = (1^st ‘𝑅)
2		rnghommul.3	. . . . . . 7 ⊢ 𝐻 = (2^nd ‘𝑅)
3		rnghommul.2	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
4		eqid 2732	. . . . . . 7 ⊢ (GId‘𝐻) = (GId‘𝐻)
5		eqid 2732	. . . . . . 7 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
6		rnghommul.4	. . . . . . 7 ⊢ 𝐾 = (2^nd ‘𝑆)
7		eqid 2732	. . . . . . 7 ⊢ ran (1^st ‘𝑆) = ran (1^st ‘𝑆)
8		eqid 2732	. . . . . . 7 ⊢ (GId‘𝐾) = (GId‘𝐾)
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 36821	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:𝑋⟶ran (1^st ‘𝑆) ∧ (𝐹‘(GId‘𝐻)) = (GId‘𝐾) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))))
10	9	biimpa 477	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋⟶ran (1^st ‘𝑆) ∧ (𝐹‘(GId‘𝐻)) = (GId‘𝐾) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))))
11	10	simp3d 1144	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))
12	11	3impa 1110	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))
13		simpr 485	. . . 4 ⊢ (((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))) → (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))
14	13	2ralimi 3123	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))
15	12, 14	syl 17	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))
16		fvoveq1 7428	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥𝐻𝑦)) = (𝐹‘(𝐴𝐻𝑦)))
17		fveq2 6888	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
18	17	oveq1d 7420	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥)𝐾(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦)))
19	16, 18	eqeq12d 2748	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐻𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦))))
20		oveq2 7413	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
21	20	fveq2d 6892	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴𝐻𝑦)) = (𝐹‘(𝐴𝐻𝐵)))
22		fveq2 6888	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
23	22	oveq2d 7421	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴)𝐾(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))
24	21, 23	eqeq12d 2748	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴𝐻𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵))))
25	19, 24	rspc2v 3621	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵))))
26	15, 25	mpan9 507	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ran crn 5676 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 1^st c1st 7969 2^nd c2nd 7970 GIdcgi 29730 RingOpscrngo 36750 RngHom crnghom 36816

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-map 8818 df-rngohom 36819

This theorem is referenced by: rngohomco 36830 rngoisocnv 36837 crngohomfo 36862 keridl 36888

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