Theorem rngohommul 37999

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

Hypotheses

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

Assertion

Ref	Expression
rngohommul	⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 = (1st ‘𝑅)
2		rnghommul.3	. . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅)
3		rnghommul.2	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
4		eqid 2736	. . . . . . 7 ⊢ (GId‘𝐻) = (GId‘𝐻)
5		eqid 2736	. . . . . . 7 ⊢ (1st ‘𝑆) = (1st ‘𝑆)
6		rnghommul.4	. . . . . . 7 ⊢ 𝐾 = (2nd ‘𝑆)
7		eqid 2736	. . . . . . 7 ⊢ ran (1st ‘𝑆) = ran (1st ‘𝑆)
8		eqid 2736	. . . . . . 7 ⊢ (GId‘𝐾) = (GId‘𝐾)
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 37994	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ↔ (𝐹:𝑋⟶ran (1st ‘𝑆) ∧ (𝐹‘(GId‘𝐻)) = (GId‘𝐾) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))))
10	9	biimpa 476	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹:𝑋⟶ran (1st ‘𝑆) ∧ (𝐹‘(GId‘𝐻)) = (GId‘𝐾) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))))
11	10	simp3d 1144	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))
12	11	3impa 1109	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))
13		simpr 484	. . . 4 ⊢ (((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))) → (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))
14	13	2ralimi 3111	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘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 RingOpsHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))
16		fvoveq1 7433	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥𝐻𝑦)) = (𝐹‘(𝐴𝐻𝑦)))
17		fveq2 6881	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
18	17	oveq1d 7425	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥)𝐾(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦)))
19	16, 18	eqeq12d 2752	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐻𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦))))
20		oveq2 7418	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
21	20	fveq2d 6885	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴𝐻𝑦)) = (𝐹‘(𝐴𝐻𝐵)))
22		fveq2 6881	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
23	22	oveq2d 7426	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴)𝐾(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))
24	21, 23	eqeq12d 2752	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴𝐻𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵))))
25	19, 24	rspc2v 3617	. 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 506	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ran crn 5660  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  1^st c1st 7991  2^nd c2nd 7992  GIdcgi 30476  RingOpscrngo 37923   RingOpsHom crngohom 37989

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8847  df-rngohom 37992

This theorem is referenced by:  rngohomco  38003  rngoisocnv  38010  crngohomfo  38035  keridl  38061