Theorem rngohommul 38904

Description: Obsolete theorem, use rhmmul 20720 instead. Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

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 2761	. . . . . . 7 ⊢ (GId‘𝐻) = (GId‘𝐻)
5		eqid 2761	. . . . . . 7 ⊢ (1st ‘𝑆) = (1st ‘𝑆)
6		rnghommul.4	. . . . . . 7 ⊢ 𝐾 = (2nd ‘𝑆)
7		eqid 2761	. . . . . . 7 ⊢ ran (1st ‘𝑆) = ran (1st ‘𝑆)
8		eqid 2761	. . . . . . 7 ⊢ (GId‘𝐾) = (GId‘𝐾)
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 38899	. . . . . 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 482	. . . . 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 1162	. . . 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 1127	. . 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 490	. . . 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 3133	. . 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 18	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))
16		fvoveq1 7443	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥𝐻𝑦)) = (𝐹‘(𝐴𝐻𝑦)))
17		fveq2 6885	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
18	17	oveq1d 7435	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥)𝐾(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦)))
19	16, 18	eqeq12d 2777	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐻𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦))))
20		oveq2 7428	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
21	20	fveq2d 6889	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴𝐻𝑦)) = (𝐹‘(𝐴𝐻𝐵)))
22		fveq2 6885	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
23	22	oveq2d 7436	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴)𝐾(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))
24	21, 23	eqeq12d 2777	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴𝐻𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵))))
25	19, 24	rspc2v 3587	. 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 516	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ran crn 5652  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  GIdcgi 31092  RingOpscrngo 38828   RingOpsHom crngohom 38894

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8849  df-rngohom 38897

This theorem is used by:  rngohomco  38908  rngoisocnv  38915  crngohomfo  38940  keridl  38966