Theorem rngohomadd 37350

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

Hypotheses

Ref	Expression
rnghomadd.1	⊢ 𝐺 = (1st ‘𝑅)
rnghomadd.2	⊢ 𝑋 = ran 𝐺
rnghomadd.3	⊢ 𝐽 = (1st ‘𝑆)

Assertion

Ref	Expression
rngohomadd	⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹‘𝐴)𝐽(𝐹‘𝐵)))

Proof of Theorem rngohomadd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnghomadd.1	. . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅)
2		eqid 2726	. . . . . . 7 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
3		rnghomadd.2	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
4		eqid 2726	. . . . . . 7 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅))
5		rnghomadd.3	. . . . . . 7 ⊢ 𝐽 = (1st ‘𝑆)
6		eqid 2726	. . . . . . 7 ⊢ (2nd ‘𝑆) = (2nd ‘𝑆)
7		eqid 2726	. . . . . . 7 ⊢ ran 𝐽 = ran 𝐽
8		eqid 2726	. . . . . . 7 ⊢ (GId‘(2nd ‘𝑆)) = (GId‘(2nd ‘𝑆))
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 37346	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ↔ (𝐹:𝑋⟶ran 𝐽 ∧ (𝐹‘(GId‘(2nd ‘𝑅))) = (GId‘(2nd ‘𝑆)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥(2nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑦))))))
10	9	biimpa 476	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹:𝑋⟶ran 𝐽 ∧ (𝐹‘(GId‘(2nd ‘𝑅))) = (GId‘(2nd ‘𝑆)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥(2nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑦)))))
11	10	simp3d 1141	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥(2nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑦))))
12	11	3impa 1107	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥(2nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑦))))
13		simpl 482	. . . 4 ⊢ (((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥(2nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑦))) → (𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
14	13	2ralimi 3117	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥(2nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
15	12, 14	syl 17	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
16		fvoveq1 7428	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥𝐺𝑦)) = (𝐹‘(𝐴𝐺𝑦)))
17		fveq2 6885	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
18	17	oveq1d 7420	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐽(𝐹‘𝑦)))
19	16, 18	eqeq12d 2742	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐺𝑦)) = ((𝐹‘𝐴)𝐽(𝐹‘𝑦))))
20		oveq2 7413	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
21	20	fveq2d 6889	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴𝐺𝑦)) = (𝐹‘(𝐴𝐺𝐵)))
22		fveq2 6885	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
23	22	oveq2d 7421	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴)𝐽(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐽(𝐹‘𝐵)))
24	21, 23	eqeq12d 2742	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴𝐺𝑦)) = ((𝐹‘𝐴)𝐽(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐺𝐵)) = ((𝐹‘𝐴)𝐽(𝐹‘𝐵))))
25	19, 24	rspc2v 3617	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹‘𝐴)𝐽(𝐹‘𝐵))))
26	15, 25	mpan9 506	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹‘𝐴)𝐽(𝐹‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ran crn 5670  ⟶wf 6533  ‘cfv 6537  (class class class)co 7405  1^st c1st 7972  2^nd c2nd 7973  GIdcgi 30252  RingOpscrngo 37275   RingOpsHom crngohom 37341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8824  df-rngohom 37344

This theorem is referenced by:  rngogrphom  37352  rngohomco  37355  rngoisocnv  37362  keridl  37413