Theorem rngohomadd 38019

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 2731	. . . . . . 7 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
3		rnghomadd.2	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
4		eqid 2731	. . . . . . 7 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅))
5		rnghomadd.3	. . . . . . 7 ⊢ 𝐽 = (1st ‘𝑆)
6		eqid 2731	. . . . . . 7 ⊢ (2nd ‘𝑆) = (2nd ‘𝑆)
7		eqid 2731	. . . . . . 7 ⊢ ran 𝐽 = ran 𝐽
8		eqid 2731	. . . . . . 7 ⊢ (GId‘(2nd ‘𝑆)) = (GId‘(2nd ‘𝑆))
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 38015	. . . . . 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 1144	. . . 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 1109	. . 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 3102	. . 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 7369	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥𝐺𝑦)) = (𝐹‘(𝐴𝐺𝑦)))
17		fveq2 6822	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
18	17	oveq1d 7361	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐽(𝐹‘𝑦)))
19	16, 18	eqeq12d 2747	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐺𝑦)) = ((𝐹‘𝐴)𝐽(𝐹‘𝑦))))
20		oveq2 7354	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
21	20	fveq2d 6826	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴𝐺𝑦)) = (𝐹‘(𝐴𝐺𝐵)))
22		fveq2 6822	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
23	22	oveq2d 7362	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴)𝐽(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐽(𝐹‘𝐵)))
24	21, 23	eqeq12d 2747	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴𝐺𝑦)) = ((𝐹‘𝐴)𝐽(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐺𝐵)) = ((𝐹‘𝐴)𝐽(𝐹‘𝐵))))
25	19, 24	rspc2v 3583	. 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 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ran crn 5615  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920  GIdcgi 30470  RingOpscrngo 37944   RingOpsHom crngohom 38010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-map 8752  df-rngohom 38013

This theorem is referenced by:  rngogrphom  38021  rngohomco  38024  rngoisocnv  38031  keridl  38082