rngohomsub - Mathbox for Jeff Madsen

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Theorem rngohomsub 36841

Description: Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)

**Hypotheses**
Ref	Expression
rnghomsub.1	⊢ 𝐺 = (1^st ‘𝑅)
rnghomsub.2	⊢ 𝑋 = ran 𝐺
rnghomsub.3	⊢ 𝐻 = ( /_𝑔 ‘𝐺)
rnghomsub.4	⊢ 𝐽 = (1^st ‘𝑆)
rnghomsub.5	⊢ 𝐾 = ( /_𝑔 ‘𝐽)

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

**Proof of Theorem rngohomsub**
Step	Hyp	Ref	Expression
1		rnghomsub.1	. . . . 5 ⊢ 𝐺 = (1^st ‘𝑅)
2	1	rngogrpo 36778	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3	2	3ad2ant1 1134	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐺 ∈ GrpOp)
4		rnghomsub.4	. . . . 5 ⊢ 𝐽 = (1^st ‘𝑆)
5	4	rngogrpo 36778	. . . 4 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
6	5	3ad2ant2 1135	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐽 ∈ GrpOp)
7	1, 4	rngogrphom 36839	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))
8	3, 6, 7	3jca 1129	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)))
9		rnghomsub.2	. . 3 ⊢ 𝑋 = ran 𝐺
10		rnghomsub.3	. . 3 ⊢ 𝐻 = ( /_𝑔 ‘𝐺)
11		rnghomsub.5	. . 3 ⊢ 𝐾 = ( /_𝑔 ‘𝐽)
12	9, 10, 11	ghomdiv 36760	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))
13	8, 12	sylan 581	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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ran crn 5678 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 GrpOpcgr 29742 /_𝑔 cgs 29745 GrpOpHom cghomOLD 36751 RingOpscrngo 36762 RngHom crnghom 36828

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-map 8822 df-grpo 29746 df-gid 29747 df-ginv 29748 df-gdiv 29749 df-ablo 29798 df-ghomOLD 36752 df-rngo 36763 df-rngohom 36831

This theorem is referenced by: (None)

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