rngohomsub - Mathbox for Jeff Madsen

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Theorem rngohomsub 35825

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 35762	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3	2	3ad2ant1 1135	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐺 ∈ GrpOp)
4		rnghomsub.4	. . . . 5 ⊢ 𝐽 = (1^st ‘𝑆)
5	4	rngogrpo 35762	. . . 4 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
6	5	3ad2ant2 1136	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐽 ∈ GrpOp)
7	1, 4	rngogrphom 35823	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))
8	3, 6, 7	3jca 1130	. 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 35744	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))
13	8, 12	sylan 583	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 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ran crn 5541 ‘cfv 6369 (class class class)co 7202 1^st c1st 7748 GrpOpcgr 28542 /_𝑔 cgs 28545 GrpOpHom cghomOLD 35735 RingOpscrngo 35746 RngHom crnghom 35812

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-map 8499 df-grpo 28546 df-gid 28547 df-ginv 28548 df-gdiv 28549 df-ablo 28598 df-ghomOLD 35736 df-rngo 35747 df-rngohom 35815

This theorem is referenced by: (None)

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