rngohomsub - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > rngohomsub		Structured version Visualization version GIF version

Theorem rngohomsub 38827

Description: Obsolete theorem, use rhmsub 20680 instead. Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

**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 RingOpsHom 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 38764	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3	2	3ad2ant1 1151	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐺 ∈ GrpOp)
4		rnghomsub.4	. . . . 5 ⊢ 𝐽 = (1^st ‘𝑆)
5	4	rngogrpo 38764	. . . 4 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
6	5	3ad2ant2 1152	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐽 ∈ GrpOp)
7	1, 4	rngogrphom 38825	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))
8	3, 6, 7	3jca 1146	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 38746	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))
13	8, 12	sylan 592	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ran crn 5648 ‘cfv 6527 (class class class)co 7408 1^st c1st 7982 GrpOpcgr 31024 /_𝑔 cgs 31027 GrpOpHom cghomOLD 38737 RingOpscrngo 38748 RingOpsHom crngohom 38814

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 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7984 df-2nd 7985 df-map 8827 df-grpo 31028 df-gid 31029 df-ginv 31030 df-gdiv 31031 df-ablo 31080 df-ghomOLD 38738 df-rngo 38749 df-rngohom 38817

This theorem is used by: (None)

W3C validator