Theorem ghomdiv 37899

Description: Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.)

Hypotheses

Ref	Expression
ghomdiv.1	⊢ 𝑋 = ran 𝐺
ghomdiv.2	⊢ 𝐷 = ( /𝑔 ‘𝐺)
ghomdiv.3	⊢ 𝐶 = ( /𝑔 ‘𝐻)

Assertion

Ref	Expression
ghomdiv	⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐷𝐵)) = ((𝐹‘𝐴)𝐶(𝐹‘𝐵)))

Proof of Theorem ghomdiv

Step	Hyp	Ref	Expression
1		simpl2 1193	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐻 ∈ GrpOp)
2		ghomdiv.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
3		eqid 2737	. . . . . . 7 ⊢ ran 𝐻 = ran 𝐻
4	2, 3	ghomf 37897	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶ran 𝐻)
5	4	ffvelcdmda 7104	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐻)
6	5	adantrr 717	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐴) ∈ ran 𝐻)
7	4	ffvelcdmda 7104	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) ∈ ran 𝐻)
8	7	adantrl 716	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐵) ∈ ran 𝐻)
9		ghomdiv.3	. . . . 5 ⊢ 𝐶 = ( /𝑔 ‘𝐻)
10	3, 9	grponpcan 30562	. . . 4 ⊢ ((𝐻 ∈ GrpOp ∧ (𝐹‘𝐴) ∈ ran 𝐻 ∧ (𝐹‘𝐵) ∈ ran 𝐻) → (((𝐹‘𝐴)𝐶(𝐹‘𝐵))𝐻(𝐹‘𝐵)) = (𝐹‘𝐴))
11	1, 6, 8, 10	syl3anc 1373	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝐹‘𝐴)𝐶(𝐹‘𝐵))𝐻(𝐹‘𝐵)) = (𝐹‘𝐴))
12		ghomdiv.2	. . . . . . 7 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
13	2, 12	grponpcan 30562	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
14	13	3expb 1121	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
15	14	3ad2antl1 1186	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
16	15	fveq2d 6910	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)) = (𝐹‘𝐴))
17	2, 12	grpodivcl 30558	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
18	17	3expb 1121	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
19		simprr 773	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
20	18, 19	jca 511	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
21	20	3ad2antl1 1186	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
22	2	ghomlinOLD 37895	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹‘𝐵)) = (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)))
23	22	eqcomd 2743	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)) = ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹‘𝐵)))
24	21, 23	syldan 591	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)) = ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹‘𝐵)))
25	11, 16, 24	3eqtr2rd 2784	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹‘𝐵)) = (((𝐹‘𝐴)𝐶(𝐹‘𝐵))𝐻(𝐹‘𝐵)))
26	18	3ad2antl1 1186	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
27	4	ffvelcdmda 7104	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝐷𝐵) ∈ 𝑋) → (𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻)
28	26, 27	syldan 591	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻)
29	3, 9	grpodivcl 30558	. . . 4 ⊢ ((𝐻 ∈ GrpOp ∧ (𝐹‘𝐴) ∈ ran 𝐻 ∧ (𝐹‘𝐵) ∈ ran 𝐻) → ((𝐹‘𝐴)𝐶(𝐹‘𝐵)) ∈ ran 𝐻)
30	1, 6, 8, 29	syl3anc 1373	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘𝐴)𝐶(𝐹‘𝐵)) ∈ ran 𝐻)
31	3	grporcan 30537	. . 3 ⊢ ((𝐻 ∈ GrpOp ∧ ((𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻 ∧ ((𝐹‘𝐴)𝐶(𝐹‘𝐵)) ∈ ran 𝐻 ∧ (𝐹‘𝐵) ∈ ran 𝐻)) → (((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹‘𝐵)) = (((𝐹‘𝐴)𝐶(𝐹‘𝐵))𝐻(𝐹‘𝐵)) ↔ (𝐹‘(𝐴𝐷𝐵)) = ((𝐹‘𝐴)𝐶(𝐹‘𝐵))))
32	1, 28, 30, 8, 31	syl13anc 1374	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹‘𝐵)) = (((𝐹‘𝐴)𝐶(𝐹‘𝐵))𝐻(𝐹‘𝐵)) ↔ (𝐹‘(𝐴𝐷𝐵)) = ((𝐹‘𝐴)𝐶(𝐹‘𝐵))))
33	25, 32	mpbid 232	1 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐷𝐵)) = ((𝐹‘𝐴)𝐶(𝐹‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ran crn 5686  ‘cfv 6561  (class class class)co 7431  GrpOpcgr 30508   /_𝑔 cgs 30511   GrpOpHom cghomOLD 37890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-grpo 30512  df-gid 30513  df-ginv 30514  df-gdiv 30515  df-ghomOLD 37891

This theorem is referenced by:  grpokerinj  37900  rngohomsub  37980