Theorem ghomdiv 38329

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 1202	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐻 ∈ GrpOp)
2		ghomdiv.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
3		eqid 2752	. . . . . . 7 ⊢ ran 𝐻 = ran 𝐻
4	2, 3	ghomf 38327	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶ran 𝐻)
5	4	ffvelcdmda 7050	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐻)
6	5	adantrr 725	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐴) ∈ ran 𝐻)
7	4	ffvelcdmda 7050	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) ∈ ran 𝐻)
8	7	adantrl 724	. . . 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 30681	. . . 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 1382	. . 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 30681	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
14	13	3expb 1129	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
15	14	3ad2antl1 1195	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
16	15	fveq2d 6856	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)) = (𝐹‘𝐴))
17	2, 12	grpodivcl 30677	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
18	17	3expb 1129	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
19		simprr 780	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
20	18, 19	jca 518	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
21	20	3ad2antl1 1195	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
22	2	ghomlinOLD 38325	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹‘𝐵)) = (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)))
23	22	eqcomd 2758	. . . 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 599	. . 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 2794	. 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 1195	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
27	4	ffvelcdmda 7050	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝐷𝐵) ∈ 𝑋) → (𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻)
28	26, 27	syldan 599	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻)
29	3, 9	grpodivcl 30677	. . . 4 ⊢ ((𝐻 ∈ GrpOp ∧ (𝐹‘𝐴) ∈ ran 𝐻 ∧ (𝐹‘𝐵) ∈ ran 𝐻) → ((𝐹‘𝐴)𝐶(𝐹‘𝐵)) ∈ ran 𝐻)
30	1, 6, 8, 29	syl3anc 1382	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘𝐴)𝐶(𝐹‘𝐵)) ∈ ran 𝐻)
31	3	grporcan 30656	. . 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 1383	. 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 234	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 208   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132  ran crn 5637  ‘cfv 6506  (class class class)co 7381  GrpOpcgr 30627   /_𝑔 cgs 30630   GrpOpHom cghomOLD 38320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-1st 7955  df-2nd 7956  df-grpo 30631  df-gid 30632  df-ginv 30633  df-gdiv 30634  df-ghomOLD 38321

This theorem is referenced by:  grpokerinj  38330  rngohomsub  38410