Theorem ghomdiv 34045

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 1244	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐻 ∈ GrpOp)
2		ghomdiv.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
3		eqid 2765	. . . . . . 7 ⊢ ran 𝐻 = ran 𝐻
4	2, 3	ghomf 34043	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶ran 𝐻)
5	4	ffvelrnda 6549	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐻)
6	5	adantrr 708	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐴) ∈ ran 𝐻)
7	4	ffvelrnda 6549	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) ∈ ran 𝐻)
8	7	adantrl 707	. . . 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 27789	. . . 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 1490	. . 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 27789	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
14	13	3expb 1149	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
15	14	3ad2antl1 1236	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
16	15	fveq2d 6379	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)) = (𝐹‘𝐴))
17	2, 12	grpodivcl 27785	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
18	17	3expb 1149	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
19		simprr 789	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
20	18, 19	jca 507	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
21	20	3ad2antl1 1236	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
22	2	ghomlinOLD 34041	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹‘𝐵)) = (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)))
23	22	eqcomd 2771	. . . 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 585	. . 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 2806	. 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 1236	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
27	4	ffvelrnda 6549	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝐷𝐵) ∈ 𝑋) → (𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻)
28	26, 27	syldan 585	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻)
29	3, 9	grpodivcl 27785	. . . 4 ⊢ ((𝐻 ∈ GrpOp ∧ (𝐹‘𝐴) ∈ ran 𝐻 ∧ (𝐹‘𝐵) ∈ ran 𝐻) → ((𝐹‘𝐴)𝐶(𝐹‘𝐵)) ∈ ran 𝐻)
30	1, 6, 8, 29	syl3anc 1490	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘𝐴)𝐶(𝐹‘𝐵)) ∈ ran 𝐻)
31	3	grporcan 27764	. . 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 1491	. 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 223	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 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155  ran crn 5278  ‘cfv 6068  (class class class)co 6842  GrpOpcgr 27735   /_𝑔 cgs 27738   GrpOpHom cghomOLD 34036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-grpo 27739  df-gid 27740  df-ginv 27741  df-gdiv 27742  df-ghomOLD 34037

This theorem is referenced by:  grpokerinj  34046  rngohomsub  34126