Theorem ghomdiv 38806

Description: Obsolete theorem, use ghmsub 19431 instead. Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

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 1211	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐻 ∈ GrpOp)
2		ghomdiv.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
3		eqid 2761	. . . . . . 7 ⊢ ran 𝐻 = ran 𝐻
4	2, 3	ghomf 38804	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶ran 𝐻)
5	4	ffvelcdmda 7082	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐻)
6	5	adantrr 730	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐴) ∈ ran 𝐻)
7	4	ffvelcdmda 7082	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) ∈ ran 𝐻)
8	7	adantrl 729	. . . 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 31138	. . . 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 1398	. . 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 31138	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
14	13	3expb 1138	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
15	14	3ad2antl1 1204	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
16	15	fveq2d 6887	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)) = (𝐹‘𝐴))
17	2, 12	grpodivcl 31134	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
18	17	3expb 1138	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
19		simprr 785	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
20	18, 19	jca 521	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
21	20	3ad2antl1 1204	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
22	2	ghomlinOLD 38802	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹‘𝐵)) = (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)))
23	22	eqcomd 2767	. . . 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 603	. . 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 2803	. 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 1204	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
27	4	ffvelcdmda 7082	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝐷𝐵) ∈ 𝑋) → (𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻)
28	26, 27	syldan 603	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻)
29	3, 9	grpodivcl 31134	. . . 4 ⊢ ((𝐻 ∈ GrpOp ∧ (𝐹‘𝐴) ∈ ran 𝐻 ∧ (𝐹‘𝐵) ∈ ran 𝐻) → ((𝐹‘𝐴)𝐶(𝐹‘𝐵)) ∈ ran 𝐻)
30	1, 6, 8, 29	syl3anc 1398	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘𝐴)𝐶(𝐹‘𝐵)) ∈ ran 𝐻)
31	3	grporcan 31113	. . 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 1399	. 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 235	1 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐷𝐵)) = ((𝐹‘𝐴)𝐶(𝐹‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ran crn 5652  ‘cfv 6537  (class class class)co 7418  GrpOpcgr 31084   /_𝑔 cgs 31087   GrpOpHom cghomOLD 38797

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-grpo 31088  df-gid 31089  df-ginv 31090  df-gdiv 31091  df-ghomOLD 38798

This theorem is used by:  grpokerinj  38807  rngohomsub  38887