Theorem ghomdiv 36050

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 1191	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐻 ∈ GrpOp)
2		ghomdiv.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
3		eqid 2738	. . . . . . 7 ⊢ ran 𝐻 = ran 𝐻
4	2, 3	ghomf 36048	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶ran 𝐻)
5	4	ffvelrnda 6961	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐻)
6	5	adantrr 714	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐴) ∈ ran 𝐻)
7	4	ffvelrnda 6961	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) ∈ ran 𝐻)
8	7	adantrl 713	. . . 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 28905	. . . 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 1370	. . 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 28905	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
14	13	3expb 1119	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
15	14	3ad2antl1 1184	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
16	15	fveq2d 6778	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)) = (𝐹‘𝐴))
17	2, 12	grpodivcl 28901	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
18	17	3expb 1119	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
19		simprr 770	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
20	18, 19	jca 512	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
21	20	3ad2antl1 1184	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
22	2	ghomlinOLD 36046	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ ((𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹‘𝐵)) = (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)))
23	22	eqcomd 2744	. . . 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 2785	. 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 1184	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
27	4	ffvelrnda 6961	. . . 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 28901	. . . 4 ⊢ ((𝐻 ∈ GrpOp ∧ (𝐹‘𝐴) ∈ ran 𝐻 ∧ (𝐹‘𝐵) ∈ ran 𝐻) → ((𝐹‘𝐴)𝐶(𝐹‘𝐵)) ∈ ran 𝐻)
30	1, 6, 8, 29	syl3anc 1370	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘𝐴)𝐶(𝐹‘𝐵)) ∈ ran 𝐻)
31	3	grporcan 28880	. . 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 1371	. 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 231	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ran crn 5590  ‘cfv 6433  (class class class)co 7275  GrpOpcgr 28851   /_𝑔 cgs 28854   GrpOpHom cghomOLD 36041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-grpo 28855  df-gid 28856  df-ginv 28857  df-gdiv 28858  df-ghomOLD 36042

This theorem is referenced by:  grpokerinj  36051  rngohomsub  36131