Theorem ablodivdiv4 27408

Description: Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
abldiv.1	⊢ 𝑋 = ran 𝐺
abldiv.3	⊢ 𝐷 = ( /𝑔 ‘𝐺)

Assertion

Ref	Expression
ablodivdiv4	⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = (𝐴𝐷(𝐵𝐺𝐶)))

Proof of Theorem ablodivdiv4

Step	Hyp	Ref	Expression
1		ablogrpo 27401	. . 3 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
2		simpl 473	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐺 ∈ GrpOp)
3		abldiv.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
4		abldiv.3	. . . . . 6 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
5	3, 4	grpodivcl 27393	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
6	5	3adant3r3 1276	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
7		simpr3 1069	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐶 ∈ 𝑋)
8		eqid 2622	. . . . 5 ⊢ (inv‘𝐺) = (inv‘𝐺)
9	3, 8, 4	grpodivval 27389	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐷𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
10	2, 6, 7, 9	syl3anc 1326	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
11	1, 10	sylan 488	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
12		simpr1 1067	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
13		simpr2 1068	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
14		simp3 1063	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
15	3, 8	grpoinvcl 27378	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐶 ∈ 𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
16	1, 14, 15	syl2an 494	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
17	12, 13, 16	3jca 1242	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋))
18	3, 4	ablodivdiv 27407	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷((inv‘𝐺)‘𝐶))) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
19	17, 18	syldan 487	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷((inv‘𝐺)‘𝐶))) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
20	3, 8, 4	grpodivinv 27390	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐷((inv‘𝐺)‘𝐶)) = (𝐵𝐺𝐶))
21	1, 20	syl3an1 1359	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐷((inv‘𝐺)‘𝐶)) = (𝐵𝐺𝐶))
22	21	3adant3r1 1274	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐷((inv‘𝐺)‘𝐶)) = (𝐵𝐺𝐶))
23	22	oveq2d 6666	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷((inv‘𝐺)‘𝐶))) = (𝐴𝐷(𝐵𝐺𝐶)))
24	11, 19, 23	3eqtr2d 2662	1 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = (𝐴𝐷(𝐵𝐺𝐶)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ran crn 5115  ‘cfv 5888  (class class class)co 6650  GrpOpcgr 27343  invcgn 27345   /_𝑔 cgs 27346  AbelOpcablo 27398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399

This theorem is referenced by:  ablodiv32  27409  ablonnncan  27410  ablo4pnp  33679