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Theorem ablodivdiv4 27254
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 = ( /𝑔𝐺)
Assertion
Ref Expression
ablodivdiv4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = (𝐴𝐷(𝐵𝐺𝐶)))

Proof of Theorem ablodivdiv4
StepHypRef Expression
1 ablogrpo 27247 . . 3 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
2 simpl 473 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐺 ∈ GrpOp)
3 abldiv.1 . . . . . 6 𝑋 = ran 𝐺
4 abldiv.3 . . . . . 6 𝐷 = ( /𝑔𝐺)
53, 4grpodivcl 27239 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
653adant3r3 1273 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
7 simpr3 1067 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐶𝑋)
8 eqid 2621 . . . . 5 (inv‘𝐺) = (inv‘𝐺)
93, 8, 4grpodivval 27235 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝐷𝐵) ∈ 𝑋𝐶𝑋) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
102, 6, 7, 9syl3anc 1323 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
111, 10sylan 488 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
12 simpr1 1065 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑋)
13 simpr2 1066 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐵𝑋)
14 simp3 1061 . . . . 5 ((𝐴𝑋𝐵𝑋𝐶𝑋) → 𝐶𝑋)
153, 8grpoinvcl 27224 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
161, 14, 15syl2an 494 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
1712, 13, 163jca 1240 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑋𝐵𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋))
183, 4ablodivdiv 27253 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷((inv‘𝐺)‘𝐶))) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
1917, 18syldan 487 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷(𝐵𝐷((inv‘𝐺)‘𝐶))) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
203, 8, 4grpodivinv 27236 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐷((inv‘𝐺)‘𝐶)) = (𝐵𝐺𝐶))
211, 20syl3an1 1356 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐷((inv‘𝐺)‘𝐶)) = (𝐵𝐺𝐶))
22213adant3r1 1271 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐷((inv‘𝐺)‘𝐶)) = (𝐵𝐺𝐶))
2322oveq2d 6620 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷(𝐵𝐷((inv‘𝐺)‘𝐶))) = (𝐴𝐷(𝐵𝐺𝐶)))
2411, 19, 233eqtr2d 2661 1 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = (𝐴𝐷(𝐵𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  ran crn 5075  cfv 5847  (class class class)co 6604  GrpOpcgr 27189  invcgn 27191   /𝑔 cgs 27192  AbelOpcablo 27244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-grpo 27193  df-gid 27194  df-ginv 27195  df-gdiv 27196  df-ablo 27245
This theorem is referenced by:  ablodiv32  27255  ablonnncan  27256  ablo4pnp  33308
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