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Theorem ablodivdiv4 29324
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 29317 . . 3 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
2 simpl 483 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐺 ∈ GrpOp)
3 abldiv.1 . . . . . 6 𝑋 = ran 𝐺
4 abldiv.3 . . . . . 6 𝐷 = ( /𝑔𝐺)
53, 4grpodivcl 29309 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
653adant3r3 1184 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
7 simpr3 1196 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐶𝑋)
8 eqid 2737 . . . . 5 (inv‘𝐺) = (inv‘𝐺)
93, 8, 4grpodivval 29305 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝐷𝐵) ∈ 𝑋𝐶𝑋) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
102, 6, 7, 9syl3anc 1371 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
111, 10sylan 580 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
12 simpr1 1194 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑋)
13 simpr2 1195 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐵𝑋)
14 simp3 1138 . . . . 5 ((𝐴𝑋𝐵𝑋𝐶𝑋) → 𝐶𝑋)
153, 8grpoinvcl 29294 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
161, 14, 15syl2an 596 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
1712, 13, 163jca 1128 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑋𝐵𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋))
183, 4ablodivdiv 29323 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷((inv‘𝐺)‘𝐶))) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
1917, 18syldan 591 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷(𝐵𝐷((inv‘𝐺)‘𝐶))) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
203, 8, 4grpodivinv 29306 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐷((inv‘𝐺)‘𝐶)) = (𝐵𝐺𝐶))
211, 20syl3an1 1163 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐷((inv‘𝐺)‘𝐶)) = (𝐵𝐺𝐶))
22213adant3r1 1182 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐷((inv‘𝐺)‘𝐶)) = (𝐵𝐺𝐶))
2322oveq2d 7367 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷(𝐵𝐷((inv‘𝐺)‘𝐶))) = (𝐴𝐷(𝐵𝐺𝐶)))
2411, 19, 233eqtr2d 2783 1 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = (𝐴𝐷(𝐵𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  ran crn 5632  cfv 6493  (class class class)co 7351  GrpOpcgr 29259  invcgn 29261   /𝑔 cgs 29262  AbelOpcablo 29314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-grpo 29263  df-gid 29264  df-ginv 29265  df-gdiv 29266  df-ablo 29315
This theorem is referenced by:  ablodiv32  29325  ablo4pnp  36270
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