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Theorem ablodivdiv4 27800
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 27793 . . 3 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
2 simpl 474 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐺 ∈ GrpOp)
3 abldiv.1 . . . . . 6 𝑋 = ran 𝐺
4 abldiv.3 . . . . . 6 𝐷 = ( /𝑔𝐺)
53, 4grpodivcl 27785 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
653adant3r3 1235 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
7 simpr3 1252 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐶𝑋)
8 eqid 2765 . . . . 5 (inv‘𝐺) = (inv‘𝐺)
93, 8, 4grpodivval 27781 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝐷𝐵) ∈ 𝑋𝐶𝑋) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
102, 6, 7, 9syl3anc 1490 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
111, 10sylan 575 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
12 simpr1 1248 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑋)
13 simpr2 1250 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐵𝑋)
14 simp3 1168 . . . . 5 ((𝐴𝑋𝐵𝑋𝐶𝑋) → 𝐶𝑋)
153, 8grpoinvcl 27770 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
161, 14, 15syl2an 589 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
1712, 13, 163jca 1158 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑋𝐵𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋))
183, 4ablodivdiv 27799 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷((inv‘𝐺)‘𝐶))) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
1917, 18syldan 585 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷(𝐵𝐷((inv‘𝐺)‘𝐶))) = ((𝐴𝐷𝐵)𝐺((inv‘𝐺)‘𝐶)))
203, 8, 4grpodivinv 27782 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐷((inv‘𝐺)‘𝐶)) = (𝐵𝐺𝐶))
211, 20syl3an1 1202 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐷((inv‘𝐺)‘𝐶)) = (𝐵𝐺𝐶))
22213adant3r1 1233 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐷((inv‘𝐺)‘𝐶)) = (𝐵𝐺𝐶))
2322oveq2d 6858 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷(𝐵𝐷((inv‘𝐺)‘𝐶))) = (𝐴𝐷(𝐵𝐺𝐶)))
2411, 19, 233eqtr2d 2805 1 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = (𝐴𝐷(𝐵𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  ran crn 5278  cfv 6068  (class class class)co 6842  GrpOpcgr 27735  invcgn 27737   /𝑔 cgs 27738  AbelOpcablo 27790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-grpo 27739  df-gid 27740  df-ginv 27741  df-gdiv 27742  df-ablo 27791
This theorem is referenced by:  ablodiv32  27801  ablo4pnp  34033
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