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Theorem ablodivdiv4 29794
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 β€˜πΊ)
Assertion
Ref Expression
ablodivdiv4 ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡)𝐷𝐢) = (𝐴𝐷(𝐡𝐺𝐢)))

Proof of Theorem ablodivdiv4
StepHypRef Expression
1 ablogrpo 29787 . . 3 (𝐺 ∈ AbelOp β†’ 𝐺 ∈ GrpOp)
2 simpl 483 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐺 ∈ GrpOp)
3 abldiv.1 . . . . . 6 𝑋 = ran 𝐺
4 abldiv.3 . . . . . 6 𝐷 = ( /𝑔 β€˜πΊ)
53, 4grpodivcl 29779 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) ∈ 𝑋)
653adant3r3 1184 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) ∈ 𝑋)
7 simpr3 1196 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐢 ∈ 𝑋)
8 eqid 2732 . . . . 5 (invβ€˜πΊ) = (invβ€˜πΊ)
93, 8, 4grpodivval 29775 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝐷𝐡) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ ((𝐴𝐷𝐡)𝐷𝐢) = ((𝐴𝐷𝐡)𝐺((invβ€˜πΊ)β€˜πΆ)))
102, 6, 7, 9syl3anc 1371 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡)𝐷𝐢) = ((𝐴𝐷𝐡)𝐺((invβ€˜πΊ)β€˜πΆ)))
111, 10sylan 580 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡)𝐷𝐢) = ((𝐴𝐷𝐡)𝐺((invβ€˜πΊ)β€˜πΆ)))
12 simpr1 1194 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
13 simpr2 1195 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
14 simp3 1138 . . . . 5 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ 𝐢 ∈ 𝑋)
153, 8grpoinvcl 29764 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐢 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
161, 14, 15syl2an 596 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
1712, 13, 163jca 1128 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋))
183, 4ablodivdiv 29793 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)) β†’ (𝐴𝐷(𝐡𝐷((invβ€˜πΊ)β€˜πΆ))) = ((𝐴𝐷𝐡)𝐺((invβ€˜πΊ)β€˜πΆ)))
1917, 18syldan 591 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐷(𝐡𝐷((invβ€˜πΊ)β€˜πΆ))) = ((𝐴𝐷𝐡)𝐺((invβ€˜πΊ)β€˜πΆ)))
203, 8, 4grpodivinv 29776 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐡𝐷((invβ€˜πΊ)β€˜πΆ)) = (𝐡𝐺𝐢))
211, 20syl3an1 1163 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐡𝐷((invβ€˜πΊ)β€˜πΆ)) = (𝐡𝐺𝐢))
22213adant3r1 1182 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐡𝐷((invβ€˜πΊ)β€˜πΆ)) = (𝐡𝐺𝐢))
2322oveq2d 7421 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐷(𝐡𝐷((invβ€˜πΊ)β€˜πΆ))) = (𝐴𝐷(𝐡𝐺𝐢)))
2411, 19, 233eqtr2d 2778 1 ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡)𝐷𝐢) = (𝐴𝐷(𝐡𝐺𝐢)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ran crn 5676  β€˜cfv 6540  (class class class)co 7405  GrpOpcgr 29729  invcgn 29731   /𝑔 cgs 29732  AbelOpcablo 29784
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-grpo 29733  df-gid 29734  df-ginv 29735  df-gdiv 29736  df-ablo 29785
This theorem is referenced by:  ablodiv32  29795  ablo4pnp  36736
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