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Theorem grporcan 28859
Description: Right cancellation law for groups. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grprcan.1 𝑋 = ran 𝐺
Assertion
Ref Expression
grporcan ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))

Proof of Theorem grporcan
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 grprcan.1 . . . . . . . 8 𝑋 = ran 𝐺
2 eqid 2739 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
31, 2grpoidinv2 28856 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → ((((GId‘𝐺)𝐺𝐶) = 𝐶 ∧ (𝐶𝐺(GId‘𝐺)) = 𝐶) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺))))
4 simpr 484 . . . . . . . . 9 (((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺)) → (𝐶𝐺𝑦) = (GId‘𝐺))
54reximi 3176 . . . . . . . 8 (∃𝑦𝑋 ((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺)) → ∃𝑦𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
65adantl 481 . . . . . . 7 (((((GId‘𝐺)𝐺𝐶) = 𝐶 ∧ (𝐶𝐺(GId‘𝐺)) = 𝐶) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺))) → ∃𝑦𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
73, 6syl 17 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → ∃𝑦𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
87ad2ant2rl 745 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → ∃𝑦𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
9 oveq1 7275 . . . . . . . . . . . 12 ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → ((𝐴𝐺𝐶)𝐺𝑦) = ((𝐵𝐺𝐶)𝐺𝑦))
109ad2antll 725 . . . . . . . . . . 11 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐴𝐺𝐶)𝐺𝑦) = ((𝐵𝐺𝐶)𝐺𝑦))
111grpoass 28844 . . . . . . . . . . . . . 14 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋𝑦𝑋)) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
12113anassrs 1358 . . . . . . . . . . . . 13 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) ∧ 𝑦𝑋) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
1312adantlrl 716 . . . . . . . . . . . 12 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ 𝑦𝑋) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
1413adantrr 713 . . . . . . . . . . 11 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
151grpoass 28844 . . . . . . . . . . . . . . 15 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋𝑦𝑋)) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
16153exp2 1352 . . . . . . . . . . . . . 14 (𝐺 ∈ GrpOp → (𝐵𝑋 → (𝐶𝑋 → (𝑦𝑋 → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦))))))
1716imp42 426 . . . . . . . . . . . . 13 (((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) ∧ 𝑦𝑋) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
1817adantllr 715 . . . . . . . . . . . 12 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ 𝑦𝑋) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
1918adantrr 713 . . . . . . . . . . 11 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
2010, 14, 193eqtr3d 2787 . . . . . . . . . 10 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(𝐶𝐺𝑦)))
2120adantrrl 720 . . . . . . . . 9 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(𝐶𝐺𝑦)))
22 oveq2 7276 . . . . . . . . . . 11 ((𝐶𝐺𝑦) = (GId‘𝐺) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
2322ad2antrl 724 . . . . . . . . . 10 ((𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
2423adantl 481 . . . . . . . . 9 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
25 oveq2 7276 . . . . . . . . . . 11 ((𝐶𝐺𝑦) = (GId‘𝐺) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
2625ad2antrl 724 . . . . . . . . . 10 ((𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
2726adantl 481 . . . . . . . . 9 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
2821, 24, 273eqtr3d 2787 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(GId‘𝐺)) = (𝐵𝐺(GId‘𝐺)))
291, 2grporid 28858 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
3029ad2antrr 722 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
311, 2grporid 28858 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
3231ad2ant2r 743 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
3332adantr 480 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
3428, 30, 333eqtr3d 2787 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → 𝐴 = 𝐵)
3534exp45 438 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (𝑦𝑋 → ((𝐶𝐺𝑦) = (GId‘𝐺) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → 𝐴 = 𝐵))))
3635rexlimdv 3213 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (∃𝑦𝑋 (𝐶𝐺𝑦) = (GId‘𝐺) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → 𝐴 = 𝐵)))
378, 36mpd 15 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → 𝐴 = 𝐵))
38 oveq1 7275 . . . 4 (𝐴 = 𝐵 → (𝐴𝐺𝐶) = (𝐵𝐺𝐶))
3937, 38impbid1 224 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))
4039exp43 436 . 2 (𝐺 ∈ GrpOp → (𝐴𝑋 → (𝐵𝑋 → (𝐶𝑋 → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵)))))
41403imp2 1347 1 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1541  wcel 2109  wrex 3066  ran crn 5589  cfv 6430  (class class class)co 7268  GrpOpcgr 28830  GIdcgi 28831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fo 6436  df-fv 6438  df-riota 7225  df-ov 7271  df-grpo 28834  df-gid 28835
This theorem is referenced by:  grpoinveu  28860  grpoid  28861  nvrcan  28965  ghomdiv  36029  rngorcan  36054  rngorz  36060
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