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Theorem grporcan 28299
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 2822 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
31, 2grpoidinv2 28296 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → ((((GId‘𝐺)𝐺𝐶) = 𝐶 ∧ (𝐶𝐺(GId‘𝐺)) = 𝐶) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺))))
4 simpr 488 . . . . . . . . 9 (((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺)) → (𝐶𝐺𝑦) = (GId‘𝐺))
54reximi 3231 . . . . . . . 8 (∃𝑦𝑋 ((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺)) → ∃𝑦𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
65adantl 485 . . . . . . 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 748 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → ∃𝑦𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
9 oveq1 7147 . . . . . . . . . . . 12 ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → ((𝐴𝐺𝐶)𝐺𝑦) = ((𝐵𝐺𝐶)𝐺𝑦))
109ad2antll 728 . . . . . . . . . . 11 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐴𝐺𝐶)𝐺𝑦) = ((𝐵𝐺𝐶)𝐺𝑦))
111grpoass 28284 . . . . . . . . . . . . . 14 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋𝑦𝑋)) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
12113anassrs 1357 . . . . . . . . . . . . 13 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) ∧ 𝑦𝑋) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
1312adantlrl 719 . . . . . . . . . . . 12 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ 𝑦𝑋) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
1413adantrr 716 . . . . . . . . . . 11 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
151grpoass 28284 . . . . . . . . . . . . . . 15 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋𝑦𝑋)) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
16153exp2 1351 . . . . . . . . . . . . . 14 (𝐺 ∈ GrpOp → (𝐵𝑋 → (𝐶𝑋 → (𝑦𝑋 → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦))))))
1716imp42 430 . . . . . . . . . . . . 13 (((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) ∧ 𝑦𝑋) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
1817adantllr 718 . . . . . . . . . . . 12 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ 𝑦𝑋) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
1918adantrr 716 . . . . . . . . . . 11 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
2010, 14, 193eqtr3d 2865 . . . . . . . . . 10 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(𝐶𝐺𝑦)))
2120adantrrl 723 . . . . . . . . 9 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(𝐶𝐺𝑦)))
22 oveq2 7148 . . . . . . . . . . 11 ((𝐶𝐺𝑦) = (GId‘𝐺) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
2322ad2antrl 727 . . . . . . . . . 10 ((𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
2423adantl 485 . . . . . . . . 9 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
25 oveq2 7148 . . . . . . . . . . 11 ((𝐶𝐺𝑦) = (GId‘𝐺) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
2625ad2antrl 727 . . . . . . . . . 10 ((𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
2726adantl 485 . . . . . . . . 9 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
2821, 24, 273eqtr3d 2865 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(GId‘𝐺)) = (𝐵𝐺(GId‘𝐺)))
291, 2grporid 28298 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
3029ad2antrr 725 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
311, 2grporid 28298 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
3231ad2ant2r 746 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
3332adantr 484 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
3428, 30, 333eqtr3d 2865 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝑦𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → 𝐴 = 𝐵)
3534exp45 442 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (𝑦𝑋 → ((𝐶𝐺𝑦) = (GId‘𝐺) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → 𝐴 = 𝐵))))
3635rexlimdv 3269 . . . . 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 7147 . . . 4 (𝐴 = 𝐵 → (𝐴𝐺𝐶) = (𝐵𝐺𝐶))
3937, 38impbid1 228 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))
4039exp43 440 . 2 (𝐺 ∈ GrpOp → (𝐴𝑋 → (𝐵𝑋 → (𝐶𝑋 → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵)))))
41403imp2 1346 1 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wrex 3131  ran crn 5533  cfv 6334  (class class class)co 7140  GrpOpcgr 28270  GIdcgi 28271
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fo 6340  df-fv 6342  df-riota 7098  df-ov 7143  df-grpo 28274  df-gid 28275
This theorem is referenced by:  grpoinveu  28300  grpoid  28301  nvrcan  28405  ghomdiv  35288  rngorcan  35313  rngorz  35319
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