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Theorem galcan 19162
Description: The action of a particular group element is left-cancelable. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypothesis
Ref Expression
galcan.1 𝑋 = (Base‘𝐺)
Assertion
Ref Expression
galcan (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 𝐶) ↔ 𝐵 = 𝐶))

Proof of Theorem galcan
StepHypRef Expression
1 oveq2 7413 . . 3 ((𝐴 𝐵) = (𝐴 𝐶) → (((invg𝐺)‘𝐴) (𝐴 𝐵)) = (((invg𝐺)‘𝐴) (𝐴 𝐶)))
2 simpl 483 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ∈ (𝐺 GrpAct 𝑌))
3 gagrp 19150 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
42, 3syl 17 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐺 ∈ Grp)
5 simpr1 1194 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐴𝑋)
6 galcan.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
7 eqid 2732 . . . . . . . 8 (+g𝐺) = (+g𝐺)
8 eqid 2732 . . . . . . . 8 (0g𝐺) = (0g𝐺)
9 eqid 2732 . . . . . . . 8 (invg𝐺) = (invg𝐺)
106, 7, 8, 9grplinv 18870 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = (0g𝐺))
114, 5, 10syl2anc 584 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = (0g𝐺))
1211oveq1d 7420 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐵) = ((0g𝐺) 𝐵))
136, 9grpinvcl 18868 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
144, 5, 13syl2anc 584 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((invg𝐺)‘𝐴) ∈ 𝑋)
15 simpr2 1195 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐵𝑌)
166, 7gaass 19155 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝐴𝑋𝐵𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐵) = (((invg𝐺)‘𝐴) (𝐴 𝐵)))
172, 14, 5, 15, 16syl13anc 1372 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐵) = (((invg𝐺)‘𝐴) (𝐴 𝐵)))
188gagrpid 19152 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐵𝑌) → ((0g𝐺) 𝐵) = 𝐵)
192, 15, 18syl2anc 584 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐵) = 𝐵)
2012, 17, 193eqtr3d 2780 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴) (𝐴 𝐵)) = 𝐵)
2111oveq1d 7420 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐶) = ((0g𝐺) 𝐶))
22 simpr3 1196 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐶𝑌)
236, 7gaass 19155 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝐴𝑋𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐶) = (((invg𝐺)‘𝐴) (𝐴 𝐶)))
242, 14, 5, 22, 23syl13anc 1372 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐶) = (((invg𝐺)‘𝐴) (𝐴 𝐶)))
258gagrpid 19152 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶𝑌) → ((0g𝐺) 𝐶) = 𝐶)
262, 22, 25syl2anc 584 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐶) = 𝐶)
2721, 24, 263eqtr3d 2780 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴) (𝐴 𝐶)) = 𝐶)
2820, 27eqeq12d 2748 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴) (𝐴 𝐵)) = (((invg𝐺)‘𝐴) (𝐴 𝐶)) ↔ 𝐵 = 𝐶))
291, 28imbitrid 243 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 𝐶) → 𝐵 = 𝐶))
30 oveq2 7413 . 2 (𝐵 = 𝐶 → (𝐴 𝐵) = (𝐴 𝐶))
3129, 30impbid1 224 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 𝐶) ↔ 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  cfv 6540  (class class class)co 7405  Basecbs 17140  +gcplusg 17193  0gc0g 17381  Grpcgrp 18815  invgcminusg 18816   GrpAct cga 19147
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-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-rmo 3376  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-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-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-ga 19148
This theorem is referenced by:  gacan  19163
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