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Theorem galcan 19080
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 7362 . . 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 19068 . . . . . . . 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 2736 . . . . . . . 8 (+g𝐺) = (+g𝐺)
8 eqid 2736 . . . . . . . 8 (0g𝐺) = (0g𝐺)
9 eqid 2736 . . . . . . . 8 (invg𝐺) = (invg𝐺)
106, 7, 8, 9grplinv 18796 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = (0g𝐺))
114, 5, 10syl2anc 584 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = (0g𝐺))
1211oveq1d 7369 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐵) = ((0g𝐺) 𝐵))
136, 9grpinvcl 18795 . . . . . . 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 19073 . . . . . 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 19070 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐵𝑌) → ((0g𝐺) 𝐵) = 𝐵)
192, 15, 18syl2anc 584 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐵) = 𝐵)
2012, 17, 193eqtr3d 2784 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴) (𝐴 𝐵)) = 𝐵)
2111oveq1d 7369 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐶) = ((0g𝐺) 𝐶))
22 simpr3 1196 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐶𝑌)
236, 7gaass 19073 . . . . . 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 19070 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶𝑌) → ((0g𝐺) 𝐶) = 𝐶)
262, 22, 25syl2anc 584 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐶) = 𝐶)
2721, 24, 263eqtr3d 2784 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴) (𝐴 𝐶)) = 𝐶)
2820, 27eqeq12d 2752 . . 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 7362 . 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 6494  (class class class)co 7354  Basecbs 17080  +gcplusg 17130  0gc0g 17318  Grpcgrp 18745  invgcminusg 18746   GrpAct cga 19065
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 2707  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-riota 7310  df-ov 7357  df-oprab 7358  df-mpo 7359  df-map 8764  df-0g 17320  df-mgm 18494  df-sgrp 18543  df-mnd 18554  df-grp 18748  df-minusg 18749  df-ga 19066
This theorem is referenced by:  gacan  19081
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