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Theorem galcan 19335
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 7439 . . 3 ((𝐴 𝐵) = (𝐴 𝐶) → (((invg𝐺)‘𝐴) (𝐴 𝐵)) = (((invg𝐺)‘𝐴) (𝐴 𝐶)))
2 simpl 482 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ∈ (𝐺 GrpAct 𝑌))
3 gagrp 19323 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
42, 3syl 17 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐺 ∈ Grp)
5 simpr1 1193 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐴𝑋)
6 galcan.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
7 eqid 2735 . . . . . . . 8 (+g𝐺) = (+g𝐺)
8 eqid 2735 . . . . . . . 8 (0g𝐺) = (0g𝐺)
9 eqid 2735 . . . . . . . 8 (invg𝐺) = (invg𝐺)
106, 7, 8, 9grplinv 19020 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = (0g𝐺))
114, 5, 10syl2anc 584 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = (0g𝐺))
1211oveq1d 7446 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐵) = ((0g𝐺) 𝐵))
136, 9grpinvcl 19018 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
144, 5, 13syl2anc 584 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((invg𝐺)‘𝐴) ∈ 𝑋)
15 simpr2 1194 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐵𝑌)
166, 7gaass 19328 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝐴𝑋𝐵𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐵) = (((invg𝐺)‘𝐴) (𝐴 𝐵)))
172, 14, 5, 15, 16syl13anc 1371 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐵) = (((invg𝐺)‘𝐴) (𝐴 𝐵)))
188gagrpid 19325 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐵𝑌) → ((0g𝐺) 𝐵) = 𝐵)
192, 15, 18syl2anc 584 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐵) = 𝐵)
2012, 17, 193eqtr3d 2783 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴) (𝐴 𝐵)) = 𝐵)
2111oveq1d 7446 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐶) = ((0g𝐺) 𝐶))
22 simpr3 1195 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐶𝑌)
236, 7gaass 19328 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝐴𝑋𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐶) = (((invg𝐺)‘𝐴) (𝐴 𝐶)))
242, 14, 5, 22, 23syl13anc 1371 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐶) = (((invg𝐺)‘𝐴) (𝐴 𝐶)))
258gagrpid 19325 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶𝑌) → ((0g𝐺) 𝐶) = 𝐶)
262, 22, 25syl2anc 584 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐶) = 𝐶)
2721, 24, 263eqtr3d 2783 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴) (𝐴 𝐶)) = 𝐶)
2820, 27eqeq12d 2751 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴) (𝐴 𝐵)) = (((invg𝐺)‘𝐴) (𝐴 𝐶)) ↔ 𝐵 = 𝐶))
291, 28imbitrid 244 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 𝐶) → 𝐵 = 𝐶))
30 oveq2 7439 . 2 (𝐵 = 𝐶 → (𝐴 𝐵) = (𝐴 𝐶))
3129, 30impbid1 225 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 𝐶) ↔ 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Grpcgrp 18964  invgcminusg 18965   GrpAct cga 19320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-ga 19321
This theorem is referenced by:  gacan  19336
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