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Theorem grplcan 18887
Description: Left cancellation law for groups. (Contributed by NM, 25-Aug-2011.)
Hypotheses
Ref Expression
grplcan.b 𝐵 = (Base‘𝐺)
grplcan.p + = (+g𝐺)
Assertion
Ref Expression
grplcan ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))

Proof of Theorem grplcan
StepHypRef Expression
1 oveq2 7419 . . . . . 6 ((𝑍 + 𝑋) = (𝑍 + 𝑌) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)))
21adantl 482 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)))
3 grplcan.b . . . . . . . . . . 11 𝐵 = (Base‘𝐺)
4 grplcan.p . . . . . . . . . . 11 + = (+g𝐺)
5 eqid 2732 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
6 eqid 2732 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
73, 4, 5, 6grplinv 18876 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
87adantlr 713 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
98oveq1d 7426 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = ((0g𝐺) + 𝑋))
103, 6grpinvcl 18874 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
1110adantrl 714 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
12 simprr 771 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → 𝑍𝐵)
13 simprl 769 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → 𝑋𝐵)
1411, 12, 133jca 1128 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑋𝐵))
153, 4grpass 18830 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑋𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)))
1614, 15syldan 591 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)))
1716anassrs 468 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)))
183, 4, 5grplid 18854 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((0g𝐺) + 𝑋) = 𝑋)
1918adantr 481 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((0g𝐺) + 𝑋) = 𝑋)
209, 17, 193eqtr3d 2780 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
2120adantrl 714 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
2221adantr 481 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
237adantrl 714 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
2423oveq1d 7426 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑌) = ((0g𝐺) + 𝑌))
2510adantrl 714 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
26 simprr 771 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
27 simprl 769 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
2825, 26, 273jca 1128 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵))
293, 4grpass 18830 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)))
3028, 29syldan 591 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)))
313, 4, 5grplid 18854 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
3231adantrr 715 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((0g𝐺) + 𝑌) = 𝑌)
3324, 30, 323eqtr3d 2780 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
3433adantlr 713 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
3534adantr 481 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
362, 22, 353eqtr3d 2780 . . . 4 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → 𝑋 = 𝑌)
3736exp53 448 . . 3 (𝐺 ∈ Grp → (𝑋𝐵 → (𝑌𝐵 → (𝑍𝐵 → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌)))))
38373imp2 1349 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌))
39 oveq2 7419 . 2 (𝑋 = 𝑌 → (𝑍 + 𝑋) = (𝑍 + 𝑌))
4038, 39impbid1 224 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  cfv 6543  (class class class)co 7411  Basecbs 17146  +gcplusg 17199  0gc0g 17387  Grpcgrp 18821  invgcminusg 18822
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7367  df-ov 7414  df-0g 17389  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-grp 18824  df-minusg 18825
This theorem is referenced by:  grpidrcan  18890  grpinvinv  18892  grplmulf1o  18899  grplactcnv  18928  conjghm  19125  conjnmzb  19129  sylow3lem2  19498  gex2abl  19721  ringcom  20099  ringlz  20109  lmodlcan  20492  lmodfopne  20515  isnumbasgrplem2  41928  rnglz  46743  grptcmon  47794
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