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Theorem grplcan 18925
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 7420 . . . . . 6 ((𝑍 + 𝑋) = (𝑍 + 𝑌) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)))
21adantl 481 . . . . 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 2731 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
6 eqid 2731 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
73, 4, 5, 6grplinv 18914 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
87adantlr 712 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
98oveq1d 7427 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = ((0g𝐺) + 𝑋))
103, 6grpinvcl 18912 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
1110adantrl 713 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
12 simprr 770 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → 𝑍𝐵)
13 simprl 768 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → 𝑋𝐵)
1411, 12, 133jca 1127 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑋𝐵))
153, 4grpass 18867 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑋𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)))
1614, 15syldan 590 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)))
1716anassrs 467 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)))
183, 4, 5grplid 18892 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((0g𝐺) + 𝑋) = 𝑋)
1918adantr 480 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((0g𝐺) + 𝑋) = 𝑋)
209, 17, 193eqtr3d 2779 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
2120adantrl 713 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
2221adantr 480 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
237adantrl 713 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
2423oveq1d 7427 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑌) = ((0g𝐺) + 𝑌))
2510adantrl 713 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
26 simprr 770 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
27 simprl 768 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
2825, 26, 273jca 1127 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵))
293, 4grpass 18867 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)))
3028, 29syldan 590 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)))
313, 4, 5grplid 18892 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
3231adantrr 714 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((0g𝐺) + 𝑌) = 𝑌)
3324, 30, 323eqtr3d 2779 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
3433adantlr 712 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
3534adantr 480 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
362, 22, 353eqtr3d 2779 . . . 4 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → 𝑋 = 𝑌)
3736exp53 447 . . 3 (𝐺 ∈ Grp → (𝑋𝐵 → (𝑌𝐵 → (𝑍𝐵 → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌)))))
38373imp2 1348 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌))
39 oveq2 7420 . 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 395  w3a 1086   = wceq 1540  wcel 2105  cfv 6543  (class class class)co 7412  Basecbs 17151  +gcplusg 17204  0gc0g 17392  Grpcgrp 18858  invgcminusg 18859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-0g 17394  df-mgm 18568  df-sgrp 18647  df-mnd 18663  df-grp 18861  df-minusg 18862
This theorem is referenced by:  grpidrcan  18928  grpinvinv  18930  grplmulf1o  18937  grplactcnv  18966  conjghm  19167  conjnmzb  19171  sylow3lem2  19541  gex2abl  19764  rnglz  20063  ringcom  20172  lmodlcan  20635  lmodfopne  20658  isnumbasgrplem2  42161  grptcmon  47816
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