MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grplcan Structured version   Visualization version   GIF version

Theorem grplcan 17685
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 6804 . . . . . 6 ((𝑍 + 𝑋) = (𝑍 + 𝑌) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)))
21adantl 467 . . . . 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 2771 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
6 eqid 2771 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
73, 4, 5, 6grplinv 17676 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
87adantlr 694 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
98oveq1d 6811 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = ((0g𝐺) + 𝑋))
103, 6grpinvcl 17675 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
1110adantrl 695 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
12 simprr 756 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → 𝑍𝐵)
13 simprl 754 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → 𝑋𝐵)
1411, 12, 133jca 1122 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑋𝐵))
153, 4grpass 17639 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑋𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)))
1614, 15syldan 579 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)))
1716anassrs 453 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)))
183, 4, 5grplid 17660 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((0g𝐺) + 𝑋) = 𝑋)
1918adantr 466 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((0g𝐺) + 𝑋) = 𝑋)
209, 17, 193eqtr3d 2813 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
2120adantrl 695 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
2221adantr 466 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
237adantrl 695 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
2423oveq1d 6811 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑌) = ((0g𝐺) + 𝑌))
2510adantrl 695 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
26 simprr 756 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
27 simprl 754 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
2825, 26, 273jca 1122 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵))
293, 4grpass 17639 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)))
3028, 29syldan 579 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)))
313, 4, 5grplid 17660 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
3231adantrr 696 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((0g𝐺) + 𝑌) = 𝑌)
3324, 30, 323eqtr3d 2813 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
3433adantlr 694 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
3534adantr 466 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
362, 22, 353eqtr3d 2813 . . . 4 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → 𝑋 = 𝑌)
3736exp53 434 . . 3 (𝐺 ∈ Grp → (𝑋𝐵 → (𝑌𝐵 → (𝑍𝐵 → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌)))))
38373imp2 1442 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌))
39 oveq2 6804 . 2 (𝑋 = 𝑌 → (𝑍 + 𝑋) = (𝑍 + 𝑌))
4038, 39impbid1 215 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Grpcgrp 17630  invgcminusg 17631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634
This theorem is referenced by:  grpidrcan  17688  grpinvinv  17690  grplmulf1o  17697  grplactcnv  17726  conjghm  17899  conjnmzb  17903  sylow3lem2  18250  gex2abl  18461  ringcom  18787  ringlz  18795  lmodlcan  19089  lmodfopne  19111  isnumbasgrplem2  38200  rnglz  42409
  Copyright terms: Public domain W3C validator