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Theorem grplcan 18099
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 7153 . . . . . 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 2818 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
6 eqid 2818 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
73, 4, 5, 6grplinv 18090 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
87adantlr 711 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
98oveq1d 7160 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = ((0g𝐺) + 𝑋))
103, 6grpinvcl 18089 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
1110adantrl 712 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
12 simprr 769 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → 𝑍𝐵)
13 simprl 767 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → 𝑋𝐵)
1411, 12, 133jca 1120 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑋𝐵))
153, 4grpass 18050 . . . . . . . . . 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 18071 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((0g𝐺) + 𝑋) = 𝑋)
1918adantr 481 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((0g𝐺) + 𝑋) = 𝑋)
209, 17, 193eqtr3d 2861 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
2120adantrl 712 . . . . . 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 712 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
2423oveq1d 7160 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑌) = ((0g𝐺) + 𝑌))
2510adantrl 712 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
26 simprr 769 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
27 simprl 767 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
2825, 26, 273jca 1120 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵))
293, 4grpass 18050 . . . . . . . . 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 18071 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
3231adantrr 713 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((0g𝐺) + 𝑌) = 𝑌)
3324, 30, 323eqtr3d 2861 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
3433adantlr 711 . . . . . 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 2861 . . . 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 1341 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌))
39 oveq2 7153 . 2 (𝑋 = 𝑌 → (𝑍 + 𝑋) = (𝑍 + 𝑌))
4038, 39impbid1 226 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701  Grpcgrp 18041  invgcminusg 18042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-riota 7103  df-ov 7148  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045
This theorem is referenced by:  grpidrcan  18102  grpinvinv  18104  grplmulf1o  18111  grplactcnv  18140  conjghm  18327  conjnmzb  18331  sylow3lem2  18682  gex2abl  18900  ringcom  19258  ringlz  19266  lmodlcan  19579  lmodfopne  19601  isnumbasgrplem2  39582  rnglz  44083
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