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Theorem grplcan 12808
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 5876 . . . . . 6 ((𝑍 + 𝑋) = (𝑍 + 𝑌) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)))
21adantl 277 . . . . 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 2177 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
6 eqid 2177 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
73, 4, 5, 6grplinv 12799 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
87adantlr 477 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
98oveq1d 5883 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = ((0g𝐺) + 𝑋))
103, 6grpinvcl 12798 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
1110adantrl 478 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
12 simprr 531 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → 𝑍𝐵)
13 simprl 529 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → 𝑋𝐵)
1411, 12, 133jca 1177 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑋𝐵))
153, 4grpass 12763 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑋𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)))
1614, 15syldan 282 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)))
1716anassrs 400 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)))
183, 4, 5grplid 12783 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((0g𝐺) + 𝑋) = 𝑋)
1918adantr 276 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → ((0g𝐺) + 𝑋) = 𝑋)
209, 17, 193eqtr3d 2218 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑍𝐵) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
2120adantrl 478 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
2221adantr 276 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
237adantrl 478 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + 𝑍) = (0g𝐺))
2423oveq1d 5883 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑌) = ((0g𝐺) + 𝑌))
2510adantrl 478 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
26 simprr 531 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
27 simprl 529 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
2825, 26, 273jca 1177 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵))
293, 4grpass 12763 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)))
3028, 29syldan 282 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((((invg𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)))
313, 4, 5grplid 12783 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
3231adantrr 479 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → ((0g𝐺) + 𝑌) = 𝑌)
3324, 30, 323eqtr3d 2218 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
3433adantlr 477 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
3534adantr 276 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
362, 22, 353eqtr3d 2218 . . . 4 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → 𝑋 = 𝑌)
3736exp53 377 . . 3 (𝐺 ∈ Grp → (𝑋𝐵 → (𝑌𝐵 → (𝑍𝐵 → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌)))))
38373imp2 1222 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌))
39 oveq2 5876 . 2 (𝑋 = 𝑌 → (𝑍 + 𝑋) = (𝑍 + 𝑌))
4038, 39impbid1 142 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  cfv 5211  (class class class)co 5868  Basecbs 12432  +gcplusg 12505  0gc0g 12640  Grpcgrp 12754  invgcminusg 12755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757  df-minusg 12758
This theorem is referenced by:  grpidrcan  12811  grpinvinv  12813  grplmulf1o  12820  grplactcnv  12848  ringcom  13027  ringlz  13035
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