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Theorem grpinveu 12932
Description: The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 24-Aug-2011.)
Hypotheses
Ref Expression
grpinveu.b 𝐵 = (Base‘𝐺)
grpinveu.p + = (+g𝐺)
grpinveu.o 0 = (0g𝐺)
Assertion
Ref Expression
grpinveu ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃!𝑦𝐵 (𝑦 + 𝑋) = 0 )
Distinct variable groups:   𝑦,𝐵   𝑦,𝐺   𝑦, +   𝑦, 0   𝑦,𝑋

Proof of Theorem grpinveu
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 grpinveu.b . . . 4 𝐵 = (Base‘𝐺)
2 grpinveu.p . . . 4 + = (+g𝐺)
3 grpinveu.o . . . 4 0 = (0g𝐺)
41, 2, 3grpinvex 12906 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
5 eqtr3 2207 . . . . . . . . . . . 12 (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → (𝑦 + 𝑋) = (𝑧 + 𝑋))
61, 2grprcan 12931 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝑦𝐵𝑧𝐵𝑋𝐵)) → ((𝑦 + 𝑋) = (𝑧 + 𝑋) ↔ 𝑦 = 𝑧))
75, 6imbitrid 154 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑦𝐵𝑧𝐵𝑋𝐵)) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
873exp2 1226 . . . . . . . . . 10 (𝐺 ∈ Grp → (𝑦𝐵 → (𝑧𝐵 → (𝑋𝐵 → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧)))))
98com24 87 . . . . . . . . 9 (𝐺 ∈ Grp → (𝑋𝐵 → (𝑧𝐵 → (𝑦𝐵 → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧)))))
109imp41 353 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑧𝐵) ∧ 𝑦𝐵) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
1110an32s 568 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) ∧ 𝑧𝐵) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
1211expd 258 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) ∧ 𝑧𝐵) → ((𝑦 + 𝑋) = 0 → ((𝑧 + 𝑋) = 0𝑦 = 𝑧)))
1312ralrimdva 2567 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) → ((𝑦 + 𝑋) = 0 → ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧)))
1413ancld 325 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) → ((𝑦 + 𝑋) = 0 → ((𝑦 + 𝑋) = 0 ∧ ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧))))
1514reximdva 2589 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (∃𝑦𝐵 (𝑦 + 𝑋) = 0 → ∃𝑦𝐵 ((𝑦 + 𝑋) = 0 ∧ ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧))))
164, 15mpd 13 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃𝑦𝐵 ((𝑦 + 𝑋) = 0 ∧ ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧)))
17 oveq1 5895 . . . 4 (𝑦 = 𝑧 → (𝑦 + 𝑋) = (𝑧 + 𝑋))
1817eqeq1d 2196 . . 3 (𝑦 = 𝑧 → ((𝑦 + 𝑋) = 0 ↔ (𝑧 + 𝑋) = 0 ))
1918reu8 2945 . 2 (∃!𝑦𝐵 (𝑦 + 𝑋) = 0 ↔ ∃𝑦𝐵 ((𝑦 + 𝑋) = 0 ∧ ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧)))
2016, 19sylibr 134 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃!𝑦𝐵 (𝑦 + 𝑋) = 0 )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 979   = wceq 1363  wcel 2158  wral 2465  wrex 2466  ∃!wreu 2467  cfv 5228  (class class class)co 5888  Basecbs 12475  +gcplusg 12550  0gc0g 12722  Grpcgrp 12896
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-riota 5844  df-ov 5891  df-inn 8933  df-2 8991  df-ndx 12478  df-slot 12479  df-base 12481  df-plusg 12563  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12837  df-grp 12899
This theorem is referenced by:  grpinvf  12941  grplinv  12944  isgrpinv  12948
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