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Theorem grpinv11 17871
 Description: The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.)
Hypotheses
Ref Expression
grpinvinv.b 𝐵 = (Base‘𝐺)
grpinvinv.n 𝑁 = (invg𝐺)
grpinv11.g (𝜑𝐺 ∈ Grp)
grpinv11.x (𝜑𝑋𝐵)
grpinv11.y (𝜑𝑌𝐵)
Assertion
Ref Expression
grpinv11 (𝜑 → ((𝑁𝑋) = (𝑁𝑌) ↔ 𝑋 = 𝑌))

Proof of Theorem grpinv11
StepHypRef Expression
1 fveq2 6446 . . . . 5 ((𝑁𝑋) = (𝑁𝑌) → (𝑁‘(𝑁𝑋)) = (𝑁‘(𝑁𝑌)))
21adantl 475 . . . 4 ((𝜑 ∧ (𝑁𝑋) = (𝑁𝑌)) → (𝑁‘(𝑁𝑋)) = (𝑁‘(𝑁𝑌)))
3 grpinv11.g . . . . . 6 (𝜑𝐺 ∈ Grp)
4 grpinv11.x . . . . . 6 (𝜑𝑋𝐵)
5 grpinvinv.b . . . . . . 7 𝐵 = (Base‘𝐺)
6 grpinvinv.n . . . . . . 7 𝑁 = (invg𝐺)
75, 6grpinvinv 17869 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁‘(𝑁𝑋)) = 𝑋)
83, 4, 7syl2anc 579 . . . . 5 (𝜑 → (𝑁‘(𝑁𝑋)) = 𝑋)
98adantr 474 . . . 4 ((𝜑 ∧ (𝑁𝑋) = (𝑁𝑌)) → (𝑁‘(𝑁𝑋)) = 𝑋)
10 grpinv11.y . . . . . 6 (𝜑𝑌𝐵)
115, 6grpinvinv 17869 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁‘(𝑁𝑌)) = 𝑌)
123, 10, 11syl2anc 579 . . . . 5 (𝜑 → (𝑁‘(𝑁𝑌)) = 𝑌)
1312adantr 474 . . . 4 ((𝜑 ∧ (𝑁𝑋) = (𝑁𝑌)) → (𝑁‘(𝑁𝑌)) = 𝑌)
142, 9, 133eqtr3d 2821 . . 3 ((𝜑 ∧ (𝑁𝑋) = (𝑁𝑌)) → 𝑋 = 𝑌)
1514ex 403 . 2 (𝜑 → ((𝑁𝑋) = (𝑁𝑌) → 𝑋 = 𝑌))
16 fveq2 6446 . 2 (𝑋 = 𝑌 → (𝑁𝑋) = (𝑁𝑌))
1715, 16impbid1 217 1 (𝜑 → ((𝑁𝑋) = (𝑁𝑌) ↔ 𝑋 = 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2106  ‘cfv 6135  Basecbs 16255  Grpcgrp 17809  invgcminusg 17810 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812  df-minusg 17813 This theorem is referenced by:  gexdvds  18383  dchrisum0re  25654  mapdpglem30  37840
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