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Theorem grpinv11 18106
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 6663 . . . . 5 ((𝑁𝑋) = (𝑁𝑌) → (𝑁‘(𝑁𝑋)) = (𝑁‘(𝑁𝑌)))
21adantl 482 . . . 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 18104 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁‘(𝑁𝑋)) = 𝑋)
83, 4, 7syl2anc 584 . . . . 5 (𝜑 → (𝑁‘(𝑁𝑋)) = 𝑋)
98adantr 481 . . . 4 ((𝜑 ∧ (𝑁𝑋) = (𝑁𝑌)) → (𝑁‘(𝑁𝑋)) = 𝑋)
10 grpinv11.y . . . . . 6 (𝜑𝑌𝐵)
115, 6grpinvinv 18104 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁‘(𝑁𝑌)) = 𝑌)
123, 10, 11syl2anc 584 . . . . 5 (𝜑 → (𝑁‘(𝑁𝑌)) = 𝑌)
1312adantr 481 . . . 4 ((𝜑 ∧ (𝑁𝑋) = (𝑁𝑌)) → (𝑁‘(𝑁𝑌)) = 𝑌)
142, 9, 133eqtr3d 2861 . . 3 ((𝜑 ∧ (𝑁𝑋) = (𝑁𝑌)) → 𝑋 = 𝑌)
1514ex 413 . 2 (𝜑 → ((𝑁𝑋) = (𝑁𝑌) → 𝑋 = 𝑌))
16 fveq2 6663 . 2 (𝑋 = 𝑌 → (𝑁𝑋) = (𝑁𝑌))
1715, 16impbid1 226 1 (𝜑 → ((𝑁𝑋) = (𝑁𝑌) ↔ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  cfv 6348  Basecbs 16471  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:  gexdvds  18638  dchrisum0re  26016  mapdpglem30  38718
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