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Theorem grpinv11 17691
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 6332 . . . . 5 ((𝑁𝑋) = (𝑁𝑌) → (𝑁‘(𝑁𝑋)) = (𝑁‘(𝑁𝑌)))
21adantl 467 . . . 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 17689 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁‘(𝑁𝑋)) = 𝑋)
83, 4, 7syl2anc 565 . . . . 5 (𝜑 → (𝑁‘(𝑁𝑋)) = 𝑋)
98adantr 466 . . . 4 ((𝜑 ∧ (𝑁𝑋) = (𝑁𝑌)) → (𝑁‘(𝑁𝑋)) = 𝑋)
10 grpinv11.y . . . . . 6 (𝜑𝑌𝐵)
115, 6grpinvinv 17689 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁‘(𝑁𝑌)) = 𝑌)
123, 10, 11syl2anc 565 . . . . 5 (𝜑 → (𝑁‘(𝑁𝑌)) = 𝑌)
1312adantr 466 . . . 4 ((𝜑 ∧ (𝑁𝑋) = (𝑁𝑌)) → (𝑁‘(𝑁𝑌)) = 𝑌)
142, 9, 133eqtr3d 2812 . . 3 ((𝜑 ∧ (𝑁𝑋) = (𝑁𝑌)) → 𝑋 = 𝑌)
1514ex 397 . 2 (𝜑 → ((𝑁𝑋) = (𝑁𝑌) → 𝑋 = 𝑌))
16 fveq2 6332 . 2 (𝑋 = 𝑌 → (𝑁𝑋) = (𝑁𝑌))
1715, 16impbid1 215 1 (𝜑 → ((𝑁𝑋) = (𝑁𝑌) ↔ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  cfv 6031  Basecbs 16063  Grpcgrp 17629  invgcminusg 17630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-minusg 17633
This theorem is referenced by:  gexdvds  18205  dchrisum0re  25422  mapdpglem30  37505
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