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Theorem grpinv11 18146
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 6643 . . . . 5 ((𝑁𝑋) = (𝑁𝑌) → (𝑁‘(𝑁𝑋)) = (𝑁‘(𝑁𝑌)))
21adantl 485 . . . 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 18144 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁‘(𝑁𝑋)) = 𝑋)
83, 4, 7syl2anc 587 . . . . 5 (𝜑 → (𝑁‘(𝑁𝑋)) = 𝑋)
98adantr 484 . . . 4 ((𝜑 ∧ (𝑁𝑋) = (𝑁𝑌)) → (𝑁‘(𝑁𝑋)) = 𝑋)
10 grpinv11.y . . . . . 6 (𝜑𝑌𝐵)
115, 6grpinvinv 18144 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁‘(𝑁𝑌)) = 𝑌)
123, 10, 11syl2anc 587 . . . . 5 (𝜑 → (𝑁‘(𝑁𝑌)) = 𝑌)
1312adantr 484 . . . 4 ((𝜑 ∧ (𝑁𝑋) = (𝑁𝑌)) → (𝑁‘(𝑁𝑌)) = 𝑌)
142, 9, 133eqtr3d 2864 . . 3 ((𝜑 ∧ (𝑁𝑋) = (𝑁𝑌)) → 𝑋 = 𝑌)
1514ex 416 . 2 (𝜑 → ((𝑁𝑋) = (𝑁𝑌) → 𝑋 = 𝑌))
16 fveq2 6643 . 2 (𝑋 = 𝑌 → (𝑁𝑋) = (𝑁𝑌))
1715, 16impbid1 228 1 (𝜑 → ((𝑁𝑋) = (𝑁𝑌) ↔ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cfv 6328  Basecbs 16461  Grpcgrp 18081  invgcminusg 18082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-riota 7088  df-ov 7133  df-0g 16693  df-mgm 17830  df-sgrp 17879  df-mnd 17890  df-grp 18084  df-minusg 18085
This theorem is referenced by:  gexdvds  18687  dchrisum0re  26075  mapdpglem30  38878
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