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Theorem grpinvnz 17258
Description: The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Hypotheses
Ref Expression
grpinvnzcl.b 𝐵 = (Base‘𝐺)
grpinvnzcl.z 0 = (0g𝐺)
grpinvnzcl.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpinvnz ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑋0 ) → (𝑁𝑋) ≠ 0 )

Proof of Theorem grpinvnz
StepHypRef Expression
1 fveq2 6088 . . . . . 6 ((𝑁𝑋) = 0 → (𝑁‘(𝑁𝑋)) = (𝑁0 ))
21adantl 481 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → (𝑁‘(𝑁𝑋)) = (𝑁0 ))
3 grpinvnzcl.b . . . . . . 7 𝐵 = (Base‘𝐺)
4 grpinvnzcl.n . . . . . . 7 𝑁 = (invg𝐺)
53, 4grpinvinv 17254 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁‘(𝑁𝑋)) = 𝑋)
65adantr 480 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → (𝑁‘(𝑁𝑋)) = 𝑋)
7 grpinvnzcl.z . . . . . . 7 0 = (0g𝐺)
87, 4grpinvid 17248 . . . . . 6 (𝐺 ∈ Grp → (𝑁0 ) = 0 )
98ad2antrr 758 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → (𝑁0 ) = 0 )
102, 6, 93eqtr3d 2652 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → 𝑋 = 0 )
1110ex 449 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) = 0𝑋 = 0 ))
1211necon3d 2803 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋0 → (𝑁𝑋) ≠ 0 ))
13123impia 1253 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑋0 ) → (𝑁𝑋) ≠ 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  cfv 5790  Basecbs 15644  0gc0g 15872  Grpcgrp 17194  invgcminusg 17195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-0g 15874  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-grp 17197  df-minusg 17198
This theorem is referenced by:  grpinvnzcl  17259
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