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Theorem grpinvnz 18162
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 6645 . . . . . 6 ((𝑁𝑋) = 0 → (𝑁‘(𝑁𝑋)) = (𝑁0 ))
21adantl 485 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → (𝑁‘(𝑁𝑋)) = (𝑁0 ))
3 grpinvnzcl.b . . . . . . 7 𝐵 = (Base‘𝐺)
4 grpinvnzcl.n . . . . . . 7 𝑁 = (invg𝐺)
53, 4grpinvinv 18158 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁‘(𝑁𝑋)) = 𝑋)
65adantr 484 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → (𝑁‘(𝑁𝑋)) = 𝑋)
7 grpinvnzcl.z . . . . . . 7 0 = (0g𝐺)
87, 4grpinvid 18152 . . . . . 6 (𝐺 ∈ Grp → (𝑁0 ) = 0 )
98ad2antrr 725 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → (𝑁0 ) = 0 )
102, 6, 93eqtr3d 2841 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → 𝑋 = 0 )
1110ex 416 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) = 0𝑋 = 0 ))
1211necon3d 3008 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋0 → (𝑁𝑋) ≠ 0 ))
13123impia 1114 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑋0 ) → (𝑁𝑋) ≠ 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  wne 2987  cfv 6324  Basecbs 16475  0gc0g 16705  Grpcgrp 18095  invgcminusg 18096
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-riota 7093  df-ov 7138  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099
This theorem is referenced by:  grpinvnzcl  18163
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