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Theorem grpinvnz 17533
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 6229 . . . . . 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 17529 . . . . . 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 17523 . . . . . 6 (𝐺 ∈ Grp → (𝑁0 ) = 0 )
98ad2antrr 762 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → (𝑁0 ) = 0 )
102, 6, 93eqtr3d 2693 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → 𝑋 = 0 )
1110ex 449 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) = 0𝑋 = 0 ))
1211necon3d 2844 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋0 → (𝑁𝑋) ≠ 0 ))
13123impia 1280 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑋0 ) → (𝑁𝑋) ≠ 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  cfv 5926  Basecbs 15904  0gc0g 16147  Grpcgrp 17469  invgcminusg 17470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473
This theorem is referenced by:  grpinvnzcl  17534
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