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Theorem grpinvnz 17957
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 6499 . . . . . 6 ((𝑁𝑋) = 0 → (𝑁‘(𝑁𝑋)) = (𝑁0 ))
21adantl 474 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → (𝑁‘(𝑁𝑋)) = (𝑁0 ))
3 grpinvnzcl.b . . . . . . 7 𝐵 = (Base‘𝐺)
4 grpinvnzcl.n . . . . . . 7 𝑁 = (invg𝐺)
53, 4grpinvinv 17953 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁‘(𝑁𝑋)) = 𝑋)
65adantr 473 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → (𝑁‘(𝑁𝑋)) = 𝑋)
7 grpinvnzcl.z . . . . . . 7 0 = (0g𝐺)
87, 4grpinvid 17947 . . . . . 6 (𝐺 ∈ Grp → (𝑁0 ) = 0 )
98ad2antrr 713 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → (𝑁0 ) = 0 )
102, 6, 93eqtr3d 2822 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ (𝑁𝑋) = 0 ) → 𝑋 = 0 )
1110ex 405 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) = 0𝑋 = 0 ))
1211necon3d 2988 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋0 → (𝑁𝑋) ≠ 0 ))
13123impia 1097 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑋0 ) → (𝑁𝑋) ≠ 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2967  cfv 6188  Basecbs 16339  0gc0g 16569  Grpcgrp 17891  invgcminusg 17892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196  df-riota 6937  df-ov 6979  df-0g 16571  df-mgm 17710  df-sgrp 17752  df-mnd 17763  df-grp 17894  df-minusg 17895
This theorem is referenced by:  grpinvnzcl  17958
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