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Theorem grpinvval 17777
Description: The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)
Hypotheses
Ref Expression
grpinvval.b 𝐵 = (Base‘𝐺)
grpinvval.p + = (+g𝐺)
grpinvval.o 0 = (0g𝐺)
grpinvval.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpinvval (𝑋𝐵 → (𝑁𝑋) = (𝑦𝐵 (𝑦 + 𝑋) = 0 ))
Distinct variable groups:   𝑦,𝐵   𝑦,𝐺   𝑦,𝑋
Allowed substitution hints:   + (𝑦)   𝑁(𝑦)   0 (𝑦)

Proof of Theorem grpinvval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq2 6886 . . . 4 (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
21eqeq1d 2801 . . 3 (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
32riotabidv 6841 . 2 (𝑥 = 𝑋 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) = (𝑦𝐵 (𝑦 + 𝑋) = 0 ))
4 grpinvval.b . . 3 𝐵 = (Base‘𝐺)
5 grpinvval.p . . 3 + = (+g𝐺)
6 grpinvval.o . . 3 0 = (0g𝐺)
7 grpinvval.n . . 3 𝑁 = (invg𝐺)
84, 5, 6, 7grpinvfval 17776 . 2 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
9 riotaex 6843 . 2 (𝑦𝐵 (𝑦 + 𝑋) = 0 ) ∈ V
103, 8, 9fvmpt 6507 1 (𝑋𝐵 → (𝑁𝑋) = (𝑦𝐵 (𝑦 + 𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  cfv 6101  crio 6838  (class class class)co 6878  Basecbs 16184  +gcplusg 16267  0gc0g 16415  invgcminusg 17739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-minusg 17742
This theorem is referenced by:  grplinv  17784  isgrpinv  17788  xrsinvgval  30193  ringinvval  30308
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