MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpinvfval Structured version   Visualization version   GIF version

Theorem grpinvfval 17283
Description: The inverse function of a group. (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
grpinvfval 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥, 0   𝑥, +
Allowed substitution hints:   + (𝑦)   𝑁(𝑥,𝑦)   0 (𝑦)

Proof of Theorem grpinvfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpinvval.n . 2 𝑁 = (invg𝐺)
2 fveq2 6103 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpinvval.b . . . . . 6 𝐵 = (Base‘𝐺)
42, 3syl6eqr 2662 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6103 . . . . . . . . 9 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 grpinvval.p . . . . . . . . 9 + = (+g𝐺)
75, 6syl6eqr 2662 . . . . . . . 8 (𝑔 = 𝐺 → (+g𝑔) = + )
87oveqd 6566 . . . . . . 7 (𝑔 = 𝐺 → (𝑦(+g𝑔)𝑥) = (𝑦 + 𝑥))
9 fveq2 6103 . . . . . . . 8 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
10 grpinvval.o . . . . . . . 8 0 = (0g𝐺)
119, 10syl6eqr 2662 . . . . . . 7 (𝑔 = 𝐺 → (0g𝑔) = 0 )
128, 11eqeq12d 2625 . . . . . 6 (𝑔 = 𝐺 → ((𝑦(+g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 + 𝑥) = 0 ))
134, 12riotaeqbidv 6514 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔)) = (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
144, 13mpteq12dv 4663 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
15 df-minusg 17249 . . . 4 invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))))
16 fvex 6113 . . . . . 6 (Base‘𝐺) ∈ V
173, 16eqeltri 2684 . . . . 5 𝐵 ∈ V
1817mptex 6390 . . . 4 (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) ∈ V
1914, 15, 18fvmpt 6191 . . 3 (𝐺 ∈ V → (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
20 fvprc 6097 . . . . 5 𝐺 ∈ V → (invg𝐺) = ∅)
21 mpt0 5934 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) = ∅
2220, 21syl6eqr 2662 . . . 4 𝐺 ∈ V → (invg𝐺) = (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
23 fvprc 6097 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
243, 23syl5eq 2656 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
2524mpteq1d 4666 . . . 4 𝐺 ∈ V → (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
2622, 25eqtr4d 2647 . . 3 𝐺 ∈ V → (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
2719, 26pm2.61i 175 . 2 (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
281, 27eqtri 2632 1 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1475  wcel 1977  Vcvv 3173  c0 3874  cmpt 4643  cfv 5804  crio 6510  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  invgcminusg 17246
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 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
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-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 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-minusg 17249
This theorem is referenced by:  grpinvval  17284  grpinvfn  17285  grpinvf  17289  grpinvpropd  17313  opprneg  18458
  Copyright terms: Public domain W3C validator