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Theorem grpinvfvalALT 18904
Description: Shorter proof of grpinvfval 18903 using ax-rep 5285. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinvval.b 𝐵 = (Base‘𝐺)
grpinvval.p + = (+g𝐺)
grpinvval.o 0 = (0g𝐺)
grpinvval.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpinvfvalALT 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 grpinvfvalALT
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpinvval.n . 2 𝑁 = (invg𝐺)
2 fveq2 6891 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpinvval.b . . . . . 6 𝐵 = (Base‘𝐺)
42, 3eqtr4di 2789 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6891 . . . . . . . . 9 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 grpinvval.p . . . . . . . . 9 + = (+g𝐺)
75, 6eqtr4di 2789 . . . . . . . 8 (𝑔 = 𝐺 → (+g𝑔) = + )
87oveqd 7429 . . . . . . 7 (𝑔 = 𝐺 → (𝑦(+g𝑔)𝑥) = (𝑦 + 𝑥))
9 fveq2 6891 . . . . . . . 8 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
10 grpinvval.o . . . . . . . 8 0 = (0g𝐺)
119, 10eqtr4di 2789 . . . . . . 7 (𝑔 = 𝐺 → (0g𝑔) = 0 )
128, 11eqeq12d 2747 . . . . . 6 (𝑔 = 𝐺 → ((𝑦(+g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 + 𝑥) = 0 ))
134, 12riotaeqbidv 7371 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔)) = (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
144, 13mpteq12dv 5239 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
15 df-minusg 18862 . . . 4 invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))))
1614, 15, 3mptfvmpt 7232 . . 3 (𝐺 ∈ V → (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
17 fvprc 6883 . . . . 5 𝐺 ∈ V → (invg𝐺) = ∅)
18 mpt0 6692 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) = ∅
1917, 18eqtr4di 2789 . . . 4 𝐺 ∈ V → (invg𝐺) = (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
20 fvprc 6883 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
213, 20eqtrid 2783 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
2221mpteq1d 5243 . . . 4 𝐺 ∈ V → (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
2319, 22eqtr4d 2774 . . 3 𝐺 ∈ V → (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
2416, 23pm2.61i 182 . 2 (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
251, 24eqtri 2759 1 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2105  Vcvv 3473  c0 4322  cmpt 5231  cfv 6543  crio 7367  (class class class)co 7412  Basecbs 17151  +gcplusg 17204  0gc0g 17392  invgcminusg 18859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-minusg 18862
This theorem is referenced by: (None)
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