Theorem grpinvfvalALT 18863

Description: Shorter proof of grpinvfval 18862 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	⊢ + = (+g‘𝐺)
grpinvval.o	⊢ 0 = (0g‘𝐺)
grpinvval.n	⊢ 𝑁 = (invg‘𝐺)

Assertion

Ref	Expression
grpinvfvalALT	⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (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.

Step	Hyp	Ref	Expression
1		grpinvval.n	. 2 ⊢ 𝑁 = (invg‘𝐺)
2		fveq2 6891	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		grpinvval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	eqtr4di 2790	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5		fveq2 6891	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
6		grpinvval.p	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
7	5, 6	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
8	7	oveqd 7425	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥))
9		fveq2 6891	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
10		grpinvval.o	. . . . . . . 8 ⊢ 0 = (0g‘𝐺)
11	9, 10	eqtr4di 2790	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
12	8, 11	eqeq12d 2748	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑦(+g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 + 𝑥) = 0 ))
13	4, 12	riotaeqbidv 7367	. . . . 5 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
14	4, 13	mpteq12dv 5239	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
15		df-minusg 18822	. . . 4 ⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))))
16	14, 15, 3	mptfvmpt 7229	. . 3 ⊢ (𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
17		fvprc 6883	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = ∅)
18		mpt0 6692	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = ∅
19	17, 18	eqtr4di 2790	. . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
20		fvprc 6883	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
21	3, 20	eqtrid 2784	. . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
22	21	mpteq1d 5243	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
23	19, 22	eqtr4d 2775	. . 3 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
24	16, 23	pm2.61i 182	. 2 ⊢ (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
25	1, 24	eqtri 2760	1 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ∅c0 4322   ↦ cmpt 5231  ‘cfv 6543  ℩crio 7363  (class class class)co 7408  Basecbs 17143  +_gcplusg 17196  0_gc0g 17384  inv_gcminusg 18819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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 7364  df-ov 7411  df-minusg 18822

This theorem is referenced by: (None)