Theorem mulgfvalALT 19286

Description: Shorter proof of mulgfval 19285 using ax-rep 5231. (Contributed by Mario Carneiro, 11-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
mulgval.b	⊢ 𝐵 = (Base‘𝐺)
mulgval.p	⊢ + = (+g‘𝐺)
mulgval.o	⊢ 0 = (0g‘𝐺)
mulgval.i	⊢ 𝐼 = (invg‘𝐺)
mulgval.t	⊢ · = (.g‘𝐺)

Assertion

Ref	Expression
mulgfvalALT	⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛)))))

Distinct variable groups:   𝑥, 0 ,𝑛   𝑥,𝐵,𝑛   𝑥, + ,𝑛   𝑥,𝐺,𝑛   𝑥,𝐼,𝑛

Allowed substitution hints:   · (𝑥, 𝑛)

Proof of Theorem mulgfvalALT

Dummy variables 𝑤 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulgval.t	. 2 ⊢ · = (.g‘𝐺)
2		eqidd 2761	. . . . 5 ⊢ (𝑤 = 𝐺 → ℤ = ℤ)
3		fveq2 6886	. . . . . 6 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4		mulgval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2813	. . . . 5 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
6		fveq2 6886	. . . . . . 7 ⊢ (𝑤 = 𝐺 → (0g‘𝑤) = (0g‘𝐺))
7		mulgval.o	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
8	6, 7	eqtr4di 2813	. . . . . 6 ⊢ (𝑤 = 𝐺 → (0g‘𝑤) = 0 )
9		seqex 14150	. . . . . . . 8 ⊢ seq1((+g‘𝑤), (ℕ × {𝑥})) ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝑤 = 𝐺 → seq1((+g‘𝑤), (ℕ × {𝑥})) ∈ V)
11		id 23	. . . . . . . . . 10 ⊢ (𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥})) → 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥})))
12		fveq2 6886	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = (+g‘𝐺))
13		mulgval.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝐺)
14	12, 13	eqtr4di 2813	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = + )
15	14	seqeq2d 14155	. . . . . . . . . 10 ⊢ (𝑤 = 𝐺 → seq1((+g‘𝑤), (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑥})))
16	11, 15	sylan9eqr 2817	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → 𝑠 = seq1( + , (ℕ × {𝑥})))
17	16	fveq1d 6888	. . . . . . . 8 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (𝑠‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
18		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → 𝑤 = 𝐺)
19	18	fveq2d 6890	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (invg‘𝑤) = (invg‘𝐺))
20		mulgval.i	. . . . . . . . . 10 ⊢ 𝐼 = (invg‘𝐺)
21	19, 20	eqtr4di 2813	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (invg‘𝑤) = 𝐼)
22	16	fveq1d 6888	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (𝑠‘ -𝑛) = (seq1( + , (ℕ × {𝑥}))‘ -𝑛))
23	21, 22	fveq12d 6893	. . . . . . . 8 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → ((invg‘𝑤)‘(𝑠‘ -𝑛)) = (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛)))
24	17, 23	ifeq12d 4503	. . . . . . 7 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑤)‘(𝑠‘ -𝑛))) = if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))
25	10, 24	csbied 3882	. . . . . 6 ⊢ (𝑤 = 𝐺 → ⦋seq1((+g‘𝑤), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑤)‘(𝑠‘ -𝑛))) = if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))
26	8, 25	ifeq12d 4503	. . . . 5 ⊢ (𝑤 = 𝐺 → if(𝑛 = 0, (0g‘𝑤), ⦋seq1((+g‘𝑤), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑤)‘(𝑠‘ -𝑛)))) = if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛)))))
27	2, 5, 26	mpoeq123dv 7496	. . . 4 ⊢ (𝑤 = 𝐺 → (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑤) ↦ if(𝑛 = 0, (0g‘𝑤), ⦋seq1((+g‘𝑤), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑤)‘(𝑠‘ -𝑛))))) = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))))
28		df-mulg 19284	. . . 4 ⊢ .g = (𝑤 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑤) ↦ if(𝑛 = 0, (0g‘𝑤), ⦋seq1((+g‘𝑤), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑤)‘(𝑠‘ -𝑛))))))
29		zex 12706	. . . . 5 ⊢ ℤ ∈ V
30	4	fvexi 6900	. . . . 5 ⊢ 𝐵 ∈ V
31	29, 30	mpoex 8093	. . . 4 ⊢ (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))) ∈ V
32	27, 28, 31	fvmpt 6994	. . 3 ⊢ (𝐺 ∈ V → (.g‘𝐺) = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))))
33		fvprc 6878	. . . 4 ⊢ (¬ 𝐺 ∈ V → (.g‘𝐺) = ∅)
34		eqid 2760	. . . . . . 7 ⊢ (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))) = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛)))))
35	7	fvexi 6900	. . . . . . . 8 ⊢ 0 ∈ V
36		fvex 6899	. . . . . . . . 9 ⊢ (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ V
37		fvex 6899	. . . . . . . . 9 ⊢ (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛)) ∈ V
38	36, 37	ifex 4532	. . . . . . . 8 ⊢ if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))) ∈ V
39	35, 38	ifex 4532	. . . . . . 7 ⊢ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛)))) ∈ V
40	34, 39	fnmpoi 8082	. . . . . 6 ⊢ (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))) Fn (ℤ × 𝐵)
41		fvprc 6878	. . . . . . . . . 10 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
42	4, 41	eqtrid 2807	. . . . . . . . 9 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
43	42	xpeq2d 5680	. . . . . . . 8 ⊢ (¬ 𝐺 ∈ V → (ℤ × 𝐵) = (ℤ × ∅))
44		xp0 5750	. . . . . . . 8 ⊢ (ℤ × ∅) = ∅
45	43, 44	eqtrdi 2811	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (ℤ × 𝐵) = ∅)
46	45	fneq2d 6634	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → ((𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))) Fn (ℤ × 𝐵) ↔ (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))) Fn ∅))
47	40, 46	mpbii 236	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))) Fn ∅)
48		fn0 6671	. . . . 5 ⊢ ((𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))) Fn ∅ ↔ (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))) = ∅)
49	47, 48	sylib 221	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))) = ∅)
50	33, 49	eqtr4d 2798	. . 3 ⊢ (¬ 𝐺 ∈ V → (.g‘𝐺) = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛))))))
51	32, 50	pm2.61i 184	. 2 ⊢ (.g‘𝐺) = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛)))))
52	1, 51	eqtri 2783	1 ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘ -𝑛)))))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⦋csb 3846  ∅c0 4278  ifcif 4481  {csn 4584   class class class wbr 5102   × cxp 5648   Fn wfn 6534  ‘cfv 6539   ∈ cmpo 7423  0cc0 11204  1c1 11205   < clt 11347   -cneg 11546  ℕcn 12339  ℤcz 12697  seqcseq 14148  Basecbs 17393  +_gcplusg 17434  0_gc0g 17616  inv_gcminusg 19151  ._gcmg 19283

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-inf2 9646  ax-cnex 11260  ax-resscn 11261

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-neg 11548  df-z 12698  df-seq 14149  df-mulg 19284

This theorem is used by: (None)