Theorem mulgfvalALT 19159

Description: Shorter proof of mulgfval 19158 using ax-rep 5240. (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 2766	. . . . 5 ⊢ (𝑤 = 𝐺 → ℤ = ℤ)
3		fveq2 6885	. . . . . 6 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4		mulgval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2818	. . . . 5 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
6		fveq2 6885	. . . . . . 7 ⊢ (𝑤 = 𝐺 → (0g‘𝑤) = (0g‘𝐺))
7		mulgval.o	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
8	6, 7	eqtr4di 2818	. . . . . 6 ⊢ (𝑤 = 𝐺 → (0g‘𝑤) = 0 )
9		seqex 14052	. . . . . . . 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 6885	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = (+g‘𝐺))
13		mulgval.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝐺)
14	12, 13	eqtr4di 2818	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = + )
15	14	seqeq2d 14057	. . . . . . . . . 10 ⊢ (𝑤 = 𝐺 → seq1((+g‘𝑤), (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑥})))
16	11, 15	sylan9eqr 2822	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → 𝑠 = seq1( + , (ℕ × {𝑥})))
17	16	fveq1d 6887	. . . . . . . 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 6889	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (invg‘𝑤) = (invg‘𝐺))
20		mulgval.i	. . . . . . . . . 10 ⊢ 𝐼 = (invg‘𝐺)
21	19, 20	eqtr4di 2818	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (invg‘𝑤) = 𝐼)
22	16	fveq1d 6887	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (𝑠‘-𝑛) = (seq1( + , (ℕ × {𝑥}))‘-𝑛))
23	21, 22	fveq12d 6892	. . . . . . . 8 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → ((invg‘𝑤)‘(𝑠‘-𝑛)) = (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))
24	17, 23	ifeq12d 4511	. . . . . . 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 3890	. . . . . 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 4511	. . . . 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 7491	. . . 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 19157	. . . 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 12611	. . . . 5 ⊢ ℤ ∈ V
30	4	fvexi 6899	. . . . 5 ⊢ 𝐵 ∈ V
31	29, 30	mpoex 8078	. . . 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 6993	. . 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 6877	. . . 4 ⊢ (¬ 𝐺 ∈ V → (.g‘𝐺) = ∅)
34		eqid 2765	. . . . . . 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 6899	. . . . . . . 8 ⊢ 0 ∈ V
36		fvex 6898	. . . . . . . . 9 ⊢ (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ V
37		fvex 6898	. . . . . . . . 9 ⊢ (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ V
38	36, 37	ifex 4540	. . . . . . . 8 ⊢ if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) ∈ V
39	35, 38	ifex 4540	. . . . . . 7 ⊢ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) ∈ V
40	34, 39	fnmpoi 8069	. . . . . 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 6877	. . . . . . . . . 10 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
42	4, 41	eqtrid 2812	. . . . . . . . 9 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
43	42	xpeq2d 5693	. . . . . . . 8 ⊢ (¬ 𝐺 ∈ V → (ℤ × 𝐵) = (ℤ × ∅))
44		xp0 5763	. . . . . . . 8 ⊢ (ℤ × ∅) = ∅
45	43, 44	eqtrdi 2816	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (ℤ × 𝐵) = ∅)
46	45	fneq2d 6633	. . . . . 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 6670	. . . . 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 2803	. . 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 2788	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 1570   ∈ wcel 2146  Vcvv 3457  ⦋csb 3854  ∅c0 4286  ifcif 4489  {csn 4591   class class class wbr 5111   × cxp 5661   Fn wfn 6535  ‘cfv 6540   ∈ cmpo 7418  0cc0 11111  1c1 11112   < clt 11254  -cneg 11453  ℕcn 12244  ℤcz 12602  seqcseq 14050  Basecbs 17286  +_gcplusg 17327  0_gc0g 17509  inv_gcminusg 19024  ._gcmg 19156

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-inf2 9613  ax-cnex 11167  ax-resscn 11168

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-neg 11455  df-z 12603  df-seq 14051  df-mulg 19157

This theorem is used by: (None)