Theorem mulgfval 18988

Description: Group multiple (exponentiation) operation. For a shorter proof using ax-rep 5284, see mulgfvalALT 18989. (Contributed by Mario Carneiro, 11-Dec-2014.) Remove dependency on ax-rep 5284. (Revised by Rohan Ridenour, 17-Aug-2023.)

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
mulgfval	⊢ · = (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 mulgfval

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 2731	. . . . 5 ⊢ (𝑤 = 𝐺 → ℤ = ℤ)
3		fveq2 6890	. . . . . 6 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4		mulgval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2788	. . . . 5 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
6		fveq2 6890	. . . . . . 7 ⊢ (𝑤 = 𝐺 → (0g‘𝑤) = (0g‘𝐺))
7		mulgval.o	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
8	6, 7	eqtr4di 2788	. . . . . 6 ⊢ (𝑤 = 𝐺 → (0g‘𝑤) = 0 )
9		fvex 6903	. . . . . . . . 9 ⊢ (+g‘𝑤) ∈ V
10		1z 12596	. . . . . . . . 9 ⊢ 1 ∈ ℤ
11	9, 10	seqexw 13986	. . . . . . . 8 ⊢ seq1((+g‘𝑤), (ℕ × {𝑥})) ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝑤 = 𝐺 → seq1((+g‘𝑤), (ℕ × {𝑥})) ∈ V)
13		id 22	. . . . . . . . . 10 ⊢ (𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥})) → 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥})))
14		fveq2 6890	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = (+g‘𝐺))
15		mulgval.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝐺)
16	14, 15	eqtr4di 2788	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = + )
17	16	seqeq2d 13977	. . . . . . . . . 10 ⊢ (𝑤 = 𝐺 → seq1((+g‘𝑤), (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑥})))
18	13, 17	sylan9eqr 2792	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → 𝑠 = seq1( + , (ℕ × {𝑥})))
19	18	fveq1d 6892	. . . . . . . 8 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (𝑠‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
20		simpl 481	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → 𝑤 = 𝐺)
21	20	fveq2d 6894	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (invg‘𝑤) = (invg‘𝐺))
22		mulgval.i	. . . . . . . . . 10 ⊢ 𝐼 = (invg‘𝐺)
23	21, 22	eqtr4di 2788	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (invg‘𝑤) = 𝐼)
24	18	fveq1d 6892	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (𝑠‘-𝑛) = (seq1( + , (ℕ × {𝑥}))‘-𝑛))
25	23, 24	fveq12d 6897	. . . . . . . 8 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → ((invg‘𝑤)‘(𝑠‘-𝑛)) = (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))
26	19, 25	ifeq12d 4548	. . . . . . 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))))
27	12, 26	csbied 3930	. . . . . 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))))
28	8, 27	ifeq12d 4548	. . . . 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)))))
29	2, 5, 28	mpoeq123dv 7486	. . . 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))))))
30		df-mulg 18987	. . . 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))))))
31		zex 12571	. . . . 5 ⊢ ℤ ∈ V
32	4	fvexi 6904	. . . . 5 ⊢ 𝐵 ∈ V
33		snex 5430	. . . . . 6 ⊢ { 0 } ∈ V
34	15	fvexi 6904	. . . . . . . . 9 ⊢ + ∈ V
35	34	rnex 7905	. . . . . . . 8 ⊢ ran + ∈ V
36	35, 32	unex 7735	. . . . . . 7 ⊢ (ran + ∪ 𝐵) ∈ V
37	22	fvexi 6904	. . . . . . . . 9 ⊢ 𝐼 ∈ V
38	37	rnex 7905	. . . . . . . 8 ⊢ ran 𝐼 ∈ V
39		p0ex 5381	. . . . . . . 8 ⊢ {∅} ∈ V
40	38, 39	unex 7735	. . . . . . 7 ⊢ (ran 𝐼 ∪ {∅}) ∈ V
41	36, 40	unex 7735	. . . . . 6 ⊢ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})) ∈ V
42	33, 41	unex 7735	. . . . 5 ⊢ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))) ∈ V
43		ssun1 4171	. . . . . . . . 9 ⊢ { 0 } ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
44	7	fvexi 6904	. . . . . . . . . 10 ⊢ 0 ∈ V
45	44	snid 4663	. . . . . . . . 9 ⊢ 0 ∈ { 0 }
46	43, 45	sselii 3978	. . . . . . . 8 ⊢ 0 ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
47	46	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → 0 ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
48		ssun2 4172	. . . . . . . . . . . . . 14 ⊢ 𝐵 ⊆ (ran + ∪ 𝐵)
49		ssun1 4171	. . . . . . . . . . . . . 14 ⊢ (ran + ∪ 𝐵) ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
50	48, 49	sstri 3990	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
51		ssun2 4172	. . . . . . . . . . . . 13 ⊢ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})) ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
52	50, 51	sstri 3990	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
53		fveq2 6890	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘1))
54	53	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘1))
55		seq1 13983	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℤ → (seq1( + , (ℕ × {𝑥}))‘1) = ((ℕ × {𝑥})‘1))
56	10, 55	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (seq1( + , (ℕ × {𝑥}))‘1) = ((ℕ × {𝑥})‘1)
57		1nn 12227	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ
58		vex 3476	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑥 ∈ V
59	58	fvconst2 7206	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℕ → ((ℕ × {𝑥})‘1) = 𝑥)
60	57, 59	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((ℕ × {𝑥})‘1) = 𝑥
61	60	eleq1i 2822	. . . . . . . . . . . . . . . 16 ⊢ (((ℕ × {𝑥})‘1) ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)
62	61	biimpri 227	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐵 → ((ℕ × {𝑥})‘1) ∈ 𝐵)
63	56, 62	eqeltrid 2835	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐵 → (seq1( + , (ℕ × {𝑥}))‘1) ∈ 𝐵)
64	63	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘1) ∈ 𝐵)
65	54, 64	eqeltrd 2831	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ 𝐵)
66	52, 65	sselid 3979	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
67	66	ad4ant24 750	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ 𝑛 ∈ (ℤ≥‘1)) ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
68		zcn 12567	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
69		npcan1 11643	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
70	68, 69	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℤ → ((𝑛 − 1) + 1) = 𝑛)
71	70	fveq2d 6894	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
72	71	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
73		seqp1 13985	. . . . . . . . . . . . . 14 ⊢ ((𝑛 − 1) ∈ (ℤ≥‘1) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))))
74		ssun1 4171	. . . . . . . . . . . . . . . . 17 ⊢ ran + ⊆ (ran + ∪ 𝐵)
75		ssun2 4172	. . . . . . . . . . . . . . . . 17 ⊢ {∅} ⊆ (ran 𝐼 ∪ {∅})
76		unss12 4181	. . . . . . . . . . . . . . . . 17 ⊢ ((ran + ⊆ (ran + ∪ 𝐵) ∧ {∅} ⊆ (ran 𝐼 ∪ {∅})) → (ran + ∪ {∅}) ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
77	74, 75, 76	mp2an 688	. . . . . . . . . . . . . . . 16 ⊢ (ran + ∪ {∅}) ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
78	77, 51	sstri 3990	. . . . . . . . . . . . . . 15 ⊢ (ran + ∪ {∅}) ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
79		df-ov 7414	. . . . . . . . . . . . . . . 16 ⊢ ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) = ( + ‘⟨(seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)), ((ℕ × {𝑥})‘((𝑛 − 1) + 1))⟩)
80		fvrn0 6920	. . . . . . . . . . . . . . . 16 ⊢ ( + ‘⟨(seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)), ((ℕ × {𝑥})‘((𝑛 − 1) + 1))⟩) ∈ (ran + ∪ {∅})
81	79, 80	eqeltri 2827	. . . . . . . . . . . . . . 15 ⊢ ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) ∈ (ran + ∪ {∅})
82	78, 81	sselii 3978	. . . . . . . . . . . . . 14 ⊢ ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
83	73, 82	eqeltrdi 2839	. . . . . . . . . . . . 13 ⊢ ((𝑛 − 1) ∈ (ℤ≥‘1) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
84	83	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
85	72, 84	eqeltrrd 2832	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
86	85	ad4ant14 748	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ 𝑛 ∈ (ℤ≥‘1)) ∧ (𝑛 − 1) ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
87		uzm1 12864	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘1) → (𝑛 = 1 ∨ (𝑛 − 1) ∈ (ℤ≥‘1)))
88	87	adantl 480	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ 𝑛 ∈ (ℤ≥‘1)) → (𝑛 = 1 ∨ (𝑛 − 1) ∈ (ℤ≥‘1)))
89	67, 86, 88	mpjaodan 955	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ 𝑛 ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
90		simpr 483	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑛 ∈ (ℤ≥‘1)) → ¬ 𝑛 ∈ (ℤ≥‘1))
91		seqfn 13982	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℤ → seq1( + , (ℕ × {𝑥})) Fn (ℤ≥‘1))
92	10, 91	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ seq1( + , (ℕ × {𝑥})) Fn (ℤ≥‘1)
93	92	fndmi 6652	. . . . . . . . . . . . 13 ⊢ dom seq1( + , (ℕ × {𝑥})) = (ℤ≥‘1)
94	93	eleq2i 2823	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ dom seq1( + , (ℕ × {𝑥})) ↔ 𝑛 ∈ (ℤ≥‘1))
95	90, 94	sylnibr 328	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑛 ∈ (ℤ≥‘1)) → ¬ 𝑛 ∈ dom seq1( + , (ℕ × {𝑥})))
96		ndmfv 6925	. . . . . . . . . . 11 ⊢ (¬ 𝑛 ∈ dom seq1( + , (ℕ × {𝑥})) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = ∅)
97	95, 96	syl 17	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑛 ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = ∅)
98		ssun2 4172	. . . . . . . . . . . . . 14 ⊢ (ran 𝐼 ∪ {∅}) ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
99	75, 98	sstri 3990	. . . . . . . . . . . . 13 ⊢ {∅} ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
100	99, 51	sstri 3990	. . . . . . . . . . . 12 ⊢ {∅} ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
101		0ex 5306	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
102	101	snid 4663	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅}
103	100, 102	sselii 3978	. . . . . . . . . . 11 ⊢ ∅ ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
104	103	a1i 11	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑛 ∈ (ℤ≥‘1)) → ∅ ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
105	97, 104	eqeltrd 2831	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑛 ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
106	89, 105	pm2.61dan 809	. . . . . . . 8 ⊢ ((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
107	98, 51	sstri 3990	. . . . . . . . . 10 ⊢ (ran 𝐼 ∪ {∅}) ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
108		fvrn0 6920	. . . . . . . . . 10 ⊢ (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ (ran 𝐼 ∪ {∅})
109	107, 108	sselii 3978	. . . . . . . . 9 ⊢ (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
110	109	a1i 11	. . . . . . . 8 ⊢ ((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
111	106, 110	ifcld 4573	. . . . . . 7 ⊢ ((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
112	47, 111	ifcld 4573	. . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
113	112	rgen2 3195	. . . . 5 ⊢ ∀𝑛 ∈ ℤ ∀𝑥 ∈ 𝐵 if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
114	31, 32, 42, 113	mpoexw 8067	. . . 4 ⊢ (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) ∈ V
115	29, 30, 114	fvmpt 6997	. . 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))))))
116		fvprc 6882	. . . 4 ⊢ (¬ 𝐺 ∈ V → (.g‘𝐺) = ∅)
117		eqid 2730	. . . . . . 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)))))
118		fvex 6903	. . . . . . . . 9 ⊢ (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ V
119		fvex 6903	. . . . . . . . 9 ⊢ (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ V
120	118, 119	ifex 4577	. . . . . . . 8 ⊢ if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) ∈ V
121	44, 120	ifex 4577	. . . . . . 7 ⊢ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) ∈ V
122	117, 121	fnmpoi 8058	. . . . . 6 ⊢ (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵)
123		fvprc 6882	. . . . . . . . . 10 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
124	4, 123	eqtrid 2782	. . . . . . . . 9 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
125	124	xpeq2d 5705	. . . . . . . 8 ⊢ (¬ 𝐺 ∈ V → (ℤ × 𝐵) = (ℤ × ∅))
126		xp0 6156	. . . . . . . 8 ⊢ (ℤ × ∅) = ∅
127	125, 126	eqtrdi 2786	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (ℤ × 𝐵) = ∅)
128	127	fneq2d 6642	. . . . . 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 ∅))
129	122, 128	mpbii 232	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn ∅)
130		fn0 6680	. . . . 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))))) = ∅)
131	129, 130	sylib 217	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) = ∅)
132	116, 131	eqtr4d 2773	. . 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))))))
133	115, 132	pm2.61i 182	. 2 ⊢ (.g‘𝐺) = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
134	1, 133	eqtri 2758	1 ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))

Syntax hints:  ¬ wn 3   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104  Vcvv 3472  ⦋csb 3892   ∪ cun 3945   ⊆ wss 3947  ∅c0 4321  ifcif 4527  {csn 4627  ⟨cop 4633   class class class wbr 5147   × cxp 5673  dom cdm 5675  ran crn 5676   Fn wfn 6537  ‘cfv 6542  (class class class)co 7411   ∈ cmpo 7413  ℂcc 11110  0cc0 11112  1c1 11113   + caddc 11115   < clt 11252   − cmin 11448  -cneg 11449  ℕcn 12216  ℤcz 12562  ℤ_≥cuz 12826  seqcseq 13970  Basecbs 17148  +_gcplusg 17201  0_gc0g 17389  inv_gcminusg 18856  ._gcmg 18986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-seq 13971  df-mulg 18987

This theorem is referenced by:  mulgval  18990  mulgfn  18991  mulgpropd  19032