Theorem mulgfval 18702

Description: Group multiple (exponentiation) operation. For a shorter proof using ax-rep 5209, see mulgfvalALT 18703. (Contributed by Mario Carneiro, 11-Dec-2014.) Remove dependency on ax-rep 5209. (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 2739	. . . . 5 ⊢ (𝑤 = 𝐺 → ℤ = ℤ)
3		fveq2 6774	. . . . . 6 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4		mulgval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2796	. . . . 5 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
6		fveq2 6774	. . . . . . 7 ⊢ (𝑤 = 𝐺 → (0g‘𝑤) = (0g‘𝐺))
7		mulgval.o	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
8	6, 7	eqtr4di 2796	. . . . . 6 ⊢ (𝑤 = 𝐺 → (0g‘𝑤) = 0 )
9		fvex 6787	. . . . . . . . 9 ⊢ (+g‘𝑤) ∈ V
10		1z 12350	. . . . . . . . 9 ⊢ 1 ∈ ℤ
11	9, 10	seqexw 13737	. . . . . . . 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 6774	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = (+g‘𝐺))
15		mulgval.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝐺)
16	14, 15	eqtr4di 2796	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = + )
17	16	seqeq2d 13728	. . . . . . . . . 10 ⊢ (𝑤 = 𝐺 → seq1((+g‘𝑤), (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑥})))
18	13, 17	sylan9eqr 2800	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → 𝑠 = seq1( + , (ℕ × {𝑥})))
19	18	fveq1d 6776	. . . . . . . 8 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (𝑠‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
20		simpl 483	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → 𝑤 = 𝐺)
21	20	fveq2d 6778	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (invg‘𝑤) = (invg‘𝐺))
22		mulgval.i	. . . . . . . . . 10 ⊢ 𝐼 = (invg‘𝐺)
23	21, 22	eqtr4di 2796	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (invg‘𝑤) = 𝐼)
24	18	fveq1d 6776	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (𝑠‘-𝑛) = (seq1( + , (ℕ × {𝑥}))‘-𝑛))
25	23, 24	fveq12d 6781	. . . . . . . 8 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → ((invg‘𝑤)‘(𝑠‘-𝑛)) = (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))
26	19, 25	ifeq12d 4480	. . . . . . 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 3870	. . . . . 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 4480	. . . . 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 7350	. . . 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 18701	. . . 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 12328	. . . . 5 ⊢ ℤ ∈ V
32	4	fvexi 6788	. . . . 5 ⊢ 𝐵 ∈ V
33		snex 5354	. . . . . 6 ⊢ { 0 } ∈ V
34	15	fvexi 6788	. . . . . . . . 9 ⊢ + ∈ V
35	34	rnex 7759	. . . . . . . 8 ⊢ ran + ∈ V
36	35, 32	unex 7596	. . . . . . 7 ⊢ (ran + ∪ 𝐵) ∈ V
37	22	fvexi 6788	. . . . . . . . 9 ⊢ 𝐼 ∈ V
38	37	rnex 7759	. . . . . . . 8 ⊢ ran 𝐼 ∈ V
39		p0ex 5307	. . . . . . . 8 ⊢ {∅} ∈ V
40	38, 39	unex 7596	. . . . . . 7 ⊢ (ran 𝐼 ∪ {∅}) ∈ V
41	36, 40	unex 7596	. . . . . 6 ⊢ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})) ∈ V
42	33, 41	unex 7596	. . . . 5 ⊢ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))) ∈ V
43		ssun1 4106	. . . . . . . . 9 ⊢ { 0 } ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
44	7	fvexi 6788	. . . . . . . . . 10 ⊢ 0 ∈ V
45	44	snid 4597	. . . . . . . . 9 ⊢ 0 ∈ { 0 }
46	43, 45	sselii 3918	. . . . . . . 8 ⊢ 0 ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
47	46	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → 0 ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
48		ssun2 4107	. . . . . . . . . . . . . 14 ⊢ 𝐵 ⊆ (ran + ∪ 𝐵)
49		ssun1 4106	. . . . . . . . . . . . . 14 ⊢ (ran + ∪ 𝐵) ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
50	48, 49	sstri 3930	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
51		ssun2 4107	. . . . . . . . . . . . 13 ⊢ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})) ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
52	50, 51	sstri 3930	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
53		fveq2 6774	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘1))
54	53	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘1))
55		seq1 13734	. . . . . . . . . . . . . . . 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 11984	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ
58		vex 3436	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑥 ∈ V
59	58	fvconst2 7079	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℕ → ((ℕ × {𝑥})‘1) = 𝑥)
60	57, 59	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((ℕ × {𝑥})‘1) = 𝑥
61	60	eleq1i 2829	. . . . . . . . . . . . . . . 16 ⊢ (((ℕ × {𝑥})‘1) ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)
62	61	biimpri 227	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐵 → ((ℕ × {𝑥})‘1) ∈ 𝐵)
63	56, 62	eqeltrid 2843	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐵 → (seq1( + , (ℕ × {𝑥}))‘1) ∈ 𝐵)
64	63	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘1) ∈ 𝐵)
65	54, 64	eqeltrd 2839	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ 𝐵)
66	52, 65	sselid 3919	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
67	66	ad4ant24 751	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ 𝑛 ∈ (ℤ≥‘1)) ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
68		zcn 12324	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
69		npcan1 11400	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
70	68, 69	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℤ → ((𝑛 − 1) + 1) = 𝑛)
71	70	fveq2d 6778	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
72	71	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
73		seqp1 13736	. . . . . . . . . . . . . 14 ⊢ ((𝑛 − 1) ∈ (ℤ≥‘1) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))))
74		ssun1 4106	. . . . . . . . . . . . . . . . 17 ⊢ ran + ⊆ (ran + ∪ 𝐵)
75		ssun2 4107	. . . . . . . . . . . . . . . . 17 ⊢ {∅} ⊆ (ran 𝐼 ∪ {∅})
76		unss12 4116	. . . . . . . . . . . . . . . . 17 ⊢ ((ran + ⊆ (ran + ∪ 𝐵) ∧ {∅} ⊆ (ran 𝐼 ∪ {∅})) → (ran + ∪ {∅}) ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
77	74, 75, 76	mp2an 689	. . . . . . . . . . . . . . . 16 ⊢ (ran + ∪ {∅}) ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
78	77, 51	sstri 3930	. . . . . . . . . . . . . . 15 ⊢ (ran + ∪ {∅}) ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
79		df-ov 7278	. . . . . . . . . . . . . . . 16 ⊢ ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) = ( + ‘⟨(seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)), ((ℕ × {𝑥})‘((𝑛 − 1) + 1))⟩)
80		fvrn0 6802	. . . . . . . . . . . . . . . 16 ⊢ ( + ‘⟨(seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)), ((ℕ × {𝑥})‘((𝑛 − 1) + 1))⟩) ∈ (ran + ∪ {∅})
81	79, 80	eqeltri 2835	. . . . . . . . . . . . . . 15 ⊢ ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) ∈ (ran + ∪ {∅})
82	78, 81	sselii 3918	. . . . . . . . . . . . . 14 ⊢ ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
83	73, 82	eqeltrdi 2847	. . . . . . . . . . . . 13 ⊢ ((𝑛 − 1) ∈ (ℤ≥‘1) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
84	83	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
85	72, 84	eqeltrrd 2840	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
86	85	ad4ant14 749	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ 𝑛 ∈ (ℤ≥‘1)) ∧ (𝑛 − 1) ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
87		uzm1 12616	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘1) → (𝑛 = 1 ∨ (𝑛 − 1) ∈ (ℤ≥‘1)))
88	87	adantl 482	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ 𝑛 ∈ (ℤ≥‘1)) → (𝑛 = 1 ∨ (𝑛 − 1) ∈ (ℤ≥‘1)))
89	67, 86, 88	mpjaodan 956	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ 𝑛 ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
90		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑛 ∈ (ℤ≥‘1)) → ¬ 𝑛 ∈ (ℤ≥‘1))
91		seqfn 13733	. . . . . . . . . . . . . . 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 6537	. . . . . . . . . . . . 13 ⊢ dom seq1( + , (ℕ × {𝑥})) = (ℤ≥‘1)
94	93	eleq2i 2830	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ dom seq1( + , (ℕ × {𝑥})) ↔ 𝑛 ∈ (ℤ≥‘1))
95	90, 94	sylnibr 329	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑛 ∈ (ℤ≥‘1)) → ¬ 𝑛 ∈ dom seq1( + , (ℕ × {𝑥})))
96		ndmfv 6804	. . . . . . . . . . 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 4107	. . . . . . . . . . . . . 14 ⊢ (ran 𝐼 ∪ {∅}) ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
99	75, 98	sstri 3930	. . . . . . . . . . . . 13 ⊢ {∅} ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
100	99, 51	sstri 3930	. . . . . . . . . . . 12 ⊢ {∅} ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
101		0ex 5231	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
102	101	snid 4597	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅}
103	100, 102	sselii 3918	. . . . . . . . . . 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 2839	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑛 ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
106	89, 105	pm2.61dan 810	. . . . . . . 8 ⊢ ((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
107	98, 51	sstri 3930	. . . . . . . . . 10 ⊢ (ran 𝐼 ∪ {∅}) ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
108		fvrn0 6802	. . . . . . . . . 10 ⊢ (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ (ran 𝐼 ∪ {∅})
109	107, 108	sselii 3918	. . . . . . . . 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 4505	. . . . . . 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 4505	. . . . . 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 3120	. . . . 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 7919	. . . 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 6875	. . 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 6766	. . . 4 ⊢ (¬ 𝐺 ∈ V → (.g‘𝐺) = ∅)
117		eqid 2738	. . . . . . 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 6787	. . . . . . . . 9 ⊢ (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ V
119		fvex 6787	. . . . . . . . 9 ⊢ (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ V
120	118, 119	ifex 4509	. . . . . . . 8 ⊢ if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) ∈ V
121	44, 120	ifex 4509	. . . . . . 7 ⊢ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) ∈ V
122	117, 121	fnmpoi 7910	. . . . . 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 6766	. . . . . . . . . 10 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
124	4, 123	eqtrid 2790	. . . . . . . . 9 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
125	124	xpeq2d 5619	. . . . . . . 8 ⊢ (¬ 𝐺 ∈ V → (ℤ × 𝐵) = (ℤ × ∅))
126		xp0 6061	. . . . . . . 8 ⊢ (ℤ × ∅) = ∅
127	125, 126	eqtrdi 2794	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (ℤ × 𝐵) = ∅)
128	127	fneq2d 6527	. . . . . 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 6564	. . . . 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 2781	. . 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 2766	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 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⦋csb 3832   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  ifcif 4459  {csn 4561  ⟨cop 4567   class class class wbr 5074   × cxp 5587  dom cdm 5589  ran crn 5590   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  ℂcc 10869  0cc0 10871  1c1 10872   + caddc 10874   < clt 11009   − cmin 11205  -cneg 11206  ℕcn 11973  ℤcz 12319  ℤ_≥cuz 12582  seqcseq 13721  Basecbs 16912  +_gcplusg 16962  0_gc0g 17150  inv_gcminusg 18578  ._gcmg 18700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-seq 13722  df-mulg 18701

This theorem is referenced by:  mulgval  18704  mulgfn  18705  mulgpropd  18745