Theorem mulgfval 19166

Description: Group multiple (exponentiation) operation. For a shorter proof using ax-rep 5243, see mulgfvalALT 19167. (Contributed by Mario Carneiro, 11-Dec-2014.) Remove dependency on ax-rep 5243. (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 2767	. . . . 5 ⊢ (𝑤 = 𝐺 → ℤ = ℤ)
3		fveq2 6888	. . . . . 6 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4		mulgval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2819	. . . . 5 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
6		fveq2 6888	. . . . . . 7 ⊢ (𝑤 = 𝐺 → (0g‘𝑤) = (0g‘𝐺))
7		mulgval.o	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
8	6, 7	eqtr4di 2819	. . . . . 6 ⊢ (𝑤 = 𝐺 → (0g‘𝑤) = 0 )
9		fvex 6901	. . . . . . . . 9 ⊢ (+g‘𝑤) ∈ V
10		1z 12642	. . . . . . . . 9 ⊢ 1 ∈ ℤ
11	9, 10	seqexw 14073	. . . . . . . 8 ⊢ seq1((+g‘𝑤), (ℕ × {𝑥})) ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝑤 = 𝐺 → seq1((+g‘𝑤), (ℕ × {𝑥})) ∈ V)
13		id 23	. . . . . . . . . 10 ⊢ (𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥})) → 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥})))
14		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = (+g‘𝐺))
15		mulgval.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝐺)
16	14, 15	eqtr4di 2819	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = + )
17	16	seqeq2d 14064	. . . . . . . . . 10 ⊢ (𝑤 = 𝐺 → seq1((+g‘𝑤), (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑥})))
18	13, 17	sylan9eqr 2823	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → 𝑠 = seq1( + , (ℕ × {𝑥})))
19	18	fveq1d 6890	. . . . . . . 8 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (𝑠‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
20		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → 𝑤 = 𝐺)
21	20	fveq2d 6892	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (invg‘𝑤) = (invg‘𝐺))
22		mulgval.i	. . . . . . . . . 10 ⊢ 𝐼 = (invg‘𝐺)
23	21, 22	eqtr4di 2819	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (invg‘𝑤) = 𝐼)
24	18	fveq1d 6890	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (𝑠‘-𝑛) = (seq1( + , (ℕ × {𝑥}))‘-𝑛))
25	23, 24	fveq12d 6895	. . . . . . . 8 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → ((invg‘𝑤)‘(𝑠‘-𝑛)) = (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))
26	19, 25	ifeq12d 4514	. . . . . . 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 3892	. . . . . 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 4514	. . . . 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 7498	. . . 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 19165	. . . 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 12618	. . . . 5 ⊢ ℤ ∈ V
32	4	fvexi 6902	. . . . 5 ⊢ 𝐵 ∈ V
33		snex 5415	. . . . . 6 ⊢ { 0 } ∈ V
34	15	fvexi 6902	. . . . . . . . 9 ⊢ + ∈ V
35	34	rnex 7916	. . . . . . . 8 ⊢ ran + ∈ V
36	35, 32	unex 7755	. . . . . . 7 ⊢ (ran + ∪ 𝐵) ∈ V
37	22	fvexi 6902	. . . . . . . . 9 ⊢ 𝐼 ∈ V
38	37	rnex 7916	. . . . . . . 8 ⊢ ran 𝐼 ∈ V
39		p0ex 5360	. . . . . . . 8 ⊢ {∅} ∈ V
40	38, 39	unex 7755	. . . . . . 7 ⊢ (ran 𝐼 ∪ {∅}) ∈ V
41	36, 40	unex 7755	. . . . . 6 ⊢ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})) ∈ V
42	33, 41	unex 7755	. . . . 5 ⊢ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))) ∈ V
43		ssun1 4134	. . . . . . . . 9 ⊢ { 0 } ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
44	7	fvexi 6902	. . . . . . . . . 10 ⊢ 0 ∈ V
45	44	snid 4633	. . . . . . . . 9 ⊢ 0 ∈ { 0 }
46	43, 45	sselii 3937	. . . . . . . 8 ⊢ 0 ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
47	46	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → 0 ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
48		ssun2 4135	. . . . . . . . . . . . . 14 ⊢ 𝐵 ⊆ (ran + ∪ 𝐵)
49		ssun1 4134	. . . . . . . . . . . . . 14 ⊢ (ran + ∪ 𝐵) ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
50	48, 49	sstri 3949	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
51		ssun2 4135	. . . . . . . . . . . . 13 ⊢ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})) ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
52	50, 51	sstri 3949	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
53		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘1))
54	53	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘1))
55		seq1 14070	. . . . . . . . . . . . . . . 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 12262	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ
58		vex 3462	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑥 ∈ V
59	58	fvconst2 7209	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℕ → ((ℕ × {𝑥})‘1) = 𝑥)
60	57, 59	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((ℕ × {𝑥})‘1) = 𝑥
61	60	eleq1i 2857	. . . . . . . . . . . . . . . 16 ⊢ (((ℕ × {𝑥})‘1) ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)
62	61	biimpri 231	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐵 → ((ℕ × {𝑥})‘1) ∈ 𝐵)
63	56, 62	eqeltrid 2870	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐵 → (seq1( + , (ℕ × {𝑥}))‘1) ∈ 𝐵)
64	63	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘1) ∈ 𝐵)
65	54, 64	eqeltrd 2866	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ 𝐵)
66	52, 65	sselid 3938	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
67	66	ad4ant24 767	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ 𝑛 ∈ (ℤ≥‘1)) ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
68		zcn 12614	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
69		npcan1 11657	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
70	68, 69	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℤ → ((𝑛 − 1) + 1) = 𝑛)
71	70	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
72	71	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
73		seqp1 14072	. . . . . . . . . . . . . 14 ⊢ ((𝑛 − 1) ∈ (ℤ≥‘1) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))))
74		ssun1 4134	. . . . . . . . . . . . . . . . 17 ⊢ ran + ⊆ (ran + ∪ 𝐵)
75		ssun2 4135	. . . . . . . . . . . . . . . . 17 ⊢ {∅} ⊆ (ran 𝐼 ∪ {∅})
76		unss12 4144	. . . . . . . . . . . . . . . . 17 ⊢ ((ran + ⊆ (ran + ∪ 𝐵) ∧ {∅} ⊆ (ran 𝐼 ∪ {∅})) → (ran + ∪ {∅}) ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
77	74, 75, 76	mp2an 705	. . . . . . . . . . . . . . . 16 ⊢ (ran + ∪ {∅}) ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
78	77, 51	sstri 3949	. . . . . . . . . . . . . . 15 ⊢ (ran + ∪ {∅}) ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
79		df-ov 7426	. . . . . . . . . . . . . . . 16 ⊢ ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) = ( + ‘⟨(seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)), ((ℕ × {𝑥})‘((𝑛 − 1) + 1))⟩)
80		fvrn0 6916	. . . . . . . . . . . . . . . 16 ⊢ ( + ‘⟨(seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)), ((ℕ × {𝑥})‘((𝑛 − 1) + 1))⟩) ∈ (ran + ∪ {∅})
81	79, 80	eqeltri 2862	. . . . . . . . . . . . . . 15 ⊢ ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) ∈ (ran + ∪ {∅})
82	78, 81	sselii 3937	. . . . . . . . . . . . . 14 ⊢ ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
83	73, 82	eqeltrdi 2874	. . . . . . . . . . . . 13 ⊢ ((𝑛 − 1) ∈ (ℤ≥‘1) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
84	83	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
85	72, 84	eqeltrrd 2867	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
86	85	ad4ant14 765	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ 𝑛 ∈ (ℤ≥‘1)) ∧ (𝑛 − 1) ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
87		uzm1 12914	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘1) → (𝑛 = 1 ∨ (𝑛 − 1) ∈ (ℤ≥‘1)))
88	87	adantl 487	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ 𝑛 ∈ (ℤ≥‘1)) → (𝑛 = 1 ∨ (𝑛 − 1) ∈ (ℤ≥‘1)))
89	67, 86, 88	mpjaodan 973	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ 𝑛 ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
90		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑛 ∈ (ℤ≥‘1)) → ¬ 𝑛 ∈ (ℤ≥‘1))
91		seqfn 14069	. . . . . . . . . . . . . . 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 6646	. . . . . . . . . . . . 13 ⊢ dom seq1( + , (ℕ × {𝑥})) = (ℤ≥‘1)
94	93	eleq2i 2858	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ dom seq1( + , (ℕ × {𝑥})) ↔ 𝑛 ∈ (ℤ≥‘1))
95	90, 94	sylnibr 332	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑛 ∈ (ℤ≥‘1)) → ¬ 𝑛 ∈ dom seq1( + , (ℕ × {𝑥})))
96		ndmfv 6920	. . . . . . . . . . 11 ⊢ (¬ 𝑛 ∈ dom seq1( + , (ℕ × {𝑥})) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = ∅)
97	95, 96	syl 18	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑛 ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = ∅)
98		ssun2 4135	. . . . . . . . . . . . . 14 ⊢ (ran 𝐼 ∪ {∅}) ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
99	75, 98	sstri 3949	. . . . . . . . . . . . 13 ⊢ {∅} ⊆ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))
100	99, 51	sstri 3949	. . . . . . . . . . . 12 ⊢ {∅} ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
101		0ex 5275	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
102	101	snid 4633	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅}
103	100, 102	sselii 3937	. . . . . . . . . . 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 2866	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑛 ∈ (ℤ≥‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
106	89, 105	pm2.61dan 825	. . . . . . . 8 ⊢ ((𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅}))))
107	98, 51	sstri 3949	. . . . . . . . . 10 ⊢ (ran 𝐼 ∪ {∅}) ⊆ ({ 0 } ∪ ((ran + ∪ 𝐵) ∪ (ran 𝐼 ∪ {∅})))
108		fvrn0 6916	. . . . . . . . . 10 ⊢ (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ (ran 𝐼 ∪ {∅})
109	107, 108	sselii 3937	. . . . . . . . 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 4539	. . . . . . 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 4539	. . . . . 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 3208	. . . . 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 8084	. . . 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 6996	. . 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 6880	. . . 4 ⊢ (¬ 𝐺 ∈ V → (.g‘𝐺) = ∅)
117		eqid 2766	. . . . . . 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 6901	. . . . . . . . 9 ⊢ (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ V
119		fvex 6901	. . . . . . . . 9 ⊢ (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ V
120	118, 119	ifex 4543	. . . . . . . 8 ⊢ if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) ∈ V
121	44, 120	ifex 4543	. . . . . . 7 ⊢ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) ∈ V
122	117, 121	fnmpoi 8076	. . . . . 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 6880	. . . . . . . . . 10 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
124	4, 123	eqtrid 2813	. . . . . . . . 9 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
125	124	xpeq2d 5696	. . . . . . . 8 ⊢ (¬ 𝐺 ∈ V → (ℤ × 𝐵) = (ℤ × ∅))
126		xp0 5766	. . . . . . . 8 ⊢ (ℤ × ∅) = ∅
127	125, 126	eqtrdi 2817	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (ℤ × 𝐵) = ∅)
128	127	fneq2d 6636	. . . . . 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 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 ∅)
130		fn0 6673	. . . . 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 221	. . . 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 2804	. . 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 184	. 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 2789	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   ∨ wo 861   = wceq 1570   ∈ wcel 2146  Vcvv 3458  ⦋csb 3856   ∪ cun 3906   ⊆ wss 3908  ∅c0 4289  ifcif 4492  {csn 4594  ⟨cop 4600   class class class wbr 5114   × cxp 5664  dom cdm 5666  ran crn 5667   Fn wfn 6538  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425  ℂcc 11116  0cc0 11118  1c1 11119   + caddc 11121   < clt 11261   − cmin 11459  -cneg 11460  ℕcn 12251  ℤcz 12609  ℤ_≥cuz 12880  seqcseq 14057  Basecbs 17294  +_gcplusg 17335  0_gc0g 17517  inv_gcminusg 19032  ._gcmg 19164

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 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-n0 12523  df-z 12610  df-uz 12881  df-seq 14058  df-mulg 19165

This theorem is used by:  mulgval  19168  mulgfn  19169  mulgpropd  19213