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Theorem mulgfval 18712
Description: Group multiple (exponentiation) operation. For a shorter proof using ax-rep 5208, see mulgfvalALT 18713. (Contributed by Mario Carneiro, 11-Dec-2014.) Remove dependency on ax-rep 5208. (Revised by Rohan Ridenour, 17-Aug-2023.)
Hypotheses
Ref Expression
mulgval.b 𝐵 = (Base‘𝐺)
mulgval.p + = (+g𝐺)
mulgval.o 0 = (0g𝐺)
mulgval.i 𝐼 = (invg𝐺)
mulgval.t · = (.g𝐺)
Assertion
Ref Expression
mulgfval · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ 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.
StepHypRef Expression
1 mulgval.t . 2 · = (.g𝐺)
2 eqidd 2739 . . . . 5 (𝑤 = 𝐺 → ℤ = ℤ)
3 fveq2 6766 . . . . . 6 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4 mulgval.b . . . . . 6 𝐵 = (Base‘𝐺)
53, 4eqtr4di 2796 . . . . 5 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
6 fveq2 6766 . . . . . . 7 (𝑤 = 𝐺 → (0g𝑤) = (0g𝐺))
7 mulgval.o . . . . . . 7 0 = (0g𝐺)
86, 7eqtr4di 2796 . . . . . 6 (𝑤 = 𝐺 → (0g𝑤) = 0 )
9 fvex 6779 . . . . . . . . 9 (+g𝑤) ∈ V
10 1z 12360 . . . . . . . . 9 1 ∈ ℤ
119, 10seqexw 13747 . . . . . . . 8 seq1((+g𝑤), (ℕ × {𝑥})) ∈ V
1211a1i 11 . . . . . . 7 (𝑤 = 𝐺 → seq1((+g𝑤), (ℕ × {𝑥})) ∈ V)
13 id 22 . . . . . . . . . 10 (𝑠 = seq1((+g𝑤), (ℕ × {𝑥})) → 𝑠 = seq1((+g𝑤), (ℕ × {𝑥})))
14 fveq2 6766 . . . . . . . . . . . 12 (𝑤 = 𝐺 → (+g𝑤) = (+g𝐺))
15 mulgval.p . . . . . . . . . . . 12 + = (+g𝐺)
1614, 15eqtr4di 2796 . . . . . . . . . . 11 (𝑤 = 𝐺 → (+g𝑤) = + )
1716seqeq2d 13738 . . . . . . . . . 10 (𝑤 = 𝐺 → seq1((+g𝑤), (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑥})))
1813, 17sylan9eqr 2800 . . . . . . . . 9 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → 𝑠 = seq1( + , (ℕ × {𝑥})))
1918fveq1d 6768 . . . . . . . 8 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (𝑠𝑛) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
20 simpl 483 . . . . . . . . . . 11 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → 𝑤 = 𝐺)
2120fveq2d 6770 . . . . . . . . . 10 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (invg𝑤) = (invg𝐺))
22 mulgval.i . . . . . . . . . 10 𝐼 = (invg𝐺)
2321, 22eqtr4di 2796 . . . . . . . . 9 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (invg𝑤) = 𝐼)
2418fveq1d 6768 . . . . . . . . 9 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (𝑠‘-𝑛) = (seq1( + , (ℕ × {𝑥}))‘-𝑛))
2523, 24fveq12d 6773 . . . . . . . 8 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → ((invg𝑤)‘(𝑠‘-𝑛)) = (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))
2619, 25ifeq12d 4480 . . . . . . 7 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))) = if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))
2712, 26csbied 3869 . . . . . 6 (𝑤 = 𝐺seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))) = if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))
288, 27ifeq12d 4480 . . . . 5 (𝑤 = 𝐺 → if(𝑛 = 0, (0g𝑤), seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛)))) = if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
292, 5, 28mpoeq123dv 7340 . . . 4 (𝑤 = 𝐺 → (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑤) ↦ if(𝑛 = 0, (0g𝑤), seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))))) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
30 df-mulg 18711 . . . 4 .g = (𝑤 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑤) ↦ if(𝑛 = 0, (0g𝑤), seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))))))
31 zex 12338 . . . . 5 ℤ ∈ V
324fvexi 6780 . . . . 5 𝐵 ∈ V
33 snex 5352 . . . . . 6 { 0 } ∈ V
3415fvexi 6780 . . . . . . . . 9 + ∈ V
3534rnex 7749 . . . . . . . 8 ran + ∈ V
3635, 32unex 7586 . . . . . . 7 (ran +𝐵) ∈ V
3722fvexi 6780 . . . . . . . . 9 𝐼 ∈ V
3837rnex 7749 . . . . . . . 8 ran 𝐼 ∈ V
39 p0ex 5305 . . . . . . . 8 {∅} ∈ V
4038, 39unex 7586 . . . . . . 7 (ran 𝐼 ∪ {∅}) ∈ V
4136, 40unex 7586 . . . . . 6 ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})) ∈ V
4233, 41unex 7586 . . . . 5 ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))) ∈ V
43 ssun1 4105 . . . . . . . . 9 { 0 } ⊆ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
447fvexi 6780 . . . . . . . . . 10 0 ∈ V
4544snid 4597 . . . . . . . . 9 0 ∈ { 0 }
4643, 45sselii 3917 . . . . . . . 8 0 ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
4746a1i 11 . . . . . . 7 ((𝑛 ∈ ℤ ∧ 𝑥𝐵) → 0 ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
48 ssun2 4106 . . . . . . . . . . . . . 14 𝐵 ⊆ (ran +𝐵)
49 ssun1 4105 . . . . . . . . . . . . . 14 (ran +𝐵) ⊆ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))
5048, 49sstri 3929 . . . . . . . . . . . . 13 𝐵 ⊆ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))
51 ssun2 4106 . . . . . . . . . . . . 13 ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})) ⊆ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
5250, 51sstri 3929 . . . . . . . . . . . 12 𝐵 ⊆ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
53 fveq2 6766 . . . . . . . . . . . . . 14 (𝑛 = 1 → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘1))
5453adantl 482 . . . . . . . . . . . . 13 ((𝑥𝐵𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘1))
55 seq1 13744 . . . . . . . . . . . . . . . 16 (1 ∈ ℤ → (seq1( + , (ℕ × {𝑥}))‘1) = ((ℕ × {𝑥})‘1))
5610, 55ax-mp 5 . . . . . . . . . . . . . . 15 (seq1( + , (ℕ × {𝑥}))‘1) = ((ℕ × {𝑥})‘1)
57 1nn 11994 . . . . . . . . . . . . . . . . . 18 1 ∈ ℕ
58 vex 3433 . . . . . . . . . . . . . . . . . . 19 𝑥 ∈ V
5958fvconst2 7071 . . . . . . . . . . . . . . . . . 18 (1 ∈ ℕ → ((ℕ × {𝑥})‘1) = 𝑥)
6057, 59ax-mp 5 . . . . . . . . . . . . . . . . 17 ((ℕ × {𝑥})‘1) = 𝑥
6160eleq1i 2829 . . . . . . . . . . . . . . . 16 (((ℕ × {𝑥})‘1) ∈ 𝐵𝑥𝐵)
6261biimpri 227 . . . . . . . . . . . . . . 15 (𝑥𝐵 → ((ℕ × {𝑥})‘1) ∈ 𝐵)
6356, 62eqeltrid 2843 . . . . . . . . . . . . . 14 (𝑥𝐵 → (seq1( + , (ℕ × {𝑥}))‘1) ∈ 𝐵)
6463adantr 481 . . . . . . . . . . . . 13 ((𝑥𝐵𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘1) ∈ 𝐵)
6554, 64eqeltrd 2839 . . . . . . . . . . . 12 ((𝑥𝐵𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ 𝐵)
6652, 65sselid 3918 . . . . . . . . . . 11 ((𝑥𝐵𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
6766ad4ant24 751 . . . . . . . . . 10 ((((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
68 zcn 12334 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
69 npcan1 11410 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
7068, 69syl 17 . . . . . . . . . . . . . 14 (𝑛 ∈ ℤ → ((𝑛 − 1) + 1) = 𝑛)
7170fveq2d 6770 . . . . . . . . . . . . 13 (𝑛 ∈ ℤ → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
7271adantr 481 . . . . . . . . . . . 12 ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ‘1)) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
73 seqp1 13746 . . . . . . . . . . . . . 14 ((𝑛 − 1) ∈ (ℤ‘1) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))))
74 ssun1 4105 . . . . . . . . . . . . . . . . 17 ran + ⊆ (ran +𝐵)
75 ssun2 4106 . . . . . . . . . . . . . . . . 17 {∅} ⊆ (ran 𝐼 ∪ {∅})
76 unss12 4115 . . . . . . . . . . . . . . . . 17 ((ran + ⊆ (ran +𝐵) ∧ {∅} ⊆ (ran 𝐼 ∪ {∅})) → (ran + ∪ {∅}) ⊆ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
7774, 75, 76mp2an 689 . . . . . . . . . . . . . . . 16 (ran + ∪ {∅}) ⊆ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))
7877, 51sstri 3929 . . . . . . . . . . . . . . 15 (ran + ∪ {∅}) ⊆ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
79 df-ov 7270 . . . . . . . . . . . . . . . 16 ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) = ( + ‘⟨(seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)), ((ℕ × {𝑥})‘((𝑛 − 1) + 1))⟩)
80 fvrn0 6794 . . . . . . . . . . . . . . . 16 ( + ‘⟨(seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)), ((ℕ × {𝑥})‘((𝑛 − 1) + 1))⟩) ∈ (ran + ∪ {∅})
8179, 80eqeltri 2835 . . . . . . . . . . . . . . 15 ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) ∈ (ran + ∪ {∅})
8278, 81sselii 3917 . . . . . . . . . . . . . 14 ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
8373, 82eqeltrdi 2847 . . . . . . . . . . . . 13 ((𝑛 − 1) ∈ (ℤ‘1) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
8483adantl 482 . . . . . . . . . . . 12 ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ‘1)) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
8572, 84eqeltrrd 2840 . . . . . . . . . . 11 ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
8685ad4ant14 749 . . . . . . . . . 10 ((((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ 𝑛 ∈ (ℤ‘1)) ∧ (𝑛 − 1) ∈ (ℤ‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
87 uzm1 12626 . . . . . . . . . . 11 (𝑛 ∈ (ℤ‘1) → (𝑛 = 1 ∨ (𝑛 − 1) ∈ (ℤ‘1)))
8887adantl 482 . . . . . . . . . 10 (((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ 𝑛 ∈ (ℤ‘1)) → (𝑛 = 1 ∨ (𝑛 − 1) ∈ (ℤ‘1)))
8967, 86, 88mpjaodan 956 . . . . . . . . 9 (((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ 𝑛 ∈ (ℤ‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
90 simpr 485 . . . . . . . . . . . 12 (((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ ¬ 𝑛 ∈ (ℤ‘1)) → ¬ 𝑛 ∈ (ℤ‘1))
91 seqfn 13743 . . . . . . . . . . . . . . 15 (1 ∈ ℤ → seq1( + , (ℕ × {𝑥})) Fn (ℤ‘1))
9210, 91ax-mp 5 . . . . . . . . . . . . . 14 seq1( + , (ℕ × {𝑥})) Fn (ℤ‘1)
9392fndmi 6529 . . . . . . . . . . . . 13 dom seq1( + , (ℕ × {𝑥})) = (ℤ‘1)
9493eleq2i 2830 . . . . . . . . . . . 12 (𝑛 ∈ dom seq1( + , (ℕ × {𝑥})) ↔ 𝑛 ∈ (ℤ‘1))
9590, 94sylnibr 329 . . . . . . . . . . 11 (((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ ¬ 𝑛 ∈ (ℤ‘1)) → ¬ 𝑛 ∈ dom seq1( + , (ℕ × {𝑥})))
96 ndmfv 6796 . . . . . . . . . . 11 𝑛 ∈ dom seq1( + , (ℕ × {𝑥})) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = ∅)
9795, 96syl 17 . . . . . . . . . 10 (((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ ¬ 𝑛 ∈ (ℤ‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = ∅)
98 ssun2 4106 . . . . . . . . . . . . . 14 (ran 𝐼 ∪ {∅}) ⊆ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))
9975, 98sstri 3929 . . . . . . . . . . . . 13 {∅} ⊆ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))
10099, 51sstri 3929 . . . . . . . . . . . 12 {∅} ⊆ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
101 0ex 5229 . . . . . . . . . . . . 13 ∅ ∈ V
102101snid 4597 . . . . . . . . . . . 12 ∅ ∈ {∅}
103100, 102sselii 3917 . . . . . . . . . . 11 ∅ ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
104103a1i 11 . . . . . . . . . 10 (((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ ¬ 𝑛 ∈ (ℤ‘1)) → ∅ ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
10597, 104eqeltrd 2839 . . . . . . . . 9 (((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ ¬ 𝑛 ∈ (ℤ‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
10689, 105pm2.61dan 810 . . . . . . . 8 ((𝑛 ∈ ℤ ∧ 𝑥𝐵) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
10798, 51sstri 3929 . . . . . . . . . 10 (ran 𝐼 ∪ {∅}) ⊆ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
108 fvrn0 6794 . . . . . . . . . 10 (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ (ran 𝐼 ∪ {∅})
109107, 108sselii 3917 . . . . . . . . 9 (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
110109a1i 11 . . . . . . . 8 ((𝑛 ∈ ℤ ∧ 𝑥𝐵) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
111106, 110ifcld 4505 . . . . . . 7 ((𝑛 ∈ ℤ ∧ 𝑥𝐵) → if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
11247, 111ifcld 4505 . . . . . 6 ((𝑛 ∈ ℤ ∧ 𝑥𝐵) → if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
113112rgen2 3127 . . . . 5 𝑛 ∈ ℤ ∀𝑥𝐵 if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
11431, 32, 42, 113mpoexw 7908 . . . 4 (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) ∈ V
11529, 30, 114fvmpt 6867 . . 3 (𝐺 ∈ V → (.g𝐺) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
116 fvprc 6758 . . . 4 𝐺 ∈ V → (.g𝐺) = ∅)
117 eqid 2738 . . . . . . 7 (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
118 fvex 6779 . . . . . . . . 9 (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ V
119 fvex 6779 . . . . . . . . 9 (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ V
120118, 119ifex 4509 . . . . . . . 8 if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) ∈ V
12144, 120ifex 4509 . . . . . . 7 if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) ∈ V
122117, 121fnmpoi 7899 . . . . . 6 (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵)
123 fvprc 6758 . . . . . . . . . 10 𝐺 ∈ V → (Base‘𝐺) = ∅)
1244, 123eqtrid 2790 . . . . . . . . 9 𝐺 ∈ V → 𝐵 = ∅)
125124xpeq2d 5614 . . . . . . . 8 𝐺 ∈ V → (ℤ × 𝐵) = (ℤ × ∅))
126 xp0 6054 . . . . . . . 8 (ℤ × ∅) = ∅
127125, 126eqtrdi 2794 . . . . . . 7 𝐺 ∈ V → (ℤ × 𝐵) = ∅)
128127fneq2d 6519 . . . . . 6 𝐺 ∈ V → ((𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵) ↔ (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn ∅))
129122, 128mpbii 232 . . . . 5 𝐺 ∈ V → (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn ∅)
130 fn0 6556 . . . . 5 ((𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn ∅ ↔ (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) = ∅)
131129, 130sylib 217 . . . 4 𝐺 ∈ V → (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) = ∅)
132116, 131eqtr4d 2781 . . 3 𝐺 ∈ V → (.g𝐺) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
133115, 132pm2.61i 182 . 2 (.g𝐺) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
1341, 133eqtri 2766 1 · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wo 844   = wceq 1539  wcel 2106  Vcvv 3429  csb 3831  cun 3884  wss 3886  c0 4256  ifcif 4459  {csn 4561  cop 4567   class class class wbr 5073   × cxp 5582  dom cdm 5584  ran crn 5585   Fn wfn 6421  cfv 6426  (class class class)co 7267  cmpo 7269  cc 10879  0cc0 10881  1c1 10882   + caddc 10884   < clt 11019  cmin 11215  -cneg 11216  cn 11983  cz 12329  cuz 12592  seqcseq 13731  Basecbs 16922  +gcplusg 16972  0gc0g 17160  invgcminusg 18588  .gcmg 18710
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 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578  ax-cnex 10937  ax-resscn 10938  ax-1cn 10939  ax-icn 10940  ax-addcl 10941  ax-addrcl 10942  ax-mulcl 10943  ax-mulrcl 10944  ax-mulcom 10945  ax-addass 10946  ax-mulass 10947  ax-distr 10948  ax-i2m1 10949  ax-1ne0 10950  ax-1rid 10951  ax-rnegex 10952  ax-rrecex 10953  ax-cnre 10954  ax-pre-lttri 10955  ax-pre-lttrn 10956  ax-pre-ltadd 10957  ax-pre-mulgt0 10958
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 3071  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3905  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 5074  df-opab 5136  df-mpt 5157  df-tr 5191  df-id 5484  df-eprel 5490  df-po 5498  df-so 5499  df-fr 5539  df-we 5541  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-pred 6195  df-ord 6262  df-on 6263  df-lim 6264  df-suc 6265  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-riota 7224  df-ov 7270  df-oprab 7271  df-mpo 7272  df-om 7703  df-1st 7820  df-2nd 7821  df-frecs 8084  df-wrecs 8115  df-recs 8189  df-rdg 8228  df-er 8485  df-en 8721  df-dom 8722  df-sdom 8723  df-pnf 11021  df-mnf 11022  df-xr 11023  df-ltxr 11024  df-le 11025  df-sub 11217  df-neg 11218  df-nn 11984  df-n0 12244  df-z 12330  df-uz 12593  df-seq 13732  df-mulg 18711
This theorem is referenced by:  mulgval  18714  mulgfn  18715  mulgpropd  18755
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