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Theorem mulgfval 18217
Description: Group multiple (exponentiation) operation. For a shorter proof using ax-rep 5173, see mulgfvalALT 18218. (Contributed by Mario Carneiro, 11-Dec-2014.) Remove dependency on ax-rep 5173. (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 2825 . . . . 5 (𝑤 = 𝐺 → ℤ = ℤ)
3 fveq2 6653 . . . . . 6 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4 mulgval.b . . . . . 6 𝐵 = (Base‘𝐺)
53, 4syl6eqr 2877 . . . . 5 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
6 fveq2 6653 . . . . . . 7 (𝑤 = 𝐺 → (0g𝑤) = (0g𝐺))
7 mulgval.o . . . . . . 7 0 = (0g𝐺)
86, 7syl6eqr 2877 . . . . . 6 (𝑤 = 𝐺 → (0g𝑤) = 0 )
9 fvex 6666 . . . . . . . . 9 (+g𝑤) ∈ V
10 1z 12000 . . . . . . . . 9 1 ∈ ℤ
119, 10seqexw 13380 . . . . . . . 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 6653 . . . . . . . . . . . 12 (𝑤 = 𝐺 → (+g𝑤) = (+g𝐺))
15 mulgval.p . . . . . . . . . . . 12 + = (+g𝐺)
1614, 15syl6eqr 2877 . . . . . . . . . . 11 (𝑤 = 𝐺 → (+g𝑤) = + )
1716seqeq2d 13371 . . . . . . . . . 10 (𝑤 = 𝐺 → seq1((+g𝑤), (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑥})))
1813, 17sylan9eqr 2881 . . . . . . . . 9 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → 𝑠 = seq1( + , (ℕ × {𝑥})))
1918fveq1d 6655 . . . . . . . 8 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (𝑠𝑛) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
20 simpl 486 . . . . . . . . . . 11 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → 𝑤 = 𝐺)
2120fveq2d 6657 . . . . . . . . . 10 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (invg𝑤) = (invg𝐺))
22 mulgval.i . . . . . . . . . 10 𝐼 = (invg𝐺)
2321, 22syl6eqr 2877 . . . . . . . . 9 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (invg𝑤) = 𝐼)
2418fveq1d 6655 . . . . . . . . 9 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (𝑠‘-𝑛) = (seq1( + , (ℕ × {𝑥}))‘-𝑛))
2523, 24fveq12d 6660 . . . . . . . 8 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → ((invg𝑤)‘(𝑠‘-𝑛)) = (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))
2619, 25ifeq12d 4468 . . . . . . 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 3901 . . . . . 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 4468 . . . . 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 7213 . . . 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 18216 . . . 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 11978 . . . . 5 ℤ ∈ V
324fvexi 6667 . . . . 5 𝐵 ∈ V
33 snex 5315 . . . . . 6 { 0 } ∈ V
3415fvexi 6667 . . . . . . . . 9 + ∈ V
3534rnex 7602 . . . . . . . 8 ran + ∈ V
3635, 32unex 7454 . . . . . . 7 (ran +𝐵) ∈ V
3722fvexi 6667 . . . . . . . . 9 𝐼 ∈ V
3837rnex 7602 . . . . . . . 8 ran 𝐼 ∈ V
39 p0ex 5268 . . . . . . . 8 {∅} ∈ V
4038, 39unex 7454 . . . . . . 7 (ran 𝐼 ∪ {∅}) ∈ V
4136, 40unex 7454 . . . . . 6 ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})) ∈ V
4233, 41unex 7454 . . . . 5 ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))) ∈ V
43 ssun1 4132 . . . . . . . . 9 { 0 } ⊆ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
447fvexi 6667 . . . . . . . . . 10 0 ∈ V
4544snid 4584 . . . . . . . . 9 0 ∈ { 0 }
4643, 45sselii 3948 . . . . . . . 8 0 ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
4746a1i 11 . . . . . . 7 ((𝑛 ∈ ℤ ∧ 𝑥𝐵) → 0 ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
48 ssun2 4133 . . . . . . . . . . . . . 14 𝐵 ⊆ (ran +𝐵)
49 ssun1 4132 . . . . . . . . . . . . . 14 (ran +𝐵) ⊆ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))
5048, 49sstri 3960 . . . . . . . . . . . . 13 𝐵 ⊆ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))
51 ssun2 4133 . . . . . . . . . . . . 13 ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})) ⊆ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
5250, 51sstri 3960 . . . . . . . . . . . 12 𝐵 ⊆ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
53 fveq2 6653 . . . . . . . . . . . . . 14 (𝑛 = 1 → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘1))
5453adantl 485 . . . . . . . . . . . . 13 ((𝑥𝐵𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘1))
55 seq1 13377 . . . . . . . . . . . . . . . 16 (1 ∈ ℤ → (seq1( + , (ℕ × {𝑥}))‘1) = ((ℕ × {𝑥})‘1))
5610, 55ax-mp 5 . . . . . . . . . . . . . . 15 (seq1( + , (ℕ × {𝑥}))‘1) = ((ℕ × {𝑥})‘1)
57 1nn 11636 . . . . . . . . . . . . . . . . . 18 1 ∈ ℕ
58 vex 3482 . . . . . . . . . . . . . . . . . . 19 𝑥 ∈ V
5958fvconst2 6949 . . . . . . . . . . . . . . . . . 18 (1 ∈ ℕ → ((ℕ × {𝑥})‘1) = 𝑥)
6057, 59ax-mp 5 . . . . . . . . . . . . . . . . 17 ((ℕ × {𝑥})‘1) = 𝑥
6160eleq1i 2906 . . . . . . . . . . . . . . . 16 (((ℕ × {𝑥})‘1) ∈ 𝐵𝑥𝐵)
6261biimpri 231 . . . . . . . . . . . . . . 15 (𝑥𝐵 → ((ℕ × {𝑥})‘1) ∈ 𝐵)
6356, 62eqeltrid 2920 . . . . . . . . . . . . . 14 (𝑥𝐵 → (seq1( + , (ℕ × {𝑥}))‘1) ∈ 𝐵)
6463adantr 484 . . . . . . . . . . . . 13 ((𝑥𝐵𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘1) ∈ 𝐵)
6554, 64eqeltrd 2916 . . . . . . . . . . . 12 ((𝑥𝐵𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ 𝐵)
6652, 65sseldi 3949 . . . . . . . . . . 11 ((𝑥𝐵𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
6766ad4ant24 753 . . . . . . . . . 10 ((((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 = 1) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
68 zcn 11974 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
69 npcan1 11052 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
7068, 69syl 17 . . . . . . . . . . . . . 14 (𝑛 ∈ ℤ → ((𝑛 − 1) + 1) = 𝑛)
7170fveq2d 6657 . . . . . . . . . . . . 13 (𝑛 ∈ ℤ → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
7271adantr 484 . . . . . . . . . . . 12 ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ‘1)) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
73 seqp1 13379 . . . . . . . . . . . . . 14 ((𝑛 − 1) ∈ (ℤ‘1) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) = ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))))
74 ssun1 4132 . . . . . . . . . . . . . . . . 17 ran + ⊆ (ran +𝐵)
75 ssun2 4133 . . . . . . . . . . . . . . . . 17 {∅} ⊆ (ran 𝐼 ∪ {∅})
76 unss12 4142 . . . . . . . . . . . . . . . . 17 ((ran + ⊆ (ran +𝐵) ∧ {∅} ⊆ (ran 𝐼 ∪ {∅})) → (ran + ∪ {∅}) ⊆ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
7774, 75, 76mp2an 691 . . . . . . . . . . . . . . . 16 (ran + ∪ {∅}) ⊆ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))
7877, 51sstri 3960 . . . . . . . . . . . . . . 15 (ran + ∪ {∅}) ⊆ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
79 df-ov 7143 . . . . . . . . . . . . . . . 16 ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) = ( + ‘⟨(seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)), ((ℕ × {𝑥})‘((𝑛 − 1) + 1))⟩)
80 fvrn0 6681 . . . . . . . . . . . . . . . 16 ( + ‘⟨(seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)), ((ℕ × {𝑥})‘((𝑛 − 1) + 1))⟩) ∈ (ran + ∪ {∅})
8179, 80eqeltri 2912 . . . . . . . . . . . . . . 15 ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) ∈ (ran + ∪ {∅})
8278, 81sselii 3948 . . . . . . . . . . . . . 14 ((seq1( + , (ℕ × {𝑥}))‘(𝑛 − 1)) + ((ℕ × {𝑥})‘((𝑛 − 1) + 1))) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
8373, 82eqeltrdi 2924 . . . . . . . . . . . . 13 ((𝑛 − 1) ∈ (ℤ‘1) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
8483adantl 485 . . . . . . . . . . . 12 ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ‘1)) → (seq1( + , (ℕ × {𝑥}))‘((𝑛 − 1) + 1)) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
8572, 84eqeltrrd 2917 . . . . . . . . . . 11 ((𝑛 ∈ ℤ ∧ (𝑛 − 1) ∈ (ℤ‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
8685ad4ant14 751 . . . . . . . . . 10 ((((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ 𝑛 ∈ (ℤ‘1)) ∧ (𝑛 − 1) ∈ (ℤ‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
87 uzm1 12264 . . . . . . . . . . 11 (𝑛 ∈ (ℤ‘1) → (𝑛 = 1 ∨ (𝑛 − 1) ∈ (ℤ‘1)))
8887adantl 485 . . . . . . . . . 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 488 . . . . . . . . . . . 12 (((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ ¬ 𝑛 ∈ (ℤ‘1)) → ¬ 𝑛 ∈ (ℤ‘1))
91 seqfn 13376 . . . . . . . . . . . . . . 15 (1 ∈ ℤ → seq1( + , (ℕ × {𝑥})) Fn (ℤ‘1))
9210, 91ax-mp 5 . . . . . . . . . . . . . 14 seq1( + , (ℕ × {𝑥})) Fn (ℤ‘1)
93 fndm 6438 . . . . . . . . . . . . . 14 (seq1( + , (ℕ × {𝑥})) Fn (ℤ‘1) → dom seq1( + , (ℕ × {𝑥})) = (ℤ‘1))
9492, 93ax-mp 5 . . . . . . . . . . . . 13 dom seq1( + , (ℕ × {𝑥})) = (ℤ‘1)
9594eleq2i 2907 . . . . . . . . . . . 12 (𝑛 ∈ dom seq1( + , (ℕ × {𝑥})) ↔ 𝑛 ∈ (ℤ‘1))
9690, 95sylnibr 332 . . . . . . . . . . 11 (((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ ¬ 𝑛 ∈ (ℤ‘1)) → ¬ 𝑛 ∈ dom seq1( + , (ℕ × {𝑥})))
97 ndmfv 6683 . . . . . . . . . . 11 𝑛 ∈ dom seq1( + , (ℕ × {𝑥})) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = ∅)
9896, 97syl 17 . . . . . . . . . 10 (((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ ¬ 𝑛 ∈ (ℤ‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = ∅)
99 ssun2 4133 . . . . . . . . . . . . . 14 (ran 𝐼 ∪ {∅}) ⊆ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))
10075, 99sstri 3960 . . . . . . . . . . . . 13 {∅} ⊆ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))
101100, 51sstri 3960 . . . . . . . . . . . 12 {∅} ⊆ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
102 0ex 5194 . . . . . . . . . . . . 13 ∅ ∈ V
103102snid 4584 . . . . . . . . . . . 12 ∅ ∈ {∅}
104101, 103sselii 3948 . . . . . . . . . . 11 ∅ ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
105104a1i 11 . . . . . . . . . 10 (((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ ¬ 𝑛 ∈ (ℤ‘1)) → ∅ ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
10698, 105eqeltrd 2916 . . . . . . . . 9 (((𝑛 ∈ ℤ ∧ 𝑥𝐵) ∧ ¬ 𝑛 ∈ (ℤ‘1)) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
10789, 106pm2.61dan 812 . . . . . . . 8 ((𝑛 ∈ ℤ ∧ 𝑥𝐵) → (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
10899, 51sstri 3960 . . . . . . . . . 10 (ran 𝐼 ∪ {∅}) ⊆ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
109 fvrn0 6681 . . . . . . . . . 10 (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ (ran 𝐼 ∪ {∅})
110108, 109sselii 3948 . . . . . . . . 9 (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅})))
111110a1i 11 . . . . . . . 8 ((𝑛 ∈ ℤ ∧ 𝑥𝐵) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
112107, 111ifcld 4493 . . . . . . 7 ((𝑛 ∈ ℤ ∧ 𝑥𝐵) → if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) ∈ ({ 0 } ∪ ((ran +𝐵) ∪ (ran 𝐼 ∪ {∅}))))
11347, 112ifcld 4493 . . . . . 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 ∪ {∅}))))
114113rgen2 3197 . . . . 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 ∪ {∅})))
11531, 32, 42, 114mpoexw 7761 . . . 4 (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) ∈ V
11629, 30, 115fvmpt 6751 . . 3 (𝐺 ∈ V → (.g𝐺) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
117 fvprc 6646 . . . 4 𝐺 ∈ V → (.g𝐺) = ∅)
118 eqid 2824 . . . . . . 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)))))
119 fvex 6666 . . . . . . . . 9 (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ V
120 fvex 6666 . . . . . . . . 9 (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ V
121119, 120ifex 4496 . . . . . . . 8 if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) ∈ V
12244, 121ifex 4496 . . . . . . 7 if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) ∈ V
123118, 122fnmpoi 7753 . . . . . 6 (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵)
124 fvprc 6646 . . . . . . . . . 10 𝐺 ∈ V → (Base‘𝐺) = ∅)
1254, 124syl5eq 2871 . . . . . . . . 9 𝐺 ∈ V → 𝐵 = ∅)
126125xpeq2d 5568 . . . . . . . 8 𝐺 ∈ V → (ℤ × 𝐵) = (ℤ × ∅))
127 xp0 5998 . . . . . . . 8 (ℤ × ∅) = ∅
128126, 127syl6eq 2875 . . . . . . 7 𝐺 ∈ V → (ℤ × 𝐵) = ∅)
129128fneq2d 6430 . . . . . 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 ∅))
130123, 129mpbii 236 . . . . 5 𝐺 ∈ V → (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn ∅)
131 fn0 6462 . . . . 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))))) = ∅)
132130, 131sylib 221 . . . 4 𝐺 ∈ V → (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) = ∅)
133117, 132eqtr4d 2862 . . 3 𝐺 ∈ V → (.g𝐺) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
134116, 133pm2.61i 185 . 2 (.g𝐺) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
1351, 134eqtri 2847 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 399  wo 844   = wceq 1538  wcel 2115  Vcvv 3479  csb 3865  cun 3916  wss 3918  c0 4274  ifcif 4448  {csn 4548  cop 4554   class class class wbr 5049   × cxp 5536  dom cdm 5538  ran crn 5539   Fn wfn 6333  cfv 6338  (class class class)co 7140  cmpo 7142  cc 10522  0cc0 10524  1c1 10525   + caddc 10527   < clt 10662  cmin 10857  -cneg 10858  cn 11625  cz 11969  cuz 12231  seqcseq 13364  Basecbs 16474  +gcplusg 16556  0gc0g 16704  invgcminusg 18095  .gcmg 18215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446  ax-cnex 10580  ax-resscn 10581  ax-1cn 10582  ax-icn 10583  ax-addcl 10584  ax-addrcl 10585  ax-mulcl 10586  ax-mulrcl 10587  ax-mulcom 10588  ax-addass 10589  ax-mulass 10590  ax-distr 10591  ax-i2m1 10592  ax-1ne0 10593  ax-1rid 10594  ax-rnegex 10595  ax-rrecex 10596  ax-cnre 10597  ax-pre-lttri 10598  ax-pre-lttrn 10599  ax-pre-ltadd 10600  ax-pre-mulgt0 10601
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7674  df-2nd 7675  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-pnf 10664  df-mnf 10665  df-xr 10666  df-ltxr 10667  df-le 10668  df-sub 10859  df-neg 10860  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-seq 13365  df-mulg 18216
This theorem is referenced by:  mulgval  18219  mulgfn  18220  mulgpropd  18260
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