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Definition df-mulg 17588
Description: Define the group multiple function, also known as group exponentiation when viewed multiplicatively. (Contributed by Mario Carneiro, 11-Dec-2014.)
Assertion
Ref Expression
df-mulg .g = (𝑔 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑔) ↦ if(𝑛 = 0, (0g𝑔), seq1((+g𝑔), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑔)‘(𝑠‘-𝑛))))))
Distinct variable group:   𝑔,𝑛,𝑠,𝑥

Detailed syntax breakdown of Definition df-mulg
StepHypRef Expression
1 cmg 17587 . 2 class .g
2 vg . . 3 setvar 𝑔
3 cvv 3231 . . 3 class V
4 vn . . . 4 setvar 𝑛
5 vx . . . 4 setvar 𝑥
6 cz 11415 . . . 4 class
72cv 1522 . . . . 5 class 𝑔
8 cbs 15904 . . . . 5 class Base
97, 8cfv 5926 . . . 4 class (Base‘𝑔)
104cv 1522 . . . . . 6 class 𝑛
11 cc0 9974 . . . . . 6 class 0
1210, 11wceq 1523 . . . . 5 wff 𝑛 = 0
13 c0g 16147 . . . . . 6 class 0g
147, 13cfv 5926 . . . . 5 class (0g𝑔)
15 vs . . . . . 6 setvar 𝑠
16 cplusg 15988 . . . . . . . 8 class +g
177, 16cfv 5926 . . . . . . 7 class (+g𝑔)
18 cn 11058 . . . . . . . 8 class
195cv 1522 . . . . . . . . 9 class 𝑥
2019csn 4210 . . . . . . . 8 class {𝑥}
2118, 20cxp 5141 . . . . . . 7 class (ℕ × {𝑥})
22 c1 9975 . . . . . . 7 class 1
2317, 21, 22cseq 12841 . . . . . 6 class seq1((+g𝑔), (ℕ × {𝑥}))
24 clt 10112 . . . . . . . 8 class <
2511, 10, 24wbr 4685 . . . . . . 7 wff 0 < 𝑛
2615cv 1522 . . . . . . . 8 class 𝑠
2710, 26cfv 5926 . . . . . . 7 class (𝑠𝑛)
2810cneg 10305 . . . . . . . . 9 class -𝑛
2928, 26cfv 5926 . . . . . . . 8 class (𝑠‘-𝑛)
30 cminusg 17470 . . . . . . . . 9 class invg
317, 30cfv 5926 . . . . . . . 8 class (invg𝑔)
3229, 31cfv 5926 . . . . . . 7 class ((invg𝑔)‘(𝑠‘-𝑛))
3325, 27, 32cif 4119 . . . . . 6 class if(0 < 𝑛, (𝑠𝑛), ((invg𝑔)‘(𝑠‘-𝑛)))
3415, 23, 33csb 3566 . . . . 5 class seq1((+g𝑔), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑔)‘(𝑠‘-𝑛)))
3512, 14, 34cif 4119 . . . 4 class if(𝑛 = 0, (0g𝑔), seq1((+g𝑔), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑔)‘(𝑠‘-𝑛))))
364, 5, 6, 9, 35cmpt2 6692 . . 3 class (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑔) ↦ if(𝑛 = 0, (0g𝑔), seq1((+g𝑔), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑔)‘(𝑠‘-𝑛)))))
372, 3, 36cmpt 4762 . 2 class (𝑔 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑔) ↦ if(𝑛 = 0, (0g𝑔), seq1((+g𝑔), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑔)‘(𝑠‘-𝑛))))))
381, 37wceq 1523 1 wff .g = (𝑔 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑔) ↦ if(𝑛 = 0, (0g𝑔), seq1((+g𝑔), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑔)‘(𝑠‘-𝑛))))))
Colors of variables: wff setvar class
This definition is referenced by:  mulgfval  17589
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