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Theorem mulgval 12862
Description: Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
Hypotheses
Ref Expression
mulgval.b 𝐵 = (Base‘𝐺)
mulgval.p + = (+g𝐺)
mulgval.o 0 = (0g𝐺)
mulgval.i 𝐼 = (invg𝐺)
mulgval.t · = (.g𝐺)
mulgval.s 𝑆 = seq1( + , (ℕ × {𝑋}))
Assertion
Ref Expression
mulgval ((𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))))

Proof of Theorem mulgval
Dummy variables 𝑥 𝑛 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulgval.b . . . 4 𝐵 = (Base‘𝐺)
21basmex 12490 . . 3 (𝑋𝐵𝐺 ∈ V)
32adantl 277 . 2 ((𝑁 ∈ ℤ ∧ 𝑋𝐵) → 𝐺 ∈ V)
4 mulgval.p . . . . 5 + = (+g𝐺)
5 mulgval.o . . . . 5 0 = (0g𝐺)
6 mulgval.i . . . . 5 𝐼 = (invg𝐺)
7 mulgval.t . . . . 5 · = (.g𝐺)
81, 4, 5, 6, 7mulgfvalg 12861 . . . 4 (𝐺 ∈ V → · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
98adantl 277 . . 3 (((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) → · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
10 simpl 109 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑛 = 𝑁)
1110eqeq1d 2186 . . . . 5 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑛 = 0 ↔ 𝑁 = 0))
1210breq2d 4012 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → (0 < 𝑛 ↔ 0 < 𝑁))
13 simpr 110 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑥 = 𝑋)
1413sneqd 3604 . . . . . . . . . 10 ((𝑛 = 𝑁𝑥 = 𝑋) → {𝑥} = {𝑋})
1514xpeq2d 4646 . . . . . . . . 9 ((𝑛 = 𝑁𝑥 = 𝑋) → (ℕ × {𝑥}) = (ℕ × {𝑋}))
1615seqeq3d 10426 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑋})))
17 mulgval.s . . . . . . . 8 𝑆 = seq1( + , (ℕ × {𝑋}))
1816, 17eqtr4di 2228 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = 𝑆)
1918, 10fveq12d 5517 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (𝑆𝑁))
2010negeqd 8129 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → -𝑛 = -𝑁)
2118, 20fveq12d 5517 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘-𝑛) = (𝑆‘-𝑁))
2221fveq2d 5514 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) = (𝐼‘(𝑆‘-𝑁)))
2312, 19, 22ifbieq12d 3560 . . . . 5 ((𝑛 = 𝑁𝑥 = 𝑋) → if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) = if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁))))
2411, 23ifbieq2d 3558 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))))
2524adantl 277 . . 3 ((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ (𝑛 = 𝑁𝑥 = 𝑋)) → if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))))
26 simpll 527 . . 3 (((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) → 𝑁 ∈ ℤ)
27 simplr 528 . . 3 (((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) → 𝑋𝐵)
28 fn0g 12673 . . . . . . 7 0g Fn V
29 funfvex 5527 . . . . . . . 8 ((Fun 0g𝐺 ∈ dom 0g) → (0g𝐺) ∈ V)
3029funfni 5311 . . . . . . 7 ((0g Fn V ∧ 𝐺 ∈ V) → (0g𝐺) ∈ V)
3128, 30mpan 424 . . . . . 6 (𝐺 ∈ V → (0g𝐺) ∈ V)
325, 31eqeltrid 2264 . . . . 5 (𝐺 ∈ V → 0 ∈ V)
3332ad2antlr 489 . . . 4 ((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ 𝑁 = 0) → 0 ∈ V)
34 nnuz 9539 . . . . . . . . 9 ℕ = (ℤ‘1)
35 1zzd 9256 . . . . . . . . 9 ((𝑋𝐵𝐺 ∈ V) → 1 ∈ ℤ)
36 fvconst2g 5725 . . . . . . . . . . . 12 ((𝑋𝐵𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) = 𝑋)
37 simpl 109 . . . . . . . . . . . 12 ((𝑋𝐵𝑢 ∈ ℕ) → 𝑋𝐵)
3836, 37eqeltrd 2254 . . . . . . . . . . 11 ((𝑋𝐵𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) ∈ 𝐵)
3938elexd 2750 . . . . . . . . . 10 ((𝑋𝐵𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) ∈ V)
4039adantlr 477 . . . . . . . . 9 (((𝑋𝐵𝐺 ∈ V) ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) ∈ V)
41 simprl 529 . . . . . . . . . 10 (((𝑋𝐵𝐺 ∈ V) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑢 ∈ V)
42 plusgslid 12538 . . . . . . . . . . . . 13 (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
4342slotex 12459 . . . . . . . . . . . 12 (𝐺 ∈ V → (+g𝐺) ∈ V)
444, 43eqeltrid 2264 . . . . . . . . . . 11 (𝐺 ∈ V → + ∈ V)
4544ad2antlr 489 . . . . . . . . . 10 (((𝑋𝐵𝐺 ∈ V) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → + ∈ V)
46 simprr 531 . . . . . . . . . 10 (((𝑋𝐵𝐺 ∈ V) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑣 ∈ V)
47 ovexg 5902 . . . . . . . . . 10 ((𝑢 ∈ V ∧ + ∈ V ∧ 𝑣 ∈ V) → (𝑢 + 𝑣) ∈ V)
4841, 45, 46, 47syl3anc 1238 . . . . . . . . 9 (((𝑋𝐵𝐺 ∈ V) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢 + 𝑣) ∈ V)
4934, 35, 40, 48seqf 10434 . . . . . . . 8 ((𝑋𝐵𝐺 ∈ V) → seq1( + , (ℕ × {𝑋})):ℕ⟶V)
5017feq1i 5353 . . . . . . . 8 (𝑆:ℕ⟶V ↔ seq1( + , (ℕ × {𝑋})):ℕ⟶V)
5149, 50sylibr 134 . . . . . . 7 ((𝑋𝐵𝐺 ∈ V) → 𝑆:ℕ⟶V)
5251ad5ant23 522 . . . . . 6 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 𝑆:ℕ⟶V)
53 simp-4l 541 . . . . . . 7 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 𝑁 ∈ ℤ)
54 simpr 110 . . . . . . 7 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 0 < 𝑁)
55 elnnz 9239 . . . . . . 7 (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
5653, 54, 55sylanbrc 417 . . . . . 6 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 𝑁 ∈ ℕ)
5752, 56ffvelcdmd 5647 . . . . 5 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → (𝑆𝑁) ∈ V)
581, 6grpinvfng 12794 . . . . . . . 8 (𝐺 ∈ V → 𝐼 Fn 𝐵)
59 basfn 12489 . . . . . . . . . 10 Base Fn V
60 funfvex 5527 . . . . . . . . . . 11 ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
6160funfni 5311 . . . . . . . . . 10 ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
6259, 61mpan 424 . . . . . . . . 9 (𝐺 ∈ V → (Base‘𝐺) ∈ V)
631, 62eqeltrid 2264 . . . . . . . 8 (𝐺 ∈ V → 𝐵 ∈ V)
64 fnex 5733 . . . . . . . 8 ((𝐼 Fn 𝐵𝐵 ∈ V) → 𝐼 ∈ V)
6558, 63, 64syl2anc 411 . . . . . . 7 (𝐺 ∈ V → 𝐼 ∈ V)
6665ad3antlr 493 . . . . . 6 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝐼 ∈ V)
6751ad5ant23 522 . . . . . . 7 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑆:ℕ⟶V)
68 znegcl 9260 . . . . . . . . 9 (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
6968ad4antr 494 . . . . . . . 8 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℤ)
70 simplr 528 . . . . . . . . . 10 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 𝑁 = 0)
71 simpr 110 . . . . . . . . . 10 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 < 𝑁)
72 ztri3or0 9271 . . . . . . . . . . 11 (𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
7372ad4antr 494 . . . . . . . . . 10 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
7470, 71, 73ecase23d 1350 . . . . . . . . 9 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 < 0)
75 zre 9233 . . . . . . . . . . 11 (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
7675ad4antr 494 . . . . . . . . . 10 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ∈ ℝ)
7776lt0neg1d 8449 . . . . . . . . 9 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ↔ 0 < -𝑁))
7874, 77mpbid 147 . . . . . . . 8 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 0 < -𝑁)
79 elnnz 9239 . . . . . . . 8 (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
8069, 78, 79sylanbrc 417 . . . . . . 7 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℕ)
8167, 80ffvelcdmd 5647 . . . . . 6 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑆‘-𝑁) ∈ V)
82 fvexg 5529 . . . . . 6 ((𝐼 ∈ V ∧ (𝑆‘-𝑁) ∈ V) → (𝐼‘(𝑆‘-𝑁)) ∈ V)
8366, 81, 82syl2anc 411 . . . . 5 (((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝐼‘(𝑆‘-𝑁)) ∈ V)
84 0zd 9241 . . . . . 6 ((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → 0 ∈ ℤ)
85 simplll 533 . . . . . 6 ((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → 𝑁 ∈ ℤ)
86 zdclt 9306 . . . . . 6 ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 0 < 𝑁)
8784, 85, 86syl2anc 411 . . . . 5 ((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → DECID 0 < 𝑁)
8857, 83, 87ifcldadc 3563 . . . 4 ((((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁))) ∈ V)
89 0zd 9241 . . . . 5 (((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) → 0 ∈ ℤ)
90 zdceq 9304 . . . . 5 ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
9126, 89, 90syl2anc 411 . . . 4 (((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) → DECID 𝑁 = 0)
9233, 88, 91ifcldadc 3563 . . 3 (((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) → if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))) ∈ V)
939, 25, 26, 27, 92ovmpod 5995 . 2 (((𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ 𝐺 ∈ V) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))))
943, 93mpdan 421 1 ((𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  DECID wdc 834  w3o 977   = wceq 1353  wcel 2148  Vcvv 2737  ifcif 3534  {csn 3591   class class class wbr 4000   × cxp 4620   Fn wfn 5206  wf 5207  cfv 5211  (class class class)co 5868  cmpo 5870  cr 7788  0cc0 7789  1c1 7790   < clt 7969  -cneg 8106  cn 8895  cz 9229  seqcseq 10418  Basecbs 12432  +gcplusg 12505  0gc0g 12640  invgcminusg 12755  .gcmg 12859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-inn 8896  df-2 8954  df-n0 9153  df-z 9230  df-uz 9505  df-seqfrec 10419  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-minusg 12758  df-mulg 12860
This theorem is referenced by:  mulg0  12864  mulgnn  12865  mulgnegnn  12869
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