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Mirrors > Home > MPE Home > Th. List > mulgval | Structured version Visualization version GIF version |
Description: Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.) |
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) |
mulgval.s | ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) |
Ref | Expression |
---|---|
mulgval | ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 476 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑛 = 𝑁) | |
2 | 1 | eqeq1d 2827 | . . 3 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑛 = 0 ↔ 𝑁 = 0)) |
3 | 1 | breq2d 4887 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (0 < 𝑛 ↔ 0 < 𝑁)) |
4 | simpr 479 | . . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
5 | 4 | sneqd 4411 | . . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → {𝑥} = {𝑋}) |
6 | 5 | xpeq2d 5376 | . . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (ℕ × {𝑥}) = (ℕ × {𝑋})) |
7 | 6 | seqeq3d 13110 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑋}))) |
8 | mulgval.s | . . . . . 6 ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) | |
9 | 7, 8 | syl6eqr 2879 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = 𝑆) |
10 | 9, 1 | fveq12d 6444 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (𝑆‘𝑁)) |
11 | 1 | negeqd 10602 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → -𝑛 = -𝑁) |
12 | 9, 11 | fveq12d 6444 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘-𝑛) = (𝑆‘-𝑁)) |
13 | 12 | fveq2d 6441 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) = (𝐼‘(𝑆‘-𝑁))) |
14 | 3, 10, 13 | ifbieq12d 4335 | . . 3 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) = if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))) |
15 | 2, 14 | ifbieq2d 4333 | . 2 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))))) |
16 | mulgval.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
17 | mulgval.p | . . 3 ⊢ + = (+g‘𝐺) | |
18 | mulgval.o | . . 3 ⊢ 0 = (0g‘𝐺) | |
19 | mulgval.i | . . 3 ⊢ 𝐼 = (invg‘𝐺) | |
20 | mulgval.t | . . 3 ⊢ · = (.g‘𝐺) | |
21 | 16, 17, 18, 19, 20 | mulgfval 17903 | . 2 ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) |
22 | 18 | fvexi 6451 | . . 3 ⊢ 0 ∈ V |
23 | fvex 6450 | . . . 4 ⊢ (𝑆‘𝑁) ∈ V | |
24 | fvex 6450 | . . . 4 ⊢ (𝐼‘(𝑆‘-𝑁)) ∈ V | |
25 | 23, 24 | ifex 4356 | . . 3 ⊢ if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))) ∈ V |
26 | 22, 25 | ifex 4356 | . 2 ⊢ if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))) ∈ V |
27 | 15, 21, 26 | ovmpt2a 7056 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ifcif 4308 {csn 4399 class class class wbr 4875 × cxp 5344 ‘cfv 6127 (class class class)co 6910 0cc0 10259 1c1 10260 < clt 10398 -cneg 10593 ℕcn 11357 ℤcz 11711 seqcseq 13102 Basecbs 16229 +gcplusg 16312 0gc0g 16460 invgcminusg 17784 .gcmg 17901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-neg 10595 df-z 11712 df-seq 13103 df-mulg 17902 |
This theorem is referenced by: mulg0 17907 mulgnn 17908 mulgnegnn 17912 subgmulg 17966 |
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