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Mirrors > Home > MPE Home > Th. List > mulgnn | Structured version Visualization version GIF version |
Description: Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulgnn.b | ⊢ 𝐵 = (Base‘𝐺) |
mulgnn.p | ⊢ + = (+g‘𝐺) |
mulgnn.t | ⊢ · = (.g‘𝐺) |
mulgnn.s | ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) |
Ref | Expression |
---|---|
mulgnn | ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝑆‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 11734 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
2 | mulgnn.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | mulgnn.p | . . . 4 ⊢ + = (+g‘𝐺) | |
4 | eqid 2825 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | eqid 2825 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
6 | mulgnn.t | . . . 4 ⊢ · = (.g‘𝐺) | |
7 | mulgnn.s | . . . 4 ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) | |
8 | 2, 3, 4, 5, 6, 7 | mulgval 17904 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁))))) |
9 | 1, 8 | sylan 575 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁))))) |
10 | nnne0 11393 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
11 | 10 | neneqd 3004 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ¬ 𝑁 = 0) |
12 | 11 | iffalsed 4319 | . . . 4 ⊢ (𝑁 ∈ ℕ → if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁)))) = if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁)))) |
13 | nngt0 11390 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
14 | 13 | iftrued 4316 | . . . 4 ⊢ (𝑁 ∈ ℕ → if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁))) = (𝑆‘𝑁)) |
15 | 12, 14 | eqtrd 2861 | . . 3 ⊢ (𝑁 ∈ ℕ → if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁)))) = (𝑆‘𝑁)) |
16 | 15 | adantr 474 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁)))) = (𝑆‘𝑁)) |
17 | 9, 16 | eqtrd 2861 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (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 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
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-nel 3103 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-riota 6871 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-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-z 11712 df-seq 13103 df-mulg 17902 |
This theorem is referenced by: mulg1 17909 mulgnnp1 17910 mulgnegnn 17912 mulgnnsubcl 17914 mulgnn0z 17927 mulgnndir 17929 submmulg 17944 subgmulg 17966 mulgnn0di 18591 gsumconst 18694 ressmulgnn 30224 |
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