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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 11998 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
2 | mulgnn.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | mulgnn.p | . . . 4 ⊢ + = (+g‘𝐺) | |
4 | eqid 2821 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | eqid 2821 | . . . 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 18222 | . . 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 582 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁))))) |
10 | nnne0 11665 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
11 | 10 | neneqd 3021 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ¬ 𝑁 = 0) |
12 | 11 | iffalsed 4477 | . . . 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 11662 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
14 | 13 | iftrued 4474 | . . . 4 ⊢ (𝑁 ∈ ℕ → if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁))) = (𝑆‘𝑁)) |
15 | 12, 14 | eqtrd 2856 | . . 3 ⊢ (𝑁 ∈ ℕ → if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁)))) = (𝑆‘𝑁)) |
16 | 15 | adantr 483 | . 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 2856 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝑆‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ifcif 4466 {csn 4560 class class class wbr 5058 × cxp 5547 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 < clt 10669 -cneg 10865 ℕcn 11632 ℤcz 11975 seqcseq 13363 Basecbs 16477 +gcplusg 16559 0gc0g 16707 invgcminusg 18098 .gcmg 18218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-seq 13364 df-mulg 18219 |
This theorem is referenced by: mulgnngsum 18227 mulg1 18229 mulgnnp1 18230 mulgnegnn 18232 mulgnnsubcl 18234 mulgnn0z 18248 mulgnndir 18250 submmulg 18265 subgmulg 18287 mulgnn0di 18940 gsumconst 19048 ressmulgnn 30665 |
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