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Mirrors > Home > MPE Home > Th. List > lgsval4a | Structured version Visualization version GIF version |
Description: Same as lgsval4 25874 for positive 𝑁. (Contributed by Mario Carneiro, 4-Feb-2015.) |
Ref | Expression |
---|---|
lgsval4.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1)) |
Ref | Expression |
---|---|
lgsval4a | ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 /L 𝑁) = (seq1( · , 𝐹)‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℤ) | |
2 | nnz 11986 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
3 | 2 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ) |
4 | nnne0 11653 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
5 | 4 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ≠ 0) |
6 | lgsval4.1 | . . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1)) | |
7 | 6 | lgsval4 25874 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /L 𝑁) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))) |
8 | 1, 3, 5, 7 | syl3anc 1367 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 /L 𝑁) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))) |
9 | nngt0 11650 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
10 | 9 | adantl 484 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 0 < 𝑁) |
11 | 0re 10624 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
12 | nnre 11626 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
13 | 12 | adantl 484 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ) |
14 | ltnsym 10719 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → ¬ 𝑁 < 0)) | |
15 | 11, 13, 14 | sylancr 589 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (0 < 𝑁 → ¬ 𝑁 < 0)) |
16 | 10, 15 | mpd 15 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ¬ 𝑁 < 0) |
17 | 16 | intnanrd 492 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ¬ (𝑁 < 0 ∧ 𝐴 < 0)) |
18 | 17 | iffalsed 4459 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) = 1) |
19 | nnnn0 11886 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
20 | 19 | adantl 484 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0) |
21 | 20 | nn0ge0d 11940 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 0 ≤ 𝑁) |
22 | 13, 21 | absidd 14762 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (abs‘𝑁) = 𝑁) |
23 | 22 | fveq2d 6655 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (seq1( · , 𝐹)‘(abs‘𝑁)) = (seq1( · , 𝐹)‘𝑁)) |
24 | 18, 23 | oveq12d 7155 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))) = (1 · (seq1( · , 𝐹)‘𝑁))) |
25 | simpr 487 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ) | |
26 | nnuz 12263 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
27 | 25, 26 | eleqtrdi 2921 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (ℤ≥‘1)) |
28 | 6 | lgsfcl3 25875 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶ℤ) |
29 | 1, 3, 5, 28 | syl3anc 1367 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐹:ℕ⟶ℤ) |
30 | elfznn 12921 | . . . . . 6 ⊢ (𝑥 ∈ (1...𝑁) → 𝑥 ∈ ℕ) | |
31 | ffvelrn 6830 | . . . . . 6 ⊢ ((𝐹:ℕ⟶ℤ ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ℤ) | |
32 | 29, 30, 31 | syl2an 597 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → (𝐹‘𝑥) ∈ ℤ) |
33 | zmulcl 12013 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 · 𝑦) ∈ ℤ) | |
34 | 33 | adantl 484 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑥 · 𝑦) ∈ ℤ) |
35 | 27, 32, 34 | seqcl 13375 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (seq1( · , 𝐹)‘𝑁) ∈ ℤ) |
36 | 35 | zcnd 12070 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (seq1( · , 𝐹)‘𝑁) ∈ ℂ) |
37 | 36 | mulid2d 10640 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (1 · (seq1( · , 𝐹)‘𝑁)) = (seq1( · , 𝐹)‘𝑁)) |
38 | 8, 24, 37 | 3eqtrd 2859 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 /L 𝑁) = (seq1( · , 𝐹)‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3011 ifcif 4448 class class class wbr 5047 ↦ cmpt 5127 ⟶wf 6332 ‘cfv 6336 (class class class)co 7137 ℝcr 10517 0cc0 10518 1c1 10519 · cmul 10523 < clt 10656 -cneg 10852 ℕcn 11619 ℕ0cn0 11879 ℤcz 11963 ℤ≥cuz 12225 ...cfz 12877 seqcseq 13354 ↑cexp 13414 abscabs 14573 ℙcprime 15993 pCnt cpc 16151 /L clgs 25851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 ax-pre-sup 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-1o 8083 df-2o 8084 df-oadd 8087 df-er 8270 df-map 8389 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-sup 8887 df-inf 8888 df-dju 9311 df-card 9349 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-div 11279 df-nn 11620 df-2 11682 df-3 11683 df-n0 11880 df-xnn0 11950 df-z 11964 df-uz 12226 df-q 12331 df-rp 12372 df-fz 12878 df-fzo 13019 df-fl 13147 df-mod 13223 df-seq 13355 df-exp 13415 df-hash 13676 df-cj 14438 df-re 14439 df-im 14440 df-sqrt 14574 df-abs 14575 df-dvds 15588 df-gcd 15822 df-prm 15994 df-phi 16081 df-pc 16152 df-lgs 25852 |
This theorem is referenced by: lgsmod 25880 |
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