lgsval4a - Intuitionistic Logic Explorer

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Theorem lgsval4a 15741

Description: Same as lgsval4 15739 for positive 𝑁. (Contributed by Mario Carneiro, 4-Feb-2015.)

**Hypothesis**
Ref	Expression
lgsval4.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))

**Assertion**
Ref	Expression
lgsval4a	⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 /_L 𝑁) = (seq1( · , 𝐹)‘𝑁))

Distinct variable groups: 𝐴,𝑛 𝑛,𝑁 Allowed substitution hint: 𝐹(𝑛)

Proof of Theorem lgsval4a Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℤ)
2		nnz 9488	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
3	2	adantl 277	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
4		nnne0 9161	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
5	4	adantl 277	. . 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 15739	. . 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 1271	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 /_L 𝑁) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))))
9		nngt0 9158	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
10	9	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 0 < 𝑁)
11		0re 8169	. . . . . . 7 ⊢ 0 ∈ ℝ
12		nnre 9140	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
13	12	adantl 277	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
14		ltnsym 8255	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → ¬ 𝑁 < 0))
15	11, 13, 14	sylancr 414	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (0 < 𝑁 → ¬ 𝑁 < 0))
16	10, 15	mpd 13	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ¬ 𝑁 < 0)
17	16	intnanrd 937	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ¬ (𝑁 < 0 ∧ 𝐴 < 0))
18	17	iffalsed 3613	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) = 1)
19		nnnn0 9399	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
20	19	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
21	20	nn0ge0d 9448	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 0 ≤ 𝑁)
22	13, 21	absidd 11718	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (abs‘𝑁) = 𝑁)
23	22	fveq2d 5639	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (seq1( · , 𝐹)‘(abs‘𝑁)) = (seq1( · , 𝐹)‘𝑁))
24	18, 23	oveq12d 6031	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))) = (1 · (seq1( · , 𝐹)‘𝑁)))
25		nnuz 9782	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
26		1zzd 9496	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℤ)
27	6	lgsfcl3 15740	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶ℤ)
28	1, 3, 5, 27	syl3anc 1271	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐹:ℕ⟶ℤ)
29	28	ffvelcdmda 5778	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ℤ)
30		zmulcl 9523	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 · 𝑦) ∈ ℤ)
31	30	adantl 277	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑥 · 𝑦) ∈ ℤ)
32	25, 26, 29, 31	seqf 10716	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → seq1( · , 𝐹):ℕ⟶ℤ)
33		simpr 110	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
34	32, 33	ffvelcdmd 5779	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (seq1( · , 𝐹)‘𝑁) ∈ ℤ)
35	34	zcnd 9593	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (seq1( · , 𝐹)‘𝑁) ∈ ℂ)
36	35	mulid2d 8188	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (1 · (seq1( · , 𝐹)‘𝑁)) = (seq1( · , 𝐹)‘𝑁))
37	8, 24, 36	3eqtrd 2266	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 /_L 𝑁) = (seq1( · , 𝐹)‘𝑁))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 ifcif 3603 class class class wbr 4086 ↦ cmpt 4148 ⟶wf 5320 ‘cfv 5324 (class class class)co 6013 ℝcr 8021 0cc0 8022 1c1 8023 · cmul 8027 < clt 8204 -cneg 8341 ℕcn 9133 ℕ₀cn0 9392 ℤcz 9469 seqcseq 10699 ↑cexp 10790 abscabs 11548 ℙcprime 12669 pCnt cpc 12847 /_L clgs 15716

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139 ax-pre-mulext 8140 ax-arch 8141 ax-caucvg 8142

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-xor 1418 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-2o 6578 df-oadd 6581 df-er 6697 df-en 6905 df-dom 6906 df-fin 6907 df-sup 7174 df-inf 7175 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-reap 8745 df-ap 8752 df-div 8843 df-inn 9134 df-2 9192 df-3 9193 df-4 9194 df-5 9195 df-6 9196 df-7 9197 df-8 9198 df-n0 9393 df-z 9470 df-uz 9746 df-q 9844 df-rp 9879 df-fz 10234 df-fzo 10368 df-fl 10520 df-mod 10575 df-seqfrec 10700 df-exp 10791 df-ihash 11028 df-cj 11393 df-re 11394 df-im 11395 df-rsqrt 11549 df-abs 11550 df-clim 11830 df-proddc 12102 df-dvds 12339 df-gcd 12515 df-prm 12670 df-phi 12773 df-pc 12848 df-lgs 15717

This theorem is referenced by: lgsmod 15745

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