lgsval4a - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > lgsval4a		GIF version

Theorem lgsval4a 15417

Description: Same as lgsval4 15415 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 9373	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
3	2	adantl 277	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
4		nnne0 9046	. . . 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 15415	. . 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 1249	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 /_L 𝑁) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))))
9		nngt0 9043	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
10	9	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 0 < 𝑁)
11		0re 8054	. . . . . . 7 ⊢ 0 ∈ ℝ
12		nnre 9025	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
13	12	adantl 277	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
14		ltnsym 8140	. . . . . . 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 933	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ¬ (𝑁 < 0 ∧ 𝐴 < 0))
18	17	iffalsed 3580	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) = 1)
19		nnnn0 9284	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
20	19	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
21	20	nn0ge0d 9333	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 0 ≤ 𝑁)
22	13, 21	absidd 11397	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (abs‘𝑁) = 𝑁)
23	22	fveq2d 5574	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (seq1( · , 𝐹)‘(abs‘𝑁)) = (seq1( · , 𝐹)‘𝑁))
24	18, 23	oveq12d 5952	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))) = (1 · (seq1( · , 𝐹)‘𝑁)))
25		nnuz 9666	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
26		1zzd 9381	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℤ)
27	6	lgsfcl3 15416	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶ℤ)
28	1, 3, 5, 27	syl3anc 1249	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐹:ℕ⟶ℤ)
29	28	ffvelcdmda 5709	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ℤ)
30		zmulcl 9408	. . . . . . 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 10590	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → seq1( · , 𝐹):ℕ⟶ℤ)
33		simpr 110	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
34	32, 33	ffvelcdmd 5710	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (seq1( · , 𝐹)‘𝑁) ∈ ℤ)
35	34	zcnd 9478	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (seq1( · , 𝐹)‘𝑁) ∈ ℂ)
36	35	mulid2d 8073	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (1 · (seq1( · , 𝐹)‘𝑁)) = (seq1( · , 𝐹)‘𝑁))
37	8, 24, 36	3eqtrd 2241	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 1372 ∈ wcel 2175 ≠ wne 2375 ifcif 3570 class class class wbr 4043 ↦ cmpt 4104 ⟶wf 5264 ‘cfv 5268 (class class class)co 5934 ℝcr 7906 0cc0 7907 1c1 7908 · cmul 7912 < clt 8089 -cneg 8226 ℕcn 9018 ℕ₀cn0 9277 ℤcz 9354 seqcseq 10573 ↑cexp 10664 abscabs 11227 ℙcprime 12348 pCnt cpc 12526 /_L clgs 15392

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 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-iinf 4634 ax-cnex 7998 ax-resscn 7999 ax-1cn 8000 ax-1re 8001 ax-icn 8002 ax-addcl 8003 ax-addrcl 8004 ax-mulcl 8005 ax-mulrcl 8006 ax-addcom 8007 ax-mulcom 8008 ax-addass 8009 ax-mulass 8010 ax-distr 8011 ax-i2m1 8012 ax-0lt1 8013 ax-1rid 8014 ax-0id 8015 ax-rnegex 8016 ax-precex 8017 ax-cnre 8018 ax-pre-ltirr 8019 ax-pre-ltwlin 8020 ax-pre-lttrn 8021 ax-pre-apti 8022 ax-pre-ltadd 8023 ax-pre-mulgt0 8024 ax-pre-mulext 8025 ax-arch 8026 ax-caucvg 8027

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-xor 1395 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4338 df-po 4341 df-iso 4342 df-iord 4411 df-on 4413 df-ilim 4414 df-suc 4416 df-iom 4637 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-rn 4684 df-res 4685 df-ima 4686 df-iota 5229 df-fun 5270 df-fn 5271 df-f 5272 df-f1 5273 df-fo 5274 df-f1o 5275 df-fv 5276 df-isom 5277 df-riota 5889 df-ov 5937 df-oprab 5938 df-mpo 5939 df-1st 6216 df-2nd 6217 df-recs 6381 df-irdg 6446 df-frec 6467 df-1o 6492 df-2o 6493 df-oadd 6496 df-er 6610 df-en 6818 df-dom 6819 df-fin 6820 df-sup 7068 df-inf 7069 df-pnf 8091 df-mnf 8092 df-xr 8093 df-ltxr 8094 df-le 8095 df-sub 8227 df-neg 8228 df-reap 8630 df-ap 8637 df-div 8728 df-inn 9019 df-2 9077 df-3 9078 df-4 9079 df-5 9080 df-6 9081 df-7 9082 df-8 9083 df-n0 9278 df-z 9355 df-uz 9631 df-q 9723 df-rp 9758 df-fz 10113 df-fzo 10247 df-fl 10394 df-mod 10449 df-seqfrec 10574 df-exp 10665 df-ihash 10902 df-cj 11072 df-re 11073 df-im 11074 df-rsqrt 11228 df-abs 11229 df-clim 11509 df-proddc 11781 df-dvds 12018 df-gcd 12194 df-prm 12349 df-phi 12452 df-pc 12527 df-lgs 15393

This theorem is referenced by: lgsmod 15421

W3C validator