lgsval4a - Intuitionistic Logic Explorer

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

Theorem lgsval4a 15179

Description: Same as lgsval4 15177 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 9339	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
3	2	adantl 277	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
4		nnne0 9012	. . . 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 15177	. . 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 9009	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
10	9	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 0 < 𝑁)
11		0re 8021	. . . . . . 7 ⊢ 0 ∈ ℝ
12		nnre 8991	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
13	12	adantl 277	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
14		ltnsym 8107	. . . . . . 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 3568	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) = 1)
19		nnnn0 9250	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
20	19	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
21	20	nn0ge0d 9299	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 0 ≤ 𝑁)
22	13, 21	absidd 11314	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (abs‘𝑁) = 𝑁)
23	22	fveq2d 5559	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (seq1( · , 𝐹)‘(abs‘𝑁)) = (seq1( · , 𝐹)‘𝑁))
24	18, 23	oveq12d 5937	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))) = (1 · (seq1( · , 𝐹)‘𝑁)))
25		nnuz 9631	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
26		1zzd 9347	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℤ)
27	6	lgsfcl3 15178	. . . . . . . 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 5694	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ℤ)
30		zmulcl 9373	. . . . . . 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 10538	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → seq1( · , 𝐹):ℕ⟶ℤ)
33		simpr 110	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
34	32, 33	ffvelcdmd 5695	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (seq1( · , 𝐹)‘𝑁) ∈ ℤ)
35	34	zcnd 9443	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (seq1( · , 𝐹)‘𝑁) ∈ ℂ)
36	35	mulid2d 8040	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (1 · (seq1( · , 𝐹)‘𝑁)) = (seq1( · , 𝐹)‘𝑁))
37	8, 24, 36	3eqtrd 2230	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 1364 ∈ wcel 2164 ≠ wne 2364 ifcif 3558 class class class wbr 4030 ↦ cmpt 4091 ⟶wf 5251 ‘cfv 5255 (class class class)co 5919 ℝcr 7873 0cc0 7874 1c1 7875 · cmul 7879 < clt 8056 -cneg 8193 ℕcn 8984 ℕ₀cn0 9243 ℤcz 9320 seqcseq 10521 ↑cexp 10612 abscabs 11144 ℙcprime 12248 pCnt cpc 12425 /_L clgs 15154

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-frec 6446 df-1o 6471 df-2o 6472 df-oadd 6475 df-er 6589 df-en 6797 df-dom 6798 df-fin 6799 df-sup 7045 df-inf 7046 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-7 9048 df-8 9049 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-fz 10078 df-fzo 10212 df-fl 10342 df-mod 10397 df-seqfrec 10522 df-exp 10613 df-ihash 10850 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-clim 11425 df-proddc 11697 df-dvds 11934 df-gcd 12083 df-prm 12249 df-phi 12352 df-pc 12426 df-lgs 15155

This theorem is referenced by: lgsmod 15183

W3C validator