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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem3 | Structured version Visualization version GIF version |
Description: The given if term is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
etransclem3.n | ⊢ (𝜑 → 𝑃 ∈ ℕ) |
etransclem3.c | ⊢ (𝜑 → 𝐶:(0...𝑀)⟶(0...𝑁)) |
etransclem3.j | ⊢ (𝜑 → 𝐽 ∈ (0...𝑀)) |
etransclem3.4 | ⊢ (𝜑 → 𝐾 ∈ ℤ) |
Ref | Expression |
---|---|
etransclem3 | ⊢ (𝜑 → if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · ((𝐾 − 𝐽)↑(𝑃 − (𝐶‘𝐽))))) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0zd 12469 | . 2 ⊢ ((𝜑 ∧ 𝑃 < (𝐶‘𝐽)) → 0 ∈ ℤ) | |
2 | 0zd 12469 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → 0 ∈ ℤ) | |
3 | etransclem3.n | . . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ) | |
4 | 3 | nnzd 12484 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℤ) |
5 | 4 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → 𝑃 ∈ ℤ) |
6 | etransclem3.c | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶:(0...𝑀)⟶(0...𝑁)) | |
7 | etransclem3.j | . . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀)) | |
8 | 6, 7 | ffvelcdmd 7032 | . . . . . . . . 9 ⊢ (𝜑 → (𝐶‘𝐽) ∈ (0...𝑁)) |
9 | 8 | elfzelzd 13396 | . . . . . . . 8 ⊢ (𝜑 → (𝐶‘𝐽) ∈ ℤ) |
10 | 4, 9 | zsubcld 12570 | . . . . . . 7 ⊢ (𝜑 → (𝑃 − (𝐶‘𝐽)) ∈ ℤ) |
11 | 10 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (𝑃 − (𝐶‘𝐽)) ∈ ℤ) |
12 | 9 | zred 12565 | . . . . . . . . 9 ⊢ (𝜑 → (𝐶‘𝐽) ∈ ℝ) |
13 | 12 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (𝐶‘𝐽) ∈ ℝ) |
14 | 5 | zred 12565 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → 𝑃 ∈ ℝ) |
15 | simpr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → ¬ 𝑃 < (𝐶‘𝐽)) | |
16 | 13, 14, 15 | nltled 11263 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (𝐶‘𝐽) ≤ 𝑃) |
17 | 14, 13 | subge0d 11703 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (0 ≤ (𝑃 − (𝐶‘𝐽)) ↔ (𝐶‘𝐽) ≤ 𝑃)) |
18 | 16, 17 | mpbird 256 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → 0 ≤ (𝑃 − (𝐶‘𝐽))) |
19 | elfzle1 13398 | . . . . . . . . 9 ⊢ ((𝐶‘𝐽) ∈ (0...𝑁) → 0 ≤ (𝐶‘𝐽)) | |
20 | 8, 19 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 0 ≤ (𝐶‘𝐽)) |
21 | 3 | nnred 12126 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
22 | 21, 12 | subge02d 11705 | . . . . . . . 8 ⊢ (𝜑 → (0 ≤ (𝐶‘𝐽) ↔ (𝑃 − (𝐶‘𝐽)) ≤ 𝑃)) |
23 | 20, 22 | mpbid 231 | . . . . . . 7 ⊢ (𝜑 → (𝑃 − (𝐶‘𝐽)) ≤ 𝑃) |
24 | 23 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (𝑃 − (𝐶‘𝐽)) ≤ 𝑃) |
25 | 2, 5, 11, 18, 24 | elfzd 13386 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (𝑃 − (𝐶‘𝐽)) ∈ (0...𝑃)) |
26 | permnn 14180 | . . . . 5 ⊢ ((𝑃 − (𝐶‘𝐽)) ∈ (0...𝑃) → ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) ∈ ℕ) | |
27 | 25, 26 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) ∈ ℕ) |
28 | 27 | nnzd 12484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) ∈ ℤ) |
29 | etransclem3.4 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
30 | 7 | elfzelzd 13396 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℤ) |
31 | 29, 30 | zsubcld 12570 | . . . . 5 ⊢ (𝜑 → (𝐾 − 𝐽) ∈ ℤ) |
32 | 31 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (𝐾 − 𝐽) ∈ ℤ) |
33 | elnn0z 12470 | . . . . 5 ⊢ ((𝑃 − (𝐶‘𝐽)) ∈ ℕ0 ↔ ((𝑃 − (𝐶‘𝐽)) ∈ ℤ ∧ 0 ≤ (𝑃 − (𝐶‘𝐽)))) | |
34 | 11, 18, 33 | sylanbrc 583 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (𝑃 − (𝐶‘𝐽)) ∈ ℕ0) |
35 | zexpcl 13936 | . . . 4 ⊢ (((𝐾 − 𝐽) ∈ ℤ ∧ (𝑃 − (𝐶‘𝐽)) ∈ ℕ0) → ((𝐾 − 𝐽)↑(𝑃 − (𝐶‘𝐽))) ∈ ℤ) | |
36 | 32, 34, 35 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → ((𝐾 − 𝐽)↑(𝑃 − (𝐶‘𝐽))) ∈ ℤ) |
37 | 28, 36 | zmulcld 12571 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · ((𝐾 − 𝐽)↑(𝑃 − (𝐶‘𝐽)))) ∈ ℤ) |
38 | 1, 37 | ifclda 4519 | 1 ⊢ (𝜑 → if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · ((𝐾 − 𝐽)↑(𝑃 − (𝐶‘𝐽))))) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2106 ifcif 4484 class class class wbr 5103 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 ℝcr 11008 0cc0 11009 · cmul 11014 < clt 11147 ≤ cle 11148 − cmin 11343 / cdiv 11770 ℕcn 12111 ℕ0cn0 12371 ℤcz 12457 ...cfz 13378 ↑cexp 13921 !cfa 14127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-rp 12870 df-fz 13379 df-seq 13861 df-exp 13922 df-fac 14128 df-bc 14157 |
This theorem is referenced by: etransclem24 44400 etransclem25 44401 etransclem26 44402 etransclem35 44411 etransclem37 44413 |
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