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Mirrors > Home > MPE Home > Th. List > elqaalem1 | Structured version Visualization version GIF version |
Description: Lemma for elqaa 25073. The function 𝑁 represents the denominators of the rational coefficients 𝐵. By multiplying them all together to make 𝑅, we get a number big enough to clear all the denominators and make 𝑅 · 𝐹 an integer polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.) |
Ref | Expression |
---|---|
elqaa.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
elqaa.2 | ⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝})) |
elqaa.3 | ⊢ (𝜑 → (𝐹‘𝐴) = 0) |
elqaa.4 | ⊢ 𝐵 = (coeff‘𝐹) |
elqaa.5 | ⊢ 𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ}, ℝ, < )) |
elqaa.6 | ⊢ 𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹)) |
Ref | Expression |
---|---|
elqaalem1 | ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → ((𝑁‘𝐾) ∈ ℕ ∧ ((𝐵‘𝐾) · (𝑁‘𝐾)) ∈ ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6677 | . . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝐵‘𝑘) = (𝐵‘𝐾)) | |
2 | 1 | oveq1d 7188 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝐵‘𝑘) · 𝑛) = ((𝐵‘𝐾) · 𝑛)) |
3 | 2 | eleq1d 2818 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (((𝐵‘𝑘) · 𝑛) ∈ ℤ ↔ ((𝐵‘𝐾) · 𝑛) ∈ ℤ)) |
4 | 3 | rabbidv 3382 | . . . . . 6 ⊢ (𝑘 = 𝐾 → {𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ} = {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}) |
5 | 4 | infeq1d 9017 | . . . . 5 ⊢ (𝑘 = 𝐾 → inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}, ℝ, < )) |
6 | elqaa.5 | . . . . 5 ⊢ 𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ}, ℝ, < )) | |
7 | ltso 10802 | . . . . . 6 ⊢ < Or ℝ | |
8 | 7 | infex 9033 | . . . . 5 ⊢ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}, ℝ, < ) ∈ V |
9 | 5, 6, 8 | fvmpt 6778 | . . . 4 ⊢ (𝐾 ∈ ℕ0 → (𝑁‘𝐾) = inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}, ℝ, < )) |
10 | 9 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → (𝑁‘𝐾) = inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}, ℝ, < )) |
11 | ssrab2 3970 | . . . . 5 ⊢ {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ} ⊆ ℕ | |
12 | nnuz 12366 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
13 | 11, 12 | sseqtri 3914 | . . . 4 ⊢ {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ} ⊆ (ℤ≥‘1) |
14 | elqaa.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝})) | |
15 | 14 | eldifad 3856 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℚ)) |
16 | 0z 12076 | . . . . . . . . 9 ⊢ 0 ∈ ℤ | |
17 | zq 12439 | . . . . . . . . 9 ⊢ (0 ∈ ℤ → 0 ∈ ℚ) | |
18 | 16, 17 | ax-mp 5 | . . . . . . . 8 ⊢ 0 ∈ ℚ |
19 | elqaa.4 | . . . . . . . . 9 ⊢ 𝐵 = (coeff‘𝐹) | |
20 | 19 | coef2 24983 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℚ) ∧ 0 ∈ ℚ) → 𝐵:ℕ0⟶ℚ) |
21 | 15, 18, 20 | sylancl 589 | . . . . . . 7 ⊢ (𝜑 → 𝐵:ℕ0⟶ℚ) |
22 | 21 | ffvelrnda 6864 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → (𝐵‘𝐾) ∈ ℚ) |
23 | qmulz 12436 | . . . . . 6 ⊢ ((𝐵‘𝐾) ∈ ℚ → ∃𝑛 ∈ ℕ ((𝐵‘𝐾) · 𝑛) ∈ ℤ) | |
24 | 22, 23 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → ∃𝑛 ∈ ℕ ((𝐵‘𝐾) · 𝑛) ∈ ℤ) |
25 | rabn0 4275 | . . . . 5 ⊢ ({𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ} ≠ ∅ ↔ ∃𝑛 ∈ ℕ ((𝐵‘𝐾) · 𝑛) ∈ ℤ) | |
26 | 24, 25 | sylibr 237 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ} ≠ ∅) |
27 | infssuzcl 12417 | . . . 4 ⊢ (({𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ} ⊆ (ℤ≥‘1) ∧ {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ} ≠ ∅) → inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}) | |
28 | 13, 26, 27 | sylancr 590 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}) |
29 | 10, 28 | eqeltrd 2834 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → (𝑁‘𝐾) ∈ {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}) |
30 | oveq2 7181 | . . . 4 ⊢ (𝑛 = (𝑁‘𝐾) → ((𝐵‘𝐾) · 𝑛) = ((𝐵‘𝐾) · (𝑁‘𝐾))) | |
31 | 30 | eleq1d 2818 | . . 3 ⊢ (𝑛 = (𝑁‘𝐾) → (((𝐵‘𝐾) · 𝑛) ∈ ℤ ↔ ((𝐵‘𝐾) · (𝑁‘𝐾)) ∈ ℤ)) |
32 | 31 | elrab 3589 | . 2 ⊢ ((𝑁‘𝐾) ∈ {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ} ↔ ((𝑁‘𝐾) ∈ ℕ ∧ ((𝐵‘𝐾) · (𝑁‘𝐾)) ∈ ℤ)) |
33 | 29, 32 | sylib 221 | 1 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → ((𝑁‘𝐾) ∈ ℕ ∧ ((𝐵‘𝐾) · (𝑁‘𝐾)) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ∃wrex 3055 {crab 3058 ∖ cdif 3841 ⊆ wss 3844 ∅c0 4212 {csn 4517 ↦ cmpt 5111 ⟶wf 6336 ‘cfv 6340 (class class class)co 7173 infcinf 8981 ℂcc 10616 ℝcr 10617 0cc0 10618 1c1 10619 · cmul 10623 < clt 10756 ℕcn 11719 ℕ0cn0 11979 ℤcz 12065 ℤ≥cuz 12327 ℚcq 12433 seqcseq 13463 0𝑝c0p 24424 Polycply 24936 coeffccoe 24938 degcdgr 24939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-inf2 9180 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-of 7428 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-map 8442 df-pm 8443 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-sup 8982 df-inf 8983 df-oi 9050 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-n0 11980 df-z 12066 df-uz 12328 df-q 12434 df-rp 12476 df-fz 12985 df-fzo 13128 df-fl 13256 df-seq 13464 df-exp 13525 df-hash 13786 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 df-clim 14938 df-rlim 14939 df-sum 15139 df-0p 24425 df-ply 24940 df-coe 24942 |
This theorem is referenced by: elqaalem2 25071 |
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