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Mirrors > Home > MPE Home > Th. List > elqaalem1 | Structured version Visualization version GIF version |
Description: Lemma for elqaa 25482. 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 6774 | . . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝐵‘𝑘) = (𝐵‘𝐾)) | |
2 | 1 | oveq1d 7290 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝐵‘𝑘) · 𝑛) = ((𝐵‘𝐾) · 𝑛)) |
3 | 2 | eleq1d 2823 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (((𝐵‘𝑘) · 𝑛) ∈ ℤ ↔ ((𝐵‘𝐾) · 𝑛) ∈ ℤ)) |
4 | 3 | rabbidv 3414 | . . . . . 6 ⊢ (𝑘 = 𝐾 → {𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ} = {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}) |
5 | 4 | infeq1d 9236 | . . . . 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 11055 | . . . . . 6 ⊢ < Or ℝ | |
8 | 7 | infex 9252 | . . . . 5 ⊢ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}, ℝ, < ) ∈ V |
9 | 5, 6, 8 | fvmpt 6875 | . . . 4 ⊢ (𝐾 ∈ ℕ0 → (𝑁‘𝐾) = inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}, ℝ, < )) |
10 | 9 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → (𝑁‘𝐾) = inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}, ℝ, < )) |
11 | ssrab2 4013 | . . . . 5 ⊢ {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ} ⊆ ℕ | |
12 | nnuz 12621 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
13 | 11, 12 | sseqtri 3957 | . . . 4 ⊢ {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ} ⊆ (ℤ≥‘1) |
14 | elqaa.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝})) | |
15 | 14 | eldifad 3899 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℚ)) |
16 | 0z 12330 | . . . . . . . . 9 ⊢ 0 ∈ ℤ | |
17 | zq 12694 | . . . . . . . . 9 ⊢ (0 ∈ ℤ → 0 ∈ ℚ) | |
18 | 16, 17 | ax-mp 5 | . . . . . . . 8 ⊢ 0 ∈ ℚ |
19 | elqaa.4 | . . . . . . . . 9 ⊢ 𝐵 = (coeff‘𝐹) | |
20 | 19 | coef2 25392 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℚ) ∧ 0 ∈ ℚ) → 𝐵:ℕ0⟶ℚ) |
21 | 15, 18, 20 | sylancl 586 | . . . . . . 7 ⊢ (𝜑 → 𝐵:ℕ0⟶ℚ) |
22 | 21 | ffvelrnda 6961 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → (𝐵‘𝐾) ∈ ℚ) |
23 | qmulz 12691 | . . . . . 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 4319 | . . . . 5 ⊢ ({𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ} ≠ ∅ ↔ ∃𝑛 ∈ ℕ ((𝐵‘𝐾) · 𝑛) ∈ ℤ) | |
26 | 24, 25 | sylibr 233 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ} ≠ ∅) |
27 | infssuzcl 12672 | . . . 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 587 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}) |
29 | 10, 28 | eqeltrd 2839 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → (𝑁‘𝐾) ∈ {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ}) |
30 | oveq2 7283 | . . . 4 ⊢ (𝑛 = (𝑁‘𝐾) → ((𝐵‘𝐾) · 𝑛) = ((𝐵‘𝐾) · (𝑁‘𝐾))) | |
31 | 30 | eleq1d 2823 | . . 3 ⊢ (𝑛 = (𝑁‘𝐾) → (((𝐵‘𝐾) · 𝑛) ∈ ℤ ↔ ((𝐵‘𝐾) · (𝑁‘𝐾)) ∈ ℤ)) |
32 | 31 | elrab 3624 | . 2 ⊢ ((𝑁‘𝐾) ∈ {𝑛 ∈ ℕ ∣ ((𝐵‘𝐾) · 𝑛) ∈ ℤ} ↔ ((𝑁‘𝐾) ∈ ℕ ∧ ((𝐵‘𝐾) · (𝑁‘𝐾)) ∈ ℤ)) |
33 | 29, 32 | sylib 217 | 1 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → ((𝑁‘𝐾) ∈ ℕ ∧ ((𝐵‘𝐾) · (𝑁‘𝐾)) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 {crab 3068 ∖ cdif 3884 ⊆ wss 3887 ∅c0 4256 {csn 4561 ↦ cmpt 5157 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 infcinf 9200 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 · cmul 10876 < clt 11009 ℕcn 11973 ℕ0cn0 12233 ℤcz 12319 ℤ≥cuz 12582 ℚcq 12688 seqcseq 13721 0𝑝c0p 24833 Polycply 25345 coeffccoe 25347 degcdgr 25348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-0p 24834 df-ply 25349 df-coe 25351 |
This theorem is referenced by: elqaalem2 25480 |
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