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Mirrors > Home > MPE Home > Th. List > ftalem6 | Structured version Visualization version GIF version |
Description: Lemma for fta 25656: Discharge the auxiliary variables in ftalem5 25653. (Contributed by Mario Carneiro, 20-Sep-2014.) (Proof shortened by AV, 28-Sep-2020.) |
Ref | Expression |
---|---|
ftalem.1 | ⊢ 𝐴 = (coeff‘𝐹) |
ftalem.2 | ⊢ 𝑁 = (deg‘𝐹) |
ftalem.3 | ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) |
ftalem.4 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
ftalem6.5 | ⊢ (𝜑 → (𝐹‘0) ≠ 0) |
Ref | Expression |
---|---|
ftalem6 | ⊢ (𝜑 → ∃𝑥 ∈ ℂ (abs‘(𝐹‘𝑥)) < (abs‘(𝐹‘0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ftalem.1 | . 2 ⊢ 𝐴 = (coeff‘𝐹) | |
2 | ftalem.2 | . 2 ⊢ 𝑁 = (deg‘𝐹) | |
3 | ftalem.3 | . 2 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) | |
4 | ftalem.4 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
5 | ftalem6.5 | . 2 ⊢ (𝜑 → (𝐹‘0) ≠ 0) | |
6 | fveq2 6669 | . . . . 5 ⊢ (𝑘 = 𝑛 → (𝐴‘𝑘) = (𝐴‘𝑛)) | |
7 | 6 | neeq1d 3075 | . . . 4 ⊢ (𝑘 = 𝑛 → ((𝐴‘𝑘) ≠ 0 ↔ (𝐴‘𝑛) ≠ 0)) |
8 | 7 | cbvrabv 3491 | . . 3 ⊢ {𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0} = {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} |
9 | 8 | infeq1i 8941 | . 2 ⊢ inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < ) |
10 | eqid 2821 | . 2 ⊢ (-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < ))) = (-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < ))) | |
11 | fveq2 6669 | . . . . . . 7 ⊢ (𝑟 = 𝑠 → (𝐴‘𝑟) = (𝐴‘𝑠)) | |
12 | oveq2 7163 | . . . . . . 7 ⊢ (𝑟 = 𝑠 → ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑟) = ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑠)) | |
13 | 11, 12 | oveq12d 7173 | . . . . . 6 ⊢ (𝑟 = 𝑠 → ((𝐴‘𝑟) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑟)) = ((𝐴‘𝑠) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑠))) |
14 | 13 | fveq2d 6673 | . . . . 5 ⊢ (𝑟 = 𝑠 → (abs‘((𝐴‘𝑟) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑟))) = (abs‘((𝐴‘𝑠) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑠)))) |
15 | 14 | cbvsumv 15052 | . . . 4 ⊢ Σ𝑟 ∈ ((inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < ) + 1)...𝑁)(abs‘((𝐴‘𝑟) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑟))) = Σ𝑠 ∈ ((inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < ) + 1)...𝑁)(abs‘((𝐴‘𝑠) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑠))) |
16 | 15 | oveq1i 7165 | . . 3 ⊢ (Σ𝑟 ∈ ((inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < ) + 1)...𝑁)(abs‘((𝐴‘𝑟) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑟))) + 1) = (Σ𝑠 ∈ ((inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < ) + 1)...𝑁)(abs‘((𝐴‘𝑠) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑠))) + 1) |
17 | 16 | oveq2i 7166 | . 2 ⊢ ((abs‘(𝐹‘0)) / (Σ𝑟 ∈ ((inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < ) + 1)...𝑁)(abs‘((𝐴‘𝑟) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑟))) + 1)) = ((abs‘(𝐹‘0)) / (Σ𝑠 ∈ ((inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < ) + 1)...𝑁)(abs‘((𝐴‘𝑠) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑠))) + 1)) |
18 | eqid 2821 | . 2 ⊢ if(1 ≤ ((abs‘(𝐹‘0)) / (Σ𝑟 ∈ ((inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < ) + 1)...𝑁)(abs‘((𝐴‘𝑟) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑟))) + 1)), 1, ((abs‘(𝐹‘0)) / (Σ𝑟 ∈ ((inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < ) + 1)...𝑁)(abs‘((𝐴‘𝑟) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑟))) + 1))) = if(1 ≤ ((abs‘(𝐹‘0)) / (Σ𝑟 ∈ ((inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < ) + 1)...𝑁)(abs‘((𝐴‘𝑟) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑟))) + 1)), 1, ((abs‘(𝐹‘0)) / (Σ𝑟 ∈ ((inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < ) + 1)...𝑁)(abs‘((𝐴‘𝑟) · ((-((𝐹‘0) / (𝐴‘inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑐(1 / inf({𝑘 ∈ ℕ ∣ (𝐴‘𝑘) ≠ 0}, ℝ, < )))↑𝑟))) + 1))) | |
19 | 1, 2, 3, 4, 5, 9, 10, 17, 18 | ftalem5 25653 | 1 ⊢ (𝜑 → ∃𝑥 ∈ ℂ (abs‘(𝐹‘𝑥)) < (abs‘(𝐹‘0))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 {crab 3142 ifcif 4466 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 infcinf 8904 ℂcc 10534 ℝcr 10535 0cc0 10536 1c1 10537 + caddc 10539 · cmul 10541 < clt 10674 ≤ cle 10675 -cneg 10870 / cdiv 11296 ℕcn 11637 ...cfz 12891 ↑cexp 13428 abscabs 14592 Σcsu 15041 Polycply 24773 coeffccoe 24775 degcdgr 24776 ↑𝑐ccxp 25138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ioc 12742 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-fac 13633 df-bc 13662 df-hash 13690 df-shft 14425 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-limsup 14827 df-clim 14844 df-rlim 14845 df-sum 15042 df-ef 15420 df-sin 15422 df-cos 15423 df-pi 15425 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-lp 21743 df-perf 21744 df-cn 21834 df-cnp 21835 df-haus 21922 df-tx 22169 df-hmeo 22362 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-xms 22929 df-ms 22930 df-tms 22931 df-cncf 23485 df-0p 24270 df-limc 24463 df-dv 24464 df-ply 24777 df-coe 24779 df-dgr 24780 df-log 25139 df-cxp 25140 |
This theorem is referenced by: ftalem7 25655 |
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