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Mirrors > Home > MPE Home > Th. List > elqaalem1 | Structured version Visualization version GIF version |
Description: Lemma for elqaa 25698. The function π represents the denominators of the rational coefficients π΅. By multiplying them all together to make π , we get a number big enough to clear all the denominators and make π Β· πΉ an integer polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.) |
Ref | Expression |
---|---|
elqaa.1 | β’ (π β π΄ β β) |
elqaa.2 | β’ (π β πΉ β ((Polyββ) β {0π})) |
elqaa.3 | β’ (π β (πΉβπ΄) = 0) |
elqaa.4 | β’ π΅ = (coeffβπΉ) |
elqaa.5 | β’ π = (π β β0 β¦ inf({π β β β£ ((π΅βπ) Β· π) β β€}, β, < )) |
elqaa.6 | β’ π = (seq0( Β· , π)β(degβπΉ)) |
Ref | Expression |
---|---|
elqaalem1 | β’ ((π β§ πΎ β β0) β ((πβπΎ) β β β§ ((π΅βπΎ) Β· (πβπΎ)) β β€)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6847 | . . . . . . . . 9 β’ (π = πΎ β (π΅βπ) = (π΅βπΎ)) | |
2 | 1 | oveq1d 7377 | . . . . . . . 8 β’ (π = πΎ β ((π΅βπ) Β· π) = ((π΅βπΎ) Β· π)) |
3 | 2 | eleq1d 2823 | . . . . . . 7 β’ (π = πΎ β (((π΅βπ) Β· π) β β€ β ((π΅βπΎ) Β· π) β β€)) |
4 | 3 | rabbidv 3418 | . . . . . 6 β’ (π = πΎ β {π β β β£ ((π΅βπ) Β· π) β β€} = {π β β β£ ((π΅βπΎ) Β· π) β β€}) |
5 | 4 | infeq1d 9420 | . . . . 5 β’ (π = πΎ β inf({π β β β£ ((π΅βπ) Β· π) β β€}, β, < ) = inf({π β β β£ ((π΅βπΎ) Β· π) β β€}, β, < )) |
6 | elqaa.5 | . . . . 5 β’ π = (π β β0 β¦ inf({π β β β£ ((π΅βπ) Β· π) β β€}, β, < )) | |
7 | ltso 11242 | . . . . . 6 β’ < Or β | |
8 | 7 | infex 9436 | . . . . 5 β’ inf({π β β β£ ((π΅βπΎ) Β· π) β β€}, β, < ) β V |
9 | 5, 6, 8 | fvmpt 6953 | . . . 4 β’ (πΎ β β0 β (πβπΎ) = inf({π β β β£ ((π΅βπΎ) Β· π) β β€}, β, < )) |
10 | 9 | adantl 483 | . . 3 β’ ((π β§ πΎ β β0) β (πβπΎ) = inf({π β β β£ ((π΅βπΎ) Β· π) β β€}, β, < )) |
11 | ssrab2 4042 | . . . . 5 β’ {π β β β£ ((π΅βπΎ) Β· π) β β€} β β | |
12 | nnuz 12813 | . . . . 5 β’ β = (β€β₯β1) | |
13 | 11, 12 | sseqtri 3985 | . . . 4 β’ {π β β β£ ((π΅βπΎ) Β· π) β β€} β (β€β₯β1) |
14 | elqaa.2 | . . . . . . . . 9 β’ (π β πΉ β ((Polyββ) β {0π})) | |
15 | 14 | eldifad 3927 | . . . . . . . 8 β’ (π β πΉ β (Polyββ)) |
16 | 0z 12517 | . . . . . . . . 9 β’ 0 β β€ | |
17 | zq 12886 | . . . . . . . . 9 β’ (0 β β€ β 0 β β) | |
18 | 16, 17 | ax-mp 5 | . . . . . . . 8 β’ 0 β β |
19 | elqaa.4 | . . . . . . . . 9 β’ π΅ = (coeffβπΉ) | |
20 | 19 | coef2 25608 | . . . . . . . 8 β’ ((πΉ β (Polyββ) β§ 0 β β) β π΅:β0βΆβ) |
21 | 15, 18, 20 | sylancl 587 | . . . . . . 7 β’ (π β π΅:β0βΆβ) |
22 | 21 | ffvelcdmda 7040 | . . . . . 6 β’ ((π β§ πΎ β β0) β (π΅βπΎ) β β) |
23 | qmulz 12883 | . . . . . 6 β’ ((π΅βπΎ) β β β βπ β β ((π΅βπΎ) Β· π) β β€) | |
24 | 22, 23 | syl 17 | . . . . 5 β’ ((π β§ πΎ β β0) β βπ β β ((π΅βπΎ) Β· π) β β€) |
25 | rabn0 4350 | . . . . 5 β’ ({π β β β£ ((π΅βπΎ) Β· π) β β€} β β β βπ β β ((π΅βπΎ) Β· π) β β€) | |
26 | 24, 25 | sylibr 233 | . . . 4 β’ ((π β§ πΎ β β0) β {π β β β£ ((π΅βπΎ) Β· π) β β€} β β ) |
27 | infssuzcl 12864 | . . . 4 β’ (({π β β β£ ((π΅βπΎ) Β· π) β β€} β (β€β₯β1) β§ {π β β β£ ((π΅βπΎ) Β· π) β β€} β β ) β inf({π β β β£ ((π΅βπΎ) Β· π) β β€}, β, < ) β {π β β β£ ((π΅βπΎ) Β· π) β β€}) | |
28 | 13, 26, 27 | sylancr 588 | . . 3 β’ ((π β§ πΎ β β0) β inf({π β β β£ ((π΅βπΎ) Β· π) β β€}, β, < ) β {π β β β£ ((π΅βπΎ) Β· π) β β€}) |
29 | 10, 28 | eqeltrd 2838 | . 2 β’ ((π β§ πΎ β β0) β (πβπΎ) β {π β β β£ ((π΅βπΎ) Β· π) β β€}) |
30 | oveq2 7370 | . . . 4 β’ (π = (πβπΎ) β ((π΅βπΎ) Β· π) = ((π΅βπΎ) Β· (πβπΎ))) | |
31 | 30 | eleq1d 2823 | . . 3 β’ (π = (πβπΎ) β (((π΅βπΎ) Β· π) β β€ β ((π΅βπΎ) Β· (πβπΎ)) β β€)) |
32 | 31 | elrab 3650 | . 2 β’ ((πβπΎ) β {π β β β£ ((π΅βπΎ) Β· π) β β€} β ((πβπΎ) β β β§ ((π΅βπΎ) Β· (πβπΎ)) β β€)) |
33 | 29, 32 | sylib 217 | 1 β’ ((π β§ πΎ β β0) β ((πβπΎ) β β β§ ((π΅βπΎ) Β· (πβπΎ)) β β€)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2944 βwrex 3074 {crab 3410 β cdif 3912 β wss 3915 β c0 4287 {csn 4591 β¦ cmpt 5193 βΆwf 6497 βcfv 6501 (class class class)co 7362 infcinf 9384 βcc 11056 βcr 11057 0cc0 11058 1c1 11059 Β· cmul 11063 < clt 11196 βcn 12160 β0cn0 12420 β€cz 12506 β€β₯cuz 12770 βcq 12880 seqcseq 13913 0πc0p 25049 Polycply 25561 coeffccoe 25563 degcdgr 25564 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-rlim 15378 df-sum 15578 df-0p 25050 df-ply 25565 df-coe 25567 |
This theorem is referenced by: elqaalem2 25696 |
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