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Mirrors > Home > MPE Home > Th. List > elqaalem1 | Structured version Visualization version GIF version |
Description: Lemma for elqaa 26275. 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 6892 | . . . . . . . . 9 β’ (π = πΎ β (π΅βπ) = (π΅βπΎ)) | |
2 | 1 | oveq1d 7431 | . . . . . . . 8 β’ (π = πΎ β ((π΅βπ) Β· π) = ((π΅βπΎ) Β· π)) |
3 | 2 | eleq1d 2810 | . . . . . . 7 β’ (π = πΎ β (((π΅βπ) Β· π) β β€ β ((π΅βπΎ) Β· π) β β€)) |
4 | 3 | rabbidv 3427 | . . . . . 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 9500 | . . . . 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 11324 | . . . . . 6 β’ < Or β | |
8 | 7 | infex 9516 | . . . . 5 β’ inf({π β β β£ ((π΅βπΎ) Β· π) β β€}, β, < ) β V |
9 | 5, 6, 8 | fvmpt 7000 | . . . 4 β’ (πΎ β β0 β (πβπΎ) = inf({π β β β£ ((π΅βπΎ) Β· π) β β€}, β, < )) |
10 | 9 | adantl 480 | . . 3 β’ ((π β§ πΎ β β0) β (πβπΎ) = inf({π β β β£ ((π΅βπΎ) Β· π) β β€}, β, < )) |
11 | ssrab2 4069 | . . . . 5 β’ {π β β β£ ((π΅βπΎ) Β· π) β β€} β β | |
12 | nnuz 12895 | . . . . 5 β’ β = (β€β₯β1) | |
13 | 11, 12 | sseqtri 4009 | . . . 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 3951 | . . . . . . . 8 β’ (π β πΉ β (Polyββ)) |
16 | 0z 12599 | . . . . . . . . 9 β’ 0 β β€ | |
17 | zq 12968 | . . . . . . . . 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 26183 | . . . . . . . 8 β’ ((πΉ β (Polyββ) β§ 0 β β) β π΅:β0βΆβ) |
21 | 15, 18, 20 | sylancl 584 | . . . . . . 7 β’ (π β π΅:β0βΆβ) |
22 | 21 | ffvelcdmda 7089 | . . . . . 6 β’ ((π β§ πΎ β β0) β (π΅βπΎ) β β) |
23 | qmulz 12965 | . . . . . 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 4381 | . . . . 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 12946 | . . . 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 585 | . . 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 2825 | . 2 β’ ((π β§ πΎ β β0) β (πβπΎ) β {π β β β£ ((π΅βπΎ) Β· π) β β€}) |
30 | oveq2 7424 | . . . 4 β’ (π = (πβπΎ) β ((π΅βπΎ) Β· π) = ((π΅βπΎ) Β· (πβπΎ))) | |
31 | 30 | eleq1d 2810 | . . 3 β’ (π = (πβπΎ) β (((π΅βπΎ) Β· π) β β€ β ((π΅βπΎ) Β· (πβπΎ)) β β€)) |
32 | 31 | elrab 3674 | . 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 394 = wceq 1533 β wcel 2098 β wne 2930 βwrex 3060 {crab 3419 β cdif 3936 β wss 3939 β c0 4318 {csn 4624 β¦ cmpt 5226 βΆwf 6539 βcfv 6543 (class class class)co 7416 infcinf 9464 βcc 11136 βcr 11137 0cc0 11138 1c1 11139 Β· cmul 11143 < clt 11278 βcn 12242 β0cn0 12502 β€cz 12588 β€β₯cuz 12852 βcq 12962 seqcseq 13998 0πc0p 25616 Polycply 26136 coeffccoe 26138 degcdgr 26139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-rlim 15465 df-sum 15665 df-0p 25617 df-ply 26140 df-coe 26142 |
This theorem is referenced by: elqaalem2 26273 |
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