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Mirrors > Home > MPE Home > Th. List > elqaalem1 | Structured version Visualization version GIF version |
Description: Lemma for elqaa 26212. 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 6885 | . . . . . . . . 9 β’ (π = πΎ β (π΅βπ) = (π΅βπΎ)) | |
2 | 1 | oveq1d 7420 | . . . . . . . 8 β’ (π = πΎ β ((π΅βπ) Β· π) = ((π΅βπΎ) Β· π)) |
3 | 2 | eleq1d 2812 | . . . . . . 7 β’ (π = πΎ β (((π΅βπ) Β· π) β β€ β ((π΅βπΎ) Β· π) β β€)) |
4 | 3 | rabbidv 3434 | . . . . . 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 9474 | . . . . 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 11298 | . . . . . 6 β’ < Or β | |
8 | 7 | infex 9490 | . . . . 5 β’ inf({π β β β£ ((π΅βπΎ) Β· π) β β€}, β, < ) β V |
9 | 5, 6, 8 | fvmpt 6992 | . . . 4 β’ (πΎ β β0 β (πβπΎ) = inf({π β β β£ ((π΅βπΎ) Β· π) β β€}, β, < )) |
10 | 9 | adantl 481 | . . 3 β’ ((π β§ πΎ β β0) β (πβπΎ) = inf({π β β β£ ((π΅βπΎ) Β· π) β β€}, β, < )) |
11 | ssrab2 4072 | . . . . 5 β’ {π β β β£ ((π΅βπΎ) Β· π) β β€} β β | |
12 | nnuz 12869 | . . . . 5 β’ β = (β€β₯β1) | |
13 | 11, 12 | sseqtri 4013 | . . . 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 3955 | . . . . . . . 8 β’ (π β πΉ β (Polyββ)) |
16 | 0z 12573 | . . . . . . . . 9 β’ 0 β β€ | |
17 | zq 12942 | . . . . . . . . 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 26120 | . . . . . . . 8 β’ ((πΉ β (Polyββ) β§ 0 β β) β π΅:β0βΆβ) |
21 | 15, 18, 20 | sylancl 585 | . . . . . . 7 β’ (π β π΅:β0βΆβ) |
22 | 21 | ffvelcdmda 7080 | . . . . . 6 β’ ((π β§ πΎ β β0) β (π΅βπΎ) β β) |
23 | qmulz 12939 | . . . . . 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 4380 | . . . . 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 12920 | . . . 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 586 | . . 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 2827 | . 2 β’ ((π β§ πΎ β β0) β (πβπΎ) β {π β β β£ ((π΅βπΎ) Β· π) β β€}) |
30 | oveq2 7413 | . . . 4 β’ (π = (πβπΎ) β ((π΅βπΎ) Β· π) = ((π΅βπΎ) Β· (πβπΎ))) | |
31 | 30 | eleq1d 2812 | . . 3 β’ (π = (πβπΎ) β (((π΅βπΎ) Β· π) β β€ β ((π΅βπΎ) Β· (πβπΎ)) β β€)) |
32 | 31 | elrab 3678 | . 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 395 = wceq 1533 β wcel 2098 β wne 2934 βwrex 3064 {crab 3426 β cdif 3940 β wss 3943 β c0 4317 {csn 4623 β¦ cmpt 5224 βΆwf 6533 βcfv 6537 (class class class)co 7405 infcinf 9438 βcc 11110 βcr 11111 0cc0 11112 1c1 11113 Β· cmul 11117 < clt 11252 βcn 12216 β0cn0 12476 β€cz 12562 β€β₯cuz 12826 βcq 12936 seqcseq 13972 0πc0p 25553 Polycply 26073 coeffccoe 26075 degcdgr 26076 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-rlim 15439 df-sum 15639 df-0p 25554 df-ply 26077 df-coe 26079 |
This theorem is referenced by: elqaalem2 26210 |
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