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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hbtlem3 | Structured version Visualization version GIF version |
Description: The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Ref | Expression |
---|---|
hbtlem.p | β’ π = (Poly1βπ ) |
hbtlem.u | β’ π = (LIdealβπ) |
hbtlem.s | β’ π = (ldgIdlSeqβπ ) |
hbtlem3.r | β’ (π β π β Ring) |
hbtlem3.i | β’ (π β πΌ β π) |
hbtlem3.j | β’ (π β π½ β π) |
hbtlem3.ij | β’ (π β πΌ β π½) |
hbtlem3.x | β’ (π β π β β0) |
Ref | Expression |
---|---|
hbtlem3 | β’ (π β ((πβπΌ)βπ) β ((πβπ½)βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbtlem3.ij | . . . 4 β’ (π β πΌ β π½) | |
2 | ssrexv 4050 | . . . 4 β’ (πΌ β π½ β (βπ β πΌ ((( deg1 βπ )βπ) β€ π β§ π = ((coe1βπ)βπ)) β βπ β π½ ((( deg1 βπ )βπ) β€ π β§ π = ((coe1βπ)βπ)))) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π β (βπ β πΌ ((( deg1 βπ )βπ) β€ π β§ π = ((coe1βπ)βπ)) β βπ β π½ ((( deg1 βπ )βπ) β€ π β§ π = ((coe1βπ)βπ)))) |
4 | 3 | ss2abdv 4059 | . 2 β’ (π β {π β£ βπ β πΌ ((( deg1 βπ )βπ) β€ π β§ π = ((coe1βπ)βπ))} β {π β£ βπ β π½ ((( deg1 βπ )βπ) β€ π β§ π = ((coe1βπ)βπ))}) |
5 | hbtlem3.r | . . 3 β’ (π β π β Ring) | |
6 | hbtlem3.i | . . 3 β’ (π β πΌ β π) | |
7 | hbtlem3.x | . . 3 β’ (π β π β β0) | |
8 | hbtlem.p | . . . 4 β’ π = (Poly1βπ ) | |
9 | hbtlem.u | . . . 4 β’ π = (LIdealβπ) | |
10 | hbtlem.s | . . . 4 β’ π = (ldgIdlSeqβπ ) | |
11 | eqid 2732 | . . . 4 β’ ( deg1 βπ ) = ( deg1 βπ ) | |
12 | 8, 9, 10, 11 | hbtlem1 41850 | . . 3 β’ ((π β Ring β§ πΌ β π β§ π β β0) β ((πβπΌ)βπ) = {π β£ βπ β πΌ ((( deg1 βπ )βπ) β€ π β§ π = ((coe1βπ)βπ))}) |
13 | 5, 6, 7, 12 | syl3anc 1371 | . 2 β’ (π β ((πβπΌ)βπ) = {π β£ βπ β πΌ ((( deg1 βπ )βπ) β€ π β§ π = ((coe1βπ)βπ))}) |
14 | hbtlem3.j | . . 3 β’ (π β π½ β π) | |
15 | 8, 9, 10, 11 | hbtlem1 41850 | . . 3 β’ ((π β Ring β§ π½ β π β§ π β β0) β ((πβπ½)βπ) = {π β£ βπ β π½ ((( deg1 βπ )βπ) β€ π β§ π = ((coe1βπ)βπ))}) |
16 | 5, 14, 7, 15 | syl3anc 1371 | . 2 β’ (π β ((πβπ½)βπ) = {π β£ βπ β π½ ((( deg1 βπ )βπ) β€ π β§ π = ((coe1βπ)βπ))}) |
17 | 4, 13, 16 | 3sstr4d 4028 | 1 β’ (π β ((πβπΌ)βπ) β ((πβπ½)βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 {cab 2709 βwrex 3070 β wss 3947 class class class wbr 5147 βcfv 6540 β€ cle 11245 β0cn0 12468 Ringcrg 20049 LIdealclidl 20775 Poly1cpl1 21692 coe1cco1 21693 deg1 cdg1 25560 ldgIdlSeqcldgis 41848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-1cn 11164 ax-addcl 11166 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-nn 12209 df-n0 12469 df-ldgis 41849 |
This theorem is referenced by: hbt 41857 |
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