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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 4051 | . . . 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 4060 | . 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 2728 | . . . 4 β’ ( deg1 βπ ) = ( deg1 βπ ) | |
12 | 8, 9, 10, 11 | hbtlem1 42578 | . . 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 1368 | . 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 42578 | . . 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 1368 | . 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 4029 | 1 β’ (π β ((πβπΌ)βπ) β ((πβπ½)βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {cab 2705 βwrex 3067 β wss 3949 class class class wbr 5152 βcfv 6553 β€ cle 11287 β0cn0 12510 Ringcrg 20180 LIdealclidl 21109 Poly1cpl1 22103 coe1cco1 22104 deg1 cdg1 26007 ldgIdlSeqcldgis 42576 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-1cn 11204 ax-addcl 11206 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-nn 12251 df-n0 12511 df-ldgis 42577 |
This theorem is referenced by: hbt 42585 |
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