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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 4016 | . . . 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 4025 | . 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 2737 | . . . 4 β’ ( deg1 βπ ) = ( deg1 βπ ) | |
12 | 8, 9, 10, 11 | hbtlem1 41479 | . . 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 1372 | . 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 41479 | . . 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 1372 | . 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 3996 | 1 β’ (π β ((πβπΌ)βπ) β ((πβπ½)βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {cab 2714 βwrex 3074 β wss 3915 class class class wbr 5110 βcfv 6501 β€ cle 11197 β0cn0 12420 Ringcrg 19971 LIdealclidl 20647 Poly1cpl1 21564 coe1cco1 21565 deg1 cdg1 25432 ldgIdlSeqcldgis 41477 |
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-pr 5389 ax-un 7677 ax-cnex 11114 ax-1cn 11116 ax-addcl 11118 |
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-ral 3066 df-rex 3075 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-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-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-ov 7365 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-nn 12161 df-n0 12421 df-ldgis 41478 |
This theorem is referenced by: hbt 41486 |
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