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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 4046 | . . . 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 4055 | . 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 2726 | . . . 4 β’ ( deg1 βπ ) = ( deg1 βπ ) | |
12 | 8, 9, 10, 11 | hbtlem1 42425 | . . 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 42425 | . . 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 4024 | 1 β’ (π β ((πβπΌ)βπ) β ((πβπ½)βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {cab 2703 βwrex 3064 β wss 3943 class class class wbr 5141 βcfv 6536 β€ cle 11250 β0cn0 12473 Ringcrg 20135 LIdealclidl 21062 Poly1cpl1 22046 coe1cco1 22047 deg1 cdg1 25937 ldgIdlSeqcldgis 42423 |
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-pr 5420 ax-un 7721 ax-cnex 11165 ax-1cn 11167 ax-addcl 11169 |
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-ral 3056 df-rex 3065 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-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-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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-nn 12214 df-n0 12474 df-ldgis 42424 |
This theorem is referenced by: hbt 42432 |
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