Mathbox for Stefan O'Rear < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hbtlem7 Structured version   Visualization version   GIF version

Theorem hbtlem7 40056
 Description: Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypotheses
Ref Expression
hbtlem.p 𝑃 = (Poly1𝑅)
hbtlem.u 𝑈 = (LIdeal‘𝑃)
hbtlem.s 𝑆 = (ldgIdlSeq‘𝑅)
hbtlem7.t 𝑇 = (LIdeal‘𝑅)
Assertion
Ref Expression
hbtlem7 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼):ℕ0𝑇)

Proof of Theorem hbtlem7
Dummy variables 𝑖 𝑗 𝑥 𝑦 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 488 . . . . . . . . 9 (((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) → 𝑦 = ((coe1𝑗)‘𝑥))
21reximi 3209 . . . . . . . 8 (∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) → ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥))
32ss2abi 3997 . . . . . . 7 {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ⊆ {𝑦 ∣ ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥)}
4 abrexexg 7648 . . . . . . 7 (𝐼𝑈 → {𝑦 ∣ ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥)} ∈ V)
5 ssexg 5194 . . . . . . 7 (({𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ⊆ {𝑦 ∣ ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥)} ∧ {𝑦 ∣ ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥)} ∈ V) → {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V)
63, 4, 5sylancr 590 . . . . . 6 (𝐼𝑈 → {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V)
76ralrimivw 3153 . . . . 5 (𝐼𝑈 → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V)
87adantl 485 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V)
9 eqid 2801 . . . . 5 (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})
109fnmpt 6464 . . . 4 (∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) Fn ℕ0)
118, 10syl 17 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) Fn ℕ0)
12 hbtlem.s . . . . . . 7 𝑆 = (ldgIdlSeq‘𝑅)
13 elex 3462 . . . . . . . 8 (𝑅 ∈ Ring → 𝑅 ∈ V)
14 fveq2 6649 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
15 hbtlem.p . . . . . . . . . . . . 13 𝑃 = (Poly1𝑅)
1614, 15eqtr4di 2854 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
1716fveq2d 6653 . . . . . . . . . . 11 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = (LIdeal‘𝑃))
18 hbtlem.u . . . . . . . . . . 11 𝑈 = (LIdeal‘𝑃)
1917, 18eqtr4di 2854 . . . . . . . . . 10 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = 𝑈)
20 fveq2 6649 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
2120fveq1d 6651 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑗) = (( deg1𝑅)‘𝑗))
2221breq1d 5043 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → ((( deg1𝑟)‘𝑗) ≤ 𝑥 ↔ (( deg1𝑅)‘𝑗) ≤ 𝑥))
2322anbi1d 632 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) ↔ ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))))
2423rexbidv 3259 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) ↔ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))))
2524abbidv 2865 . . . . . . . . . . 11 (𝑟 = 𝑅 → {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} = {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})
2625mpteq2dv 5129 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
2719, 26mpteq12dv 5118 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})))
28 df-ldgis 40053 . . . . . . . . 9 ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})))
2927, 28, 18mptfvmpt 6972 . . . . . . . 8 (𝑅 ∈ V → (ldgIdlSeq‘𝑅) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})))
3013, 29syl 17 . . . . . . 7 (𝑅 ∈ Ring → (ldgIdlSeq‘𝑅) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})))
3112, 30syl5eq 2848 . . . . . 6 (𝑅 ∈ Ring → 𝑆 = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})))
3231fveq1d 6651 . . . . 5 (𝑅 ∈ Ring → (𝑆𝐼) = ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))‘𝐼))
33 rexeq 3362 . . . . . . . 8 (𝑖 = 𝐼 → (∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) ↔ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))))
3433abbidv 2865 . . . . . . 7 (𝑖 = 𝐼 → {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} = {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})
3534mpteq2dv 5129 . . . . . 6 (𝑖 = 𝐼 → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
36 eqid 2801 . . . . . 6 (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
37 nn0ex 11895 . . . . . . 7 0 ∈ V
3837mptex 6967 . . . . . 6 (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) ∈ V
3935, 36, 38fvmpt 6749 . . . . 5 (𝐼𝑈 → ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
4032, 39sylan9eq 2856 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
4140fneq1d 6420 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → ((𝑆𝐼) Fn ℕ0 ↔ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) Fn ℕ0))
4211, 41mpbird 260 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼) Fn ℕ0)
43 hbtlem7.t . . . . 5 𝑇 = (LIdeal‘𝑅)
4415, 18, 12, 43hbtlem2 40055 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑥 ∈ ℕ0) → ((𝑆𝐼)‘𝑥) ∈ 𝑇)
45443expa 1115 . . 3 (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ 𝑥 ∈ ℕ0) → ((𝑆𝐼)‘𝑥) ∈ 𝑇)
4645ralrimiva 3152 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → ∀𝑥 ∈ ℕ0 ((𝑆𝐼)‘𝑥) ∈ 𝑇)
47 ffnfv 6863 . 2 ((𝑆𝐼):ℕ0𝑇 ↔ ((𝑆𝐼) Fn ℕ0 ∧ ∀𝑥 ∈ ℕ0 ((𝑆𝐼)‘𝑥) ∈ 𝑇))
4842, 46, 47sylanbrc 586 1 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼):ℕ0𝑇)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  {cab 2779  ∀wral 3109  ∃wrex 3110  Vcvv 3444   ⊆ wss 3884   class class class wbr 5033   ↦ cmpt 5113   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328   ≤ cle 10669  ℕ0cn0 11889  Ringcrg 19293  LIdealclidl 19938  Poly1cpl1 20809  coe1cco1 20810   deg1 cdg1 24658  ldgIdlSeqcldgis 40052 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609  ax-mulf 10610 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-ofr 7394  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-fzo 13033  df-seq 13369  df-hash 13691  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-starv 16575  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-unif 16583  df-0g 16710  df-gsum 16711  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-submnd 17952  df-grp 18101  df-minusg 18102  df-sbg 18103  df-mulg 18220  df-subg 18271  df-ghm 18351  df-cntz 18442  df-cmn 18903  df-abl 18904  df-mgp 19236  df-ur 19248  df-ring 19295  df-cring 19296  df-subrg 19529  df-lmod 19632  df-lss 19700  df-sra 19940  df-rgmod 19941  df-lidl 19942  df-cnfld 20095  df-ascl 20547  df-psr 20597  df-mvr 20598  df-mpl 20599  df-opsr 20601  df-psr1 20812  df-vr1 20813  df-ply1 20814  df-coe1 20815  df-mdeg 24659  df-deg1 24660  df-ldgis 40053 This theorem is referenced by:  hbt  40061
 Copyright terms: Public domain W3C validator