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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  0cn0 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
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