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Theorem hbtlem7 41262
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 486 . . . . . . . . 9 (((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) → 𝑦 = ((coe1𝑗)‘𝑥))
21reximi 3084 . . . . . . . 8 (∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) → ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥))
32ss2abi 4015 . . . . . . 7 {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ⊆ {𝑦 ∣ ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥)}
4 abrexexg 7876 . . . . . . 7 (𝐼𝑈 → {𝑦 ∣ ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥)} ∈ V)
5 ssexg 5272 . . . . . . 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 588 . . . . . 6 (𝐼𝑈 → {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V)
76ralrimivw 3144 . . . . 5 (𝐼𝑈 → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V)
87adantl 483 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V)
9 eqid 2737 . . . . 5 (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})
109fnmpt 6629 . . . 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 3460 . . . . . . . 8 (𝑅 ∈ Ring → 𝑅 ∈ V)
14 fveq2 6830 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
15 hbtlem.p . . . . . . . . . . . . 13 𝑃 = (Poly1𝑅)
1614, 15eqtr4di 2795 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
1716fveq2d 6834 . . . . . . . . . . 11 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = (LIdeal‘𝑃))
18 hbtlem.u . . . . . . . . . . 11 𝑈 = (LIdeal‘𝑃)
1917, 18eqtr4di 2795 . . . . . . . . . 10 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = 𝑈)
20 fveq2 6830 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
2120fveq1d 6832 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑗) = (( deg1𝑅)‘𝑗))
2221breq1d 5107 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → ((( deg1𝑟)‘𝑗) ≤ 𝑥 ↔ (( deg1𝑅)‘𝑗) ≤ 𝑥))
2322anbi1d 631 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) ↔ ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))))
2423rexbidv 3172 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) ↔ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))))
2524abbidv 2806 . . . . . . . . . . 11 (𝑟 = 𝑅 → {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} = {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})
2625mpteq2dv 5199 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
2719, 26mpteq12dv 5188 . . . . . . . . 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 41259 . . . . . . . . 9 ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})))
2927, 28, 18mptfvmpt 7165 . . . . . . . 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, 30eqtrid 2789 . . . . . 6 (𝑅 ∈ Ring → 𝑆 = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})))
3231fveq1d 6832 . . . . 5 (𝑅 ∈ Ring → (𝑆𝐼) = ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))‘𝐼))
33 rexeq 3307 . . . . . . . 8 (𝑖 = 𝐼 → (∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) ↔ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))))
3433abbidv 2806 . . . . . . 7 (𝑖 = 𝐼 → {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} = {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})
3534mpteq2dv 5199 . . . . . 6 (𝑖 = 𝐼 → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
36 eqid 2737 . . . . . 6 (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
37 nn0ex 12345 . . . . . . 7 0 ∈ V
3837mptex 7160 . . . . . 6 (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) ∈ V
3935, 36, 38fvmpt 6936 . . . . 5 (𝐼𝑈 → ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
4032, 39sylan9eq 2797 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
4140fneq1d 6583 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → ((𝑆𝐼) Fn ℕ0 ↔ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) Fn ℕ0))
4211, 41mpbird 257 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼) Fn ℕ0)
43 hbtlem7.t . . . . 5 𝑇 = (LIdeal‘𝑅)
4415, 18, 12, 43hbtlem2 41261 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑥 ∈ ℕ0) → ((𝑆𝐼)‘𝑥) ∈ 𝑇)
45443expa 1118 . . 3 (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ 𝑥 ∈ ℕ0) → ((𝑆𝐼)‘𝑥) ∈ 𝑇)
4645ralrimiva 3140 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → ∀𝑥 ∈ ℕ0 ((𝑆𝐼)‘𝑥) ∈ 𝑇)
47 ffnfv 7053 . 2 ((𝑆𝐼):ℕ0𝑇 ↔ ((𝑆𝐼) Fn ℕ0 ∧ ∀𝑥 ∈ ℕ0 ((𝑆𝐼)‘𝑥) ∈ 𝑇))
4842, 46, 47sylanbrc 584 1 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼):ℕ0𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1541  wcel 2106  {cab 2714  wral 3062  wrex 3071  Vcvv 3442  wss 3902   class class class wbr 5097  cmpt 5180   Fn wfn 6479  wf 6480  cfv 6484  cle 11116  0cn0 12339  Ringcrg 19878  LIdealclidl 20538  Poly1cpl1 21454  coe1cco1 21455   deg1 cdg1 25322  ldgIdlSeqcldgis 41258
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655  ax-cnex 11033  ax-resscn 11034  ax-1cn 11035  ax-icn 11036  ax-addcl 11037  ax-addrcl 11038  ax-mulcl 11039  ax-mulrcl 11040  ax-mulcom 11041  ax-addass 11042  ax-mulass 11043  ax-distr 11044  ax-i2m1 11045  ax-1ne0 11046  ax-1rid 11047  ax-rnegex 11048  ax-rrecex 11049  ax-cnre 11050  ax-pre-lttri 11051  ax-pre-lttrn 11052  ax-pre-ltadd 11053  ax-pre-mulgt0 11054  ax-pre-sup 11055  ax-addf 11056  ax-mulf 11057
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4858  df-int 4900  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-se 5581  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-isom 6493  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-of 7600  df-ofr 7601  df-om 7786  df-1st 7904  df-2nd 7905  df-supp 8053  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-1o 8372  df-er 8574  df-map 8693  df-pm 8694  df-ixp 8762  df-en 8810  df-dom 8811  df-sdom 8812  df-fin 8813  df-fsupp 9232  df-sup 9304  df-oi 9372  df-card 9801  df-pnf 11117  df-mnf 11118  df-xr 11119  df-ltxr 11120  df-le 11121  df-sub 11313  df-neg 11314  df-nn 12080  df-2 12142  df-3 12143  df-4 12144  df-5 12145  df-6 12146  df-7 12147  df-8 12148  df-9 12149  df-n0 12340  df-z 12426  df-dec 12544  df-uz 12689  df-fz 13346  df-fzo 13489  df-seq 13828  df-hash 14151  df-struct 16946  df-sets 16963  df-slot 16981  df-ndx 16993  df-base 17011  df-ress 17040  df-plusg 17073  df-mulr 17074  df-starv 17075  df-sca 17076  df-vsca 17077  df-ip 17078  df-tset 17079  df-ple 17080  df-ds 17082  df-unif 17083  df-0g 17250  df-gsum 17251  df-mre 17393  df-mrc 17394  df-acs 17396  df-mgm 18424  df-sgrp 18473  df-mnd 18484  df-mhm 18528  df-submnd 18529  df-grp 18677  df-minusg 18678  df-sbg 18679  df-mulg 18798  df-subg 18849  df-ghm 18929  df-cntz 19020  df-cmn 19484  df-abl 19485  df-mgp 19816  df-ur 19833  df-ring 19880  df-cring 19881  df-subrg 20127  df-lmod 20231  df-lss 20300  df-sra 20540  df-rgmod 20541  df-lidl 20542  df-cnfld 20704  df-ascl 21168  df-psr 21218  df-mvr 21219  df-mpl 21220  df-opsr 21222  df-psr1 21457  df-vr1 21458  df-ply1 21459  df-coe1 21460  df-mdeg 25323  df-deg1 25324  df-ldgis 41259
This theorem is referenced by:  hbt  41267
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