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Theorem hbtlem7 38214
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 471 . . . . . . . . 9 (((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) → 𝑦 = ((coe1𝑗)‘𝑥))
21reximi 3158 . . . . . . . 8 (∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) → ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥))
32ss2abi 3821 . . . . . . 7 {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ⊆ {𝑦 ∣ ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥)}
4 abrexexg 7286 . . . . . . 7 (𝐼𝑈 → {𝑦 ∣ ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥)} ∈ V)
5 ssexg 4935 . . . . . . 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 567 . . . . . 6 (𝐼𝑈 → {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V)
76ralrimivw 3115 . . . . 5 (𝐼𝑈 → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V)
87adantl 467 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V)
9 eqid 2770 . . . . 5 (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})
109fnmpt 6160 . . . 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 3361 . . . . . . . 8 (𝑅 ∈ Ring → 𝑅 ∈ V)
14 fveq2 6332 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
15 hbtlem.p . . . . . . . . . . . . 13 𝑃 = (Poly1𝑅)
1614, 15syl6eqr 2822 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
1716fveq2d 6336 . . . . . . . . . . 11 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = (LIdeal‘𝑃))
18 hbtlem.u . . . . . . . . . . 11 𝑈 = (LIdeal‘𝑃)
1917, 18syl6eqr 2822 . . . . . . . . . 10 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = 𝑈)
20 fveq2 6332 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
2120fveq1d 6334 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑗) = (( deg1𝑅)‘𝑗))
2221breq1d 4794 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → ((( deg1𝑟)‘𝑗) ≤ 𝑥 ↔ (( deg1𝑅)‘𝑗) ≤ 𝑥))
2322anbi1d 607 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) ↔ ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))))
2423rexbidv 3199 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) ↔ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))))
2524abbidv 2889 . . . . . . . . . . 11 (𝑟 = 𝑅 → {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} = {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})
2625mpteq2dv 4877 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
2719, 26mpteq12dv 4865 . . . . . . . . 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 38211 . . . . . . . . 9 ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})))
29 fvex 6342 . . . . . . . . . . 11 (LIdeal‘𝑃) ∈ V
3018, 29eqeltri 2845 . . . . . . . . . 10 𝑈 ∈ V
3130mptex 6629 . . . . . . . . 9 (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})) ∈ V
3227, 28, 31fvmpt 6424 . . . . . . . 8 (𝑅 ∈ V → (ldgIdlSeq‘𝑅) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})))
3313, 32syl 17 . . . . . . 7 (𝑅 ∈ Ring → (ldgIdlSeq‘𝑅) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})))
3412, 33syl5eq 2816 . . . . . 6 (𝑅 ∈ Ring → 𝑆 = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})))
3534fveq1d 6334 . . . . 5 (𝑅 ∈ Ring → (𝑆𝐼) = ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))‘𝐼))
36 rexeq 3287 . . . . . . . 8 (𝑖 = 𝐼 → (∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) ↔ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))))
3736abbidv 2889 . . . . . . 7 (𝑖 = 𝐼 → {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} = {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})
3837mpteq2dv 4877 . . . . . 6 (𝑖 = 𝐼 → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
39 eqid 2770 . . . . . 6 (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
40 nn0ex 11499 . . . . . . 7 0 ∈ V
4140mptex 6629 . . . . . 6 (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) ∈ V
4238, 39, 41fvmpt 6424 . . . . 5 (𝐼𝑈 → ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
4335, 42sylan9eq 2824 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
4443fneq1d 6121 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → ((𝑆𝐼) Fn ℕ0 ↔ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) Fn ℕ0))
4511, 44mpbird 247 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼) Fn ℕ0)
46 hbtlem7.t . . . . 5 𝑇 = (LIdeal‘𝑅)
4715, 18, 12, 46hbtlem2 38213 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑥 ∈ ℕ0) → ((𝑆𝐼)‘𝑥) ∈ 𝑇)
48473expa 1110 . . 3 (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ 𝑥 ∈ ℕ0) → ((𝑆𝐼)‘𝑥) ∈ 𝑇)
4948ralrimiva 3114 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → ∀𝑥 ∈ ℕ0 ((𝑆𝐼)‘𝑥) ∈ 𝑇)
50 ffnfv 6530 . 2 ((𝑆𝐼):ℕ0𝑇 ↔ ((𝑆𝐼) Fn ℕ0 ∧ ∀𝑥 ∈ ℕ0 ((𝑆𝐼)‘𝑥) ∈ 𝑇))
5145, 49, 50sylanbrc 564 1 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼):ℕ0𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  {cab 2756  wral 3060  wrex 3061  Vcvv 3349  wss 3721   class class class wbr 4784  cmpt 4861   Fn wfn 6026  wf 6027  cfv 6031  cle 10276  0cn0 11493  Ringcrg 18754  LIdealclidl 19384  Poly1cpl1 19761  coe1cco1 19762   deg1 cdg1 24033  ldgIdlSeqcldgis 38210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215  ax-addf 10216  ax-mulf 10217
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-ofr 7044  df-om 7212  df-1st 7314  df-2nd 7315  df-supp 7446  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-ixp 8062  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-fsupp 8431  df-sup 8503  df-oi 8570  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-fz 12533  df-fzo 12673  df-seq 13008  df-hash 13321  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-mulr 16162  df-starv 16163  df-sca 16164  df-vsca 16165  df-ip 16166  df-tset 16167  df-ple 16168  df-ds 16171  df-unif 16172  df-0g 16309  df-gsum 16310  df-mre 16453  df-mrc 16454  df-acs 16456  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-mhm 17542  df-submnd 17543  df-grp 17632  df-minusg 17633  df-sbg 17634  df-mulg 17748  df-subg 17798  df-ghm 17865  df-cntz 17956  df-cmn 18401  df-abl 18402  df-mgp 18697  df-ur 18709  df-ring 18756  df-cring 18757  df-subrg 18987  df-lmod 19074  df-lss 19142  df-sra 19386  df-rgmod 19387  df-lidl 19388  df-ascl 19528  df-psr 19570  df-mvr 19571  df-mpl 19572  df-opsr 19574  df-psr1 19764  df-vr1 19765  df-ply1 19766  df-coe1 19767  df-cnfld 19961  df-mdeg 24034  df-deg1 24035  df-ldgis 38211
This theorem is referenced by:  hbt  38219
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