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Theorem hbtlem7 37162
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 477 . . . . . . . . 9 (((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) → 𝑦 = ((coe1𝑗)‘𝑥))
21reximi 3010 . . . . . . . 8 (∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) → ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥))
32ss2abi 3658 . . . . . . 7 {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ⊆ {𝑦 ∣ ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥)}
4 abrexexg 7091 . . . . . . 7 (𝐼𝑈 → {𝑦 ∣ ∃𝑗𝐼 𝑦 = ((coe1𝑗)‘𝑥)} ∈ V)
5 ssexg 4769 . . . . . . 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 694 . . . . . 6 (𝐼𝑈 → {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V)
76ralrimivw 2966 . . . . 5 (𝐼𝑈 → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V)
87adantl 482 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} ∈ V)
9 eqid 2626 . . . . 5 (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})
109fnmpt 5979 . . . 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 3203 . . . . . . . 8 (𝑅 ∈ Ring → 𝑅 ∈ V)
14 fveq2 6150 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
15 hbtlem.p . . . . . . . . . . . . 13 𝑃 = (Poly1𝑅)
1614, 15syl6eqr 2678 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
1716fveq2d 6154 . . . . . . . . . . 11 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = (LIdeal‘𝑃))
18 hbtlem.u . . . . . . . . . . 11 𝑈 = (LIdeal‘𝑃)
1917, 18syl6eqr 2678 . . . . . . . . . 10 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = 𝑈)
20 fveq2 6150 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
2120fveq1d 6152 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑗) = (( deg1𝑅)‘𝑗))
2221breq1d 4628 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → ((( deg1𝑟)‘𝑗) ≤ 𝑥 ↔ (( deg1𝑅)‘𝑗) ≤ 𝑥))
2322anbi1d 740 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) ↔ ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))))
2423rexbidv 3050 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) ↔ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))))
2524abbidv 2744 . . . . . . . . . . 11 (𝑟 = 𝑅 → {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} = {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})
2625mpteq2dv 4710 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
2719, 26mpteq12dv 4698 . . . . . . . . 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 37159 . . . . . . . . 9 ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑟)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})))
29 fvex 6160 . . . . . . . . . . 11 (LIdeal‘𝑃) ∈ V
3018, 29eqeltri 2700 . . . . . . . . . 10 𝑈 ∈ V
3130mptex 6441 . . . . . . . . 9 (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})) ∈ V
3227, 28, 31fvmpt 6240 . . . . . . . 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 2672 . . . . . 6 (𝑅 ∈ Ring → 𝑆 = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})))
3534fveq1d 6152 . . . . 5 (𝑅 ∈ Ring → (𝑆𝐼) = ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))‘𝐼))
36 rexeq 3133 . . . . . . . 8 (𝑖 = 𝐼 → (∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥)) ↔ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))))
3736abbidv 2744 . . . . . . 7 (𝑖 = 𝐼 → {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))} = {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})
3837mpteq2dv 4710 . . . . . 6 (𝑖 = 𝐼 → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
39 eqid 2626 . . . . . 6 (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))})) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
40 nn0ex 11243 . . . . . . 7 0 ∈ V
4140mptex 6441 . . . . . 6 (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}) ∈ V
4238, 39, 41fvmpt 6240 . . . . 5 (𝐼𝑈 → ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝑖 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
4335, 42sylan9eq 2680 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗𝐼 ((( deg1𝑅)‘𝑗) ≤ 𝑥𝑦 = ((coe1𝑗)‘𝑥))}))
4443fneq1d 5941 . . 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 37161 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑥 ∈ ℕ0) → ((𝑆𝐼)‘𝑥) ∈ 𝑇)
48473expa 1262 . . 3 (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ 𝑥 ∈ ℕ0) → ((𝑆𝐼)‘𝑥) ∈ 𝑇)
4948ralrimiva 2965 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → ∀𝑥 ∈ ℕ0 ((𝑆𝐼)‘𝑥) ∈ 𝑇)
50 ffnfv 6344 . 2 ((𝑆𝐼):ℕ0𝑇 ↔ ((𝑆𝐼) Fn ℕ0 ∧ ∀𝑥 ∈ ℕ0 ((𝑆𝐼)‘𝑥) ∈ 𝑇))
5145, 49, 50sylanbrc 697 1 ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼):ℕ0𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  {cab 2612  wral 2912  wrex 2913  Vcvv 3191  wss 3560   class class class wbr 4618  cmpt 4678   Fn wfn 5845  wf 5846  cfv 5850  cle 10020  0cn0 11237  Ringcrg 18463  LIdealclidl 19084  Poly1cpl1 19461  coe1cco1 19462   deg1 cdg1 23713  ldgIdlSeqcldgis 37158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959  ax-addf 9960  ax-mulf 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-ofr 6852  df-om 7014  df-1st 7116  df-2nd 7117  df-supp 7242  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-fsupp 8221  df-sup 8293  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-z 11323  df-dec 11438  df-uz 11632  df-fz 12266  df-fzo 12404  df-seq 12739  df-hash 13055  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-mulr 15871  df-starv 15872  df-sca 15873  df-vsca 15874  df-ip 15875  df-tset 15876  df-ple 15877  df-ds 15880  df-unif 15881  df-0g 16018  df-gsum 16019  df-mre 16162  df-mrc 16163  df-acs 16165  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mhm 17251  df-submnd 17252  df-grp 17341  df-minusg 17342  df-sbg 17343  df-mulg 17457  df-subg 17507  df-ghm 17574  df-cntz 17666  df-cmn 18111  df-abl 18112  df-mgp 18406  df-ur 18418  df-ring 18465  df-cring 18466  df-subrg 18694  df-lmod 18781  df-lss 18847  df-sra 19086  df-rgmod 19087  df-lidl 19088  df-ascl 19228  df-psr 19270  df-mvr 19271  df-mpl 19272  df-opsr 19274  df-psr1 19464  df-vr1 19465  df-ply1 19466  df-coe1 19467  df-cnfld 19661  df-mdeg 23714  df-deg1 23715  df-ldgis 37159
This theorem is referenced by:  hbt  37167
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