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Theorem hbtlem6 40060
Description: There is a finite set of polynomials matching any single stage of the image. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Hypotheses
Ref Expression
hbtlem.p 𝑃 = (Poly1𝑅)
hbtlem.u 𝑈 = (LIdeal‘𝑃)
hbtlem.s 𝑆 = (ldgIdlSeq‘𝑅)
hbtlem6.n 𝑁 = (RSpan‘𝑃)
hbtlem6.r (𝜑𝑅 ∈ LNoeR)
hbtlem6.i (𝜑𝐼𝑈)
hbtlem6.x (𝜑𝑋 ∈ ℕ0)
Assertion
Ref Expression
hbtlem6 (𝜑 → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
Distinct variable groups:   𝜑,𝑘   𝑘,𝐼   𝑅,𝑘   𝑆,𝑘   𝑘,𝑋
Allowed substitution hints:   𝑃(𝑘)   𝑈(𝑘)   𝑁(𝑘)

Proof of Theorem hbtlem6
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbtlem6.r . . 3 (𝜑𝑅 ∈ LNoeR)
2 lnrring 40043 . . . . 5 (𝑅 ∈ LNoeR → 𝑅 ∈ Ring)
31, 2syl 17 . . . 4 (𝜑𝑅 ∈ Ring)
4 hbtlem6.i . . . 4 (𝜑𝐼𝑈)
5 hbtlem6.x . . . 4 (𝜑𝑋 ∈ ℕ0)
6 hbtlem.p . . . . 5 𝑃 = (Poly1𝑅)
7 hbtlem.u . . . . 5 𝑈 = (LIdeal‘𝑃)
8 hbtlem.s . . . . 5 𝑆 = (ldgIdlSeq‘𝑅)
9 eqid 2801 . . . . 5 (LIdeal‘𝑅) = (LIdeal‘𝑅)
106, 7, 8, 9hbtlem2 40055 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
113, 4, 5, 10syl3anc 1368 . . 3 (𝜑 → ((𝑆𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
12 eqid 2801 . . . 4 (RSpan‘𝑅) = (RSpan‘𝑅)
139, 12lnr2i 40047 . . 3 ((𝑅 ∈ LNoeR ∧ ((𝑆𝐼)‘𝑋) ∈ (LIdeal‘𝑅)) → ∃𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎))
141, 11, 13syl2anc 587 . 2 (𝜑 → ∃𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎))
15 elfpw 8814 . . . . 5 (𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin) ↔ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin))
16 fvex 6662 . . . . . . . . 9 ((coe1𝑏)‘𝑋) ∈ V
17 eqid 2801 . . . . . . . . 9 (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋))
1816, 17fnmpti 6467 . . . . . . . 8 (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) Fn {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}
1918a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) Fn {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋})
20 simprl 770 . . . . . . . 8 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ((𝑆𝐼)‘𝑋))
21 eqid 2801 . . . . . . . . . . . 12 ( deg1𝑅) = ( deg1𝑅)
226, 7, 8, 21hbtlem1 40054 . . . . . . . . . . 11 ((𝑅 ∈ LNoeR ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))})
231, 4, 5, 22syl3anc 1368 . . . . . . . . . 10 (𝜑 → ((𝑆𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))})
2417rnmpt 5795 . . . . . . . . . . 11 ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑑 ∣ ∃𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1𝑏)‘𝑋)}
25 fveq2 6649 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → (( deg1𝑅)‘𝑐) = (( deg1𝑅)‘𝑏))
2625breq1d 5043 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → ((( deg1𝑅)‘𝑐) ≤ 𝑋 ↔ (( deg1𝑅)‘𝑏) ≤ 𝑋))
2726rexrab 3638 . . . . . . . . . . . 12 (∃𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1𝑏)‘𝑋) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋)))
2827abbii 2866 . . . . . . . . . . 11 {𝑑 ∣ ∃𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1𝑏)‘𝑋)} = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))}
2924, 28eqtri 2824 . . . . . . . . . 10 ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))}
3023, 29eqtr4di 2854 . . . . . . . . 9 (𝜑 → ((𝑆𝐼)‘𝑋) = ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
3130adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → ((𝑆𝐼)‘𝑋) = ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
3220, 31sseqtrd 3958 . . . . . . 7 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
33 simprr 772 . . . . . . 7 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ∈ Fin)
34 fipreima 8818 . . . . . . 7 (((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) Fn {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑎 ⊆ ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ∧ 𝑎 ∈ Fin) → ∃𝑘 ∈ (𝒫 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∩ Fin)((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = 𝑎)
3519, 32, 33, 34syl3anc 1368 . . . . . 6 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → ∃𝑘 ∈ (𝒫 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∩ Fin)((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = 𝑎)
36 elfpw 8814 . . . . . . . . . 10 (𝑘 ∈ (𝒫 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∩ Fin) ↔ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin))
37 ssrab2 4010 . . . . . . . . . . . . . . . . 17 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼
38 sstr2 3925 . . . . . . . . . . . . . . . . 17 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → ({𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼𝑘𝐼))
3937, 38mpi 20 . . . . . . . . . . . . . . . 16 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → 𝑘𝐼)
4039adantl 485 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}) → 𝑘𝐼)
41 velpw 4505 . . . . . . . . . . . . . . 15 (𝑘 ∈ 𝒫 𝐼𝑘𝐼)
4240, 41sylibr 237 . . . . . . . . . . . . . 14 ((𝜑𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}) → 𝑘 ∈ 𝒫 𝐼)
4342adantrr 716 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ 𝒫 𝐼)
44 simprr 772 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ Fin)
4543, 44elind 4124 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ (𝒫 𝐼 ∩ Fin))
463adantr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑅 ∈ Ring)
476ply1ring 20880 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
483, 47syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ Ring)
4948adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑃 ∈ Ring)
50 simprl 770 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋})
5150, 37sstrdi 3930 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘𝐼)
52 eqid 2801 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑃) = (Base‘𝑃)
5352, 7lidlss 19979 . . . . . . . . . . . . . . . . . 18 (𝐼𝑈𝐼 ⊆ (Base‘𝑃))
544, 53syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝐼 ⊆ (Base‘𝑃))
5554adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝐼 ⊆ (Base‘𝑃))
5651, 55sstrd 3928 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ (Base‘𝑃))
57 hbtlem6.n . . . . . . . . . . . . . . . 16 𝑁 = (RSpan‘𝑃)
5857, 52, 7rspcl 19991 . . . . . . . . . . . . . . 15 ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → (𝑁𝑘) ∈ 𝑈)
5949, 56, 58syl2anc 587 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → (𝑁𝑘) ∈ 𝑈)
605adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑋 ∈ ℕ0)
616, 7, 8, 9hbtlem2 40055 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑁𝑘) ∈ 𝑈𝑋 ∈ ℕ0) → ((𝑆‘(𝑁𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
6246, 59, 60, 61syl3anc 1368 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
63 df-ima 5536 . . . . . . . . . . . . . . 15 ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = ran ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘)
6457, 52rspssid 19992 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → 𝑘 ⊆ (𝑁𝑘))
6549, 56, 64syl2anc 587 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ (𝑁𝑘))
66 ssrab 4003 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘𝐼 ∧ ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋))
6766simprbi 500 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋)
6867ad2antrl 727 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋)
69 ssrab 4003 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ⊆ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘 ⊆ (𝑁𝑘) ∧ ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋))
7065, 68, 69sylanbrc 586 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋})
7170resmptd 5879 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) = (𝑏𝑘 ↦ ((coe1𝑏)‘𝑋)))
72 resmpt 5876 . . . . . . . . . . . . . . . . . . 19 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) = (𝑏𝑘 ↦ ((coe1𝑏)‘𝑋)))
7372ad2antrl 727 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) = (𝑏𝑘 ↦ ((coe1𝑏)‘𝑋)))
7471, 73eqtr4d 2839 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) = ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘))
75 resss 5847 . . . . . . . . . . . . . . . . 17 ((𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) ⊆ (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋))
7674, 75eqsstrrdi 3973 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) ⊆ (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
77 rnss 5777 . . . . . . . . . . . . . . . 16 (((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) ⊆ (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) → ran ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) ⊆ ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
7876, 77syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ran ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) ⊆ ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
7963, 78eqsstrid 3966 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) ⊆ ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
806, 7, 8, 21hbtlem1 40054 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ (𝑁𝑘) ∈ 𝑈𝑋 ∈ ℕ0) → ((𝑆‘(𝑁𝑘))‘𝑋) = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))})
8146, 59, 60, 80syl3anc 1368 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁𝑘))‘𝑋) = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))})
82 eqid 2801 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋))
8382rnmpt 5795 . . . . . . . . . . . . . . . 16 ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑒 ∣ ∃𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1𝑏)‘𝑋)}
8426rexrab 3638 . . . . . . . . . . . . . . . . 17 (∃𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1𝑏)‘𝑋) ↔ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋)))
8584abbii 2866 . . . . . . . . . . . . . . . 16 {𝑒 ∣ ∃𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1𝑏)‘𝑋)} = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))}
8683, 85eqtri 2824 . . . . . . . . . . . . . . 15 ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))}
8781, 86eqtr4di 2854 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁𝑘))‘𝑋) = ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
8879, 87sseqtrrd 3959 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
8912, 9rspssp 19995 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ ((𝑆‘(𝑁𝑘))‘𝑋) ∈ (LIdeal‘𝑅) ∧ ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)) → ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘)) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
9046, 62, 88, 89syl3anc 1368 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘)) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
9145, 90jca 515 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → (𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘)) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)))
92 fveq2 6649 . . . . . . . . . . . . 13 (((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = 𝑎 → ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘)) = ((RSpan‘𝑅)‘𝑎))
9392sseq1d 3949 . . . . . . . . . . . 12 (((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = 𝑎 → (((RSpan‘𝑅)‘((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘)) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋) ↔ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)))
9493anbi2d 631 . . . . . . . . . . 11 (((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = 𝑎 → ((𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘)) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)) ↔ (𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))))
9591, 94syl5ibcom 248 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → (((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = 𝑎 → (𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))))
9636, 95sylan2b 596 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝒫 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∩ Fin)) → (((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = 𝑎 → (𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))))
9796expimpd 457 . . . . . . . 8 (𝜑 → ((𝑘 ∈ (𝒫 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∩ Fin) ∧ ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = 𝑎) → (𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))))
9897adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → ((𝑘 ∈ (𝒫 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∩ Fin) ∧ ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = 𝑎) → (𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))))
9998reximdv2 3233 . . . . . 6 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → (∃𝑘 ∈ (𝒫 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∩ Fin)((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = 𝑎 → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)))
10035, 99mpd 15 . . . . 5 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
10115, 100sylan2b 596 . . . 4 ((𝜑𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
102 sseq1 3943 . . . . 5 (((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → (((𝑆𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋) ↔ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)))
103102rexbidv 3259 . . . 4 (((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → (∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋) ↔ ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)))
104101, 103syl5ibrcom 250 . . 3 ((𝜑𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)) → (((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)))
105104rexlimdva 3246 . 2 (𝜑 → (∃𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)))
10614, 105mpd 15 1 (𝜑 → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  {cab 2779  wral 3109  wrex 3110  {crab 3113  cin 3883  wss 3884  𝒫 cpw 4500   class class class wbr 5033  cmpt 5113  ran crn 5524  cres 5525  cima 5526   Fn wfn 6323  cfv 6328  Fincfn 8496  cle 10669  0cn0 11889  Basecbs 16478  Ringcrg 19293  LIdealclidl 19938  RSpancrsp 19939  Poly1cpl1 20809  coe1cco1 20810   deg1 cdg1 24658  LNoeRclnr 40040  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-lsp 19740  df-sra 19940  df-rgmod 19941  df-lidl 19942  df-rsp 19943  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-lfig 39999  df-lnm 40007  df-lnr 40041  df-ldgis 40053
This theorem is referenced by:  hbt  40061
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