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Theorem hbtlem6 37207
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 37190 . . . . 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 2621 . . . . 5 (LIdeal‘𝑅) = (LIdeal‘𝑅)
106, 7, 8, 9hbtlem2 37202 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
113, 4, 5, 10syl3anc 1323 . . 3 (𝜑 → ((𝑆𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
12 eqid 2621 . . . 4 (RSpan‘𝑅) = (RSpan‘𝑅)
139, 12lnr2i 37194 . . 3 ((𝑅 ∈ LNoeR ∧ ((𝑆𝐼)‘𝑋) ∈ (LIdeal‘𝑅)) → ∃𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎))
141, 11, 13syl2anc 692 . 2 (𝜑 → ∃𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎))
15 elfpw 8219 . . . . 5 (𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin) ↔ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin))
16 fvex 6163 . . . . . . . . 9 ((coe1𝑏)‘𝑋) ∈ V
17 eqid 2621 . . . . . . . . 9 (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋))
1816, 17fnmpti 5984 . . . . . . . 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 793 . . . . . . . 8 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ((𝑆𝐼)‘𝑋))
21 eqid 2621 . . . . . . . . . . . 12 ( deg1𝑅) = ( deg1𝑅)
226, 7, 8, 21hbtlem1 37201 . . . . . . . . . . 11 ((𝑅 ∈ LNoeR ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))})
231, 4, 5, 22syl3anc 1323 . . . . . . . . . 10 (𝜑 → ((𝑆𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))})
2417rnmpt 5336 . . . . . . . . . . 11 ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑑 ∣ ∃𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1𝑏)‘𝑋)}
25 fveq2 6153 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → (( deg1𝑅)‘𝑐) = (( deg1𝑅)‘𝑏))
2625breq1d 4628 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → ((( deg1𝑅)‘𝑐) ≤ 𝑋 ↔ (( deg1𝑅)‘𝑏) ≤ 𝑋))
2726rexrab 3356 . . . . . . . . . . . 12 (∃𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1𝑏)‘𝑋) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋)))
2827abbii 2736 . . . . . . . . . . 11 {𝑑 ∣ ∃𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1𝑏)‘𝑋)} = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))}
2924, 28eqtri 2643 . . . . . . . . . 10 ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))}
3023, 29syl6eqr 2673 . . . . . . . . 9 (𝜑 → ((𝑆𝐼)‘𝑋) = ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
3130adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → ((𝑆𝐼)‘𝑋) = ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
3220, 31sseqtrd 3625 . . . . . . 7 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
33 simprr 795 . . . . . . 7 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ∈ Fin)
34 fipreima 8223 . . . . . . 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 1323 . . . . . 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 8219 . . . . . . . . . 10 (𝑘 ∈ (𝒫 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∩ Fin) ↔ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin))
37 ssrab2 3671 . . . . . . . . . . . . . . . . 17 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼
38 sstr2 3594 . . . . . . . . . . . . . . . . 17 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → ({𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼𝑘𝐼))
3937, 38mpi 20 . . . . . . . . . . . . . . . 16 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → 𝑘𝐼)
4039adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}) → 𝑘𝐼)
41 selpw 4142 . . . . . . . . . . . . . . 15 (𝑘 ∈ 𝒫 𝐼𝑘𝐼)
4240, 41sylibr 224 . . . . . . . . . . . . . 14 ((𝜑𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}) → 𝑘 ∈ 𝒫 𝐼)
4342adantrr 752 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ 𝒫 𝐼)
44 simprr 795 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ Fin)
4543, 44elind 3781 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ (𝒫 𝐼 ∩ Fin))
463adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑅 ∈ Ring)
476ply1ring 19546 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
483, 47syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ Ring)
4948adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑃 ∈ Ring)
50 simprl 793 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋})
5150, 37syl6ss 3599 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘𝐼)
52 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑃) = (Base‘𝑃)
5352, 7lidlss 19138 . . . . . . . . . . . . . . . . . 18 (𝐼𝑈𝐼 ⊆ (Base‘𝑃))
544, 53syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝐼 ⊆ (Base‘𝑃))
5554adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝐼 ⊆ (Base‘𝑃))
5651, 55sstrd 3597 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ (Base‘𝑃))
57 hbtlem6.n . . . . . . . . . . . . . . . 16 𝑁 = (RSpan‘𝑃)
5857, 52, 7rspcl 19150 . . . . . . . . . . . . . . 15 ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → (𝑁𝑘) ∈ 𝑈)
5949, 56, 58syl2anc 692 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → (𝑁𝑘) ∈ 𝑈)
605adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑋 ∈ ℕ0)
616, 7, 8, 9hbtlem2 37202 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑁𝑘) ∈ 𝑈𝑋 ∈ ℕ0) → ((𝑆‘(𝑁𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
6246, 59, 60, 61syl3anc 1323 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
63 df-ima 5092 . . . . . . . . . . . . . . 15 ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = ran ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘)
6457, 52rspssid 19151 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → 𝑘 ⊆ (𝑁𝑘))
6549, 56, 64syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ (𝑁𝑘))
66 ssrab 3664 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘𝐼 ∧ ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋))
6766simprbi 480 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋)
6867ad2antrl 763 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋)
69 ssrab 3664 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ⊆ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘 ⊆ (𝑁𝑘) ∧ ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋))
7065, 68, 69sylanbrc 697 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋})
7170resmptd 5416 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) = (𝑏𝑘 ↦ ((coe1𝑏)‘𝑋)))
72 resmpt 5413 . . . . . . . . . . . . . . . . . . 19 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) = (𝑏𝑘 ↦ ((coe1𝑏)‘𝑋)))
7372ad2antrl 763 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) = (𝑏𝑘 ↦ ((coe1𝑏)‘𝑋)))
7471, 73eqtr4d 2658 . . . . . . . . . . . . . . . . 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 5386 . . . . . . . . . . . . . . . . 17 ((𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) ⊆ (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋))
7674, 75syl6eqssr 3640 . . . . . . . . . . . . . . . 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 5319 . . . . . . . . . . . . . . . 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, 78syl5eqss 3633 . . . . . . . . . . . . . 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 37201 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ (𝑁𝑘) ∈ 𝑈𝑋 ∈ ℕ0) → ((𝑆‘(𝑁𝑘))‘𝑋) = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))})
8146, 59, 60, 80syl3anc 1323 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁𝑘))‘𝑋) = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))})
82 eqid 2621 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋))
8382rnmpt 5336 . . . . . . . . . . . . . . . 16 ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑒 ∣ ∃𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1𝑏)‘𝑋)}
8426rexrab 3356 . . . . . . . . . . . . . . . . 17 (∃𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1𝑏)‘𝑋) ↔ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋)))
8584abbii 2736 . . . . . . . . . . . . . . . 16 {𝑒 ∣ ∃𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1𝑏)‘𝑋)} = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))}
8683, 85eqtri 2643 . . . . . . . . . . . . . . 15 ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))}
8781, 86syl6eqr 2673 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁𝑘))‘𝑋) = ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
8879, 87sseqtr4d 3626 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
8912, 9rspssp 19154 . . . . . . . . . . . . 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 1323 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘)) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
9145, 90jca 554 . . . . . . . . . . 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 6153 . . . . . . . . . . . . 13 (((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = 𝑎 → ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘)) = ((RSpan‘𝑅)‘𝑎))
9392sseq1d 3616 . . . . . . . . . . . 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 739 . . . . . . . . . . 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 235 . . . . . . . . . 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 492 . . . . . . . . 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 628 . . . . . . . 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 481 . . . . . . 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 3009 . . . . . 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 492 . . . 4 ((𝜑𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
102 sseq1 3610 . . . . 5 (((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → (((𝑆𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋) ↔ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)))
103102rexbidv 3046 . . . 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 237 . . 3 ((𝜑𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)) → (((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)))
105104rexlimdva 3025 . 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 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  {crab 2911  cin 3558  wss 3559  𝒫 cpw 4135   class class class wbr 4618  cmpt 4678  ran crn 5080  cres 5081  cima 5082   Fn wfn 5847  cfv 5852  Fincfn 7906  cle 10026  0cn0 11243  Basecbs 15788  Ringcrg 18475  LIdealclidl 19098  RSpancrsp 19099  Poly1cpl1 19475  coe1cco1 19476   deg1 cdg1 23731  LNoeRclnr 37187  ldgIdlSeqcldgis 37199
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965  ax-addf 9966  ax-mulf 9967
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  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 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-ofr 6858  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7860  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-fsupp 8227  df-sup 8299  df-oi 8366  df-card 8716  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-3 11031  df-4 11032  df-5 11033  df-6 11034  df-7 11035  df-8 11036  df-9 11037  df-n0 11244  df-z 11329  df-dec 11445  df-uz 11639  df-fz 12276  df-fzo 12414  df-seq 12749  df-hash 13065  df-struct 15790  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-mulr 15883  df-starv 15884  df-sca 15885  df-vsca 15886  df-ip 15887  df-tset 15888  df-ple 15889  df-ds 15892  df-unif 15893  df-0g 16030  df-gsum 16031  df-mre 16174  df-mrc 16175  df-acs 16177  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-mhm 17263  df-submnd 17264  df-grp 17353  df-minusg 17354  df-sbg 17355  df-mulg 17469  df-subg 17519  df-ghm 17586  df-cntz 17678  df-cmn 18123  df-abl 18124  df-mgp 18418  df-ur 18430  df-ring 18477  df-cring 18478  df-subrg 18706  df-lmod 18793  df-lss 18861  df-lsp 18900  df-sra 19100  df-rgmod 19101  df-lidl 19102  df-rsp 19103  df-ascl 19242  df-psr 19284  df-mvr 19285  df-mpl 19286  df-opsr 19288  df-psr1 19478  df-vr1 19479  df-ply1 19480  df-coe1 19481  df-cnfld 19675  df-mdeg 23732  df-deg1 23733  df-lfig 37145  df-lnm 37153  df-lnr 37188  df-ldgis 37200
This theorem is referenced by:  hbt  37208
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