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Theorem hbtlem6 38201
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 38184 . . . . 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 2760 . . . . 5 (LIdeal‘𝑅) = (LIdeal‘𝑅)
106, 7, 8, 9hbtlem2 38196 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
113, 4, 5, 10syl3anc 1477 . . 3 (𝜑 → ((𝑆𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
12 eqid 2760 . . . 4 (RSpan‘𝑅) = (RSpan‘𝑅)
139, 12lnr2i 38188 . . 3 ((𝑅 ∈ LNoeR ∧ ((𝑆𝐼)‘𝑋) ∈ (LIdeal‘𝑅)) → ∃𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎))
141, 11, 13syl2anc 696 . 2 (𝜑 → ∃𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎))
15 elfpw 8433 . . . . 5 (𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin) ↔ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin))
16 fvex 6362 . . . . . . . . 9 ((coe1𝑏)‘𝑋) ∈ V
17 eqid 2760 . . . . . . . . 9 (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋))
1816, 17fnmpti 6183 . . . . . . . 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 811 . . . . . . . 8 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ((𝑆𝐼)‘𝑋))
21 eqid 2760 . . . . . . . . . . . 12 ( deg1𝑅) = ( deg1𝑅)
226, 7, 8, 21hbtlem1 38195 . . . . . . . . . . 11 ((𝑅 ∈ LNoeR ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))})
231, 4, 5, 22syl3anc 1477 . . . . . . . . . 10 (𝜑 → ((𝑆𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))})
2417rnmpt 5526 . . . . . . . . . . 11 ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑑 ∣ ∃𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1𝑏)‘𝑋)}
25 fveq2 6352 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → (( deg1𝑅)‘𝑐) = (( deg1𝑅)‘𝑏))
2625breq1d 4814 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → ((( deg1𝑅)‘𝑐) ≤ 𝑋 ↔ (( deg1𝑅)‘𝑏) ≤ 𝑋))
2726rexrab 3511 . . . . . . . . . . . 12 (∃𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1𝑏)‘𝑋) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋)))
2827abbii 2877 . . . . . . . . . . 11 {𝑑 ∣ ∃𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1𝑏)‘𝑋)} = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))}
2924, 28eqtri 2782 . . . . . . . . . 10 ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))}
3023, 29syl6eqr 2812 . . . . . . . . 9 (𝜑 → ((𝑆𝐼)‘𝑋) = ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
3130adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → ((𝑆𝐼)‘𝑋) = ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
3220, 31sseqtrd 3782 . . . . . . 7 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
33 simprr 813 . . . . . . 7 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ∈ Fin)
34 fipreima 8437 . . . . . . 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 1477 . . . . . 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 8433 . . . . . . . . . 10 (𝑘 ∈ (𝒫 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∩ Fin) ↔ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin))
37 ssrab2 3828 . . . . . . . . . . . . . . . . 17 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼
38 sstr2 3751 . . . . . . . . . . . . . . . . 17 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → ({𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼𝑘𝐼))
3937, 38mpi 20 . . . . . . . . . . . . . . . 16 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → 𝑘𝐼)
4039adantl 473 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}) → 𝑘𝐼)
41 selpw 4309 . . . . . . . . . . . . . . 15 (𝑘 ∈ 𝒫 𝐼𝑘𝐼)
4240, 41sylibr 224 . . . . . . . . . . . . . 14 ((𝜑𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}) → 𝑘 ∈ 𝒫 𝐼)
4342adantrr 755 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ 𝒫 𝐼)
44 simprr 813 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ Fin)
4543, 44elind 3941 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ (𝒫 𝐼 ∩ Fin))
463adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑅 ∈ Ring)
476ply1ring 19820 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
483, 47syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ Ring)
4948adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑃 ∈ Ring)
50 simprl 811 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋})
5150, 37syl6ss 3756 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘𝐼)
52 eqid 2760 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑃) = (Base‘𝑃)
5352, 7lidlss 19412 . . . . . . . . . . . . . . . . . 18 (𝐼𝑈𝐼 ⊆ (Base‘𝑃))
544, 53syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝐼 ⊆ (Base‘𝑃))
5554adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝐼 ⊆ (Base‘𝑃))
5651, 55sstrd 3754 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ (Base‘𝑃))
57 hbtlem6.n . . . . . . . . . . . . . . . 16 𝑁 = (RSpan‘𝑃)
5857, 52, 7rspcl 19424 . . . . . . . . . . . . . . 15 ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → (𝑁𝑘) ∈ 𝑈)
5949, 56, 58syl2anc 696 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → (𝑁𝑘) ∈ 𝑈)
605adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑋 ∈ ℕ0)
616, 7, 8, 9hbtlem2 38196 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑁𝑘) ∈ 𝑈𝑋 ∈ ℕ0) → ((𝑆‘(𝑁𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
6246, 59, 60, 61syl3anc 1477 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
63 df-ima 5279 . . . . . . . . . . . . . . 15 ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = ran ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘)
6457, 52rspssid 19425 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → 𝑘 ⊆ (𝑁𝑘))
6549, 56, 64syl2anc 696 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ (𝑁𝑘))
66 ssrab 3821 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘𝐼 ∧ ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋))
6766simprbi 483 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋)
6867ad2antrl 766 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋)
69 ssrab 3821 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ⊆ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘 ⊆ (𝑁𝑘) ∧ ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋))
7065, 68, 69sylanbrc 701 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋})
7170resmptd 5610 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) = (𝑏𝑘 ↦ ((coe1𝑏)‘𝑋)))
72 resmpt 5607 . . . . . . . . . . . . . . . . . . 19 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) = (𝑏𝑘 ↦ ((coe1𝑏)‘𝑋)))
7372ad2antrl 766 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) = (𝑏𝑘 ↦ ((coe1𝑏)‘𝑋)))
7471, 73eqtr4d 2797 . . . . . . . . . . . . . . . . 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 5580 . . . . . . . . . . . . . . . . 17 ((𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) ⊆ (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋))
7674, 75syl6eqssr 3797 . . . . . . . . . . . . . . . 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 5509 . . . . . . . . . . . . . . . 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 3790 . . . . . . . . . . . . . 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 38195 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ (𝑁𝑘) ∈ 𝑈𝑋 ∈ ℕ0) → ((𝑆‘(𝑁𝑘))‘𝑋) = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))})
8146, 59, 60, 80syl3anc 1477 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁𝑘))‘𝑋) = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))})
82 eqid 2760 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋))
8382rnmpt 5526 . . . . . . . . . . . . . . . 16 ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑒 ∣ ∃𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1𝑏)‘𝑋)}
8426rexrab 3511 . . . . . . . . . . . . . . . . 17 (∃𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1𝑏)‘𝑋) ↔ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋)))
8584abbii 2877 . . . . . . . . . . . . . . . 16 {𝑒 ∣ ∃𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1𝑏)‘𝑋)} = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))}
8683, 85eqtri 2782 . . . . . . . . . . . . . . 15 ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))}
8781, 86syl6eqr 2812 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁𝑘))‘𝑋) = ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
8879, 87sseqtr4d 3783 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
8912, 9rspssp 19428 . . . . . . . . . . . . 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 1477 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘)) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
9145, 90jca 555 . . . . . . . . . . 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 6352 . . . . . . . . . . . . 13 (((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = 𝑎 → ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘)) = ((RSpan‘𝑅)‘𝑎))
9392sseq1d 3773 . . . . . . . . . . . 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 742 . . . . . . . . . . 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 493 . . . . . . . . 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 630 . . . . . . . 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 472 . . . . . . 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 3152 . . . . . 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 493 . . . 4 ((𝜑𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
102 sseq1 3767 . . . . 5 (((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → (((𝑆𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋) ↔ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)))
103102rexbidv 3190 . . . 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 3169 . 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 383   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wrex 3051  {crab 3054  cin 3714  wss 3715  𝒫 cpw 4302   class class class wbr 4804  cmpt 4881  ran crn 5267  cres 5268  cima 5269   Fn wfn 6044  cfv 6049  Fincfn 8121  cle 10267  0cn0 11484  Basecbs 16059  Ringcrg 18747  LIdealclidl 19372  RSpancrsp 19373  Poly1cpl1 19749  coe1cco1 19750   deg1 cdg1 24013  LNoeRclnr 38181  ldgIdlSeqcldgis 38193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-addf 10207  ax-mulf 10208
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-ofr 7063  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-sup 8513  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-fzo 12660  df-seq 12996  df-hash 13312  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-starv 16158  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-unif 16167  df-0g 16304  df-gsum 16305  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-mulg 17742  df-subg 17792  df-ghm 17859  df-cntz 17950  df-cmn 18395  df-abl 18396  df-mgp 18690  df-ur 18702  df-ring 18749  df-cring 18750  df-subrg 18980  df-lmod 19067  df-lss 19135  df-lsp 19174  df-sra 19374  df-rgmod 19375  df-lidl 19376  df-rsp 19377  df-ascl 19516  df-psr 19558  df-mvr 19559  df-mpl 19560  df-opsr 19562  df-psr1 19752  df-vr1 19753  df-ply1 19754  df-coe1 19755  df-cnfld 19949  df-mdeg 24014  df-deg1 24015  df-lfig 38140  df-lnm 38148  df-lnr 38182  df-ldgis 38194
This theorem is referenced by:  hbt  38202
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