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Theorem hbtlem6 38400
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 38383 . . . . 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 2765 . . . . 5 (LIdeal‘𝑅) = (LIdeal‘𝑅)
106, 7, 8, 9hbtlem2 38395 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
113, 4, 5, 10syl3anc 1490 . . 3 (𝜑 → ((𝑆𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
12 eqid 2765 . . . 4 (RSpan‘𝑅) = (RSpan‘𝑅)
139, 12lnr2i 38387 . . 3 ((𝑅 ∈ LNoeR ∧ ((𝑆𝐼)‘𝑋) ∈ (LIdeal‘𝑅)) → ∃𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎))
141, 11, 13syl2anc 579 . 2 (𝜑 → ∃𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎))
15 elfpw 8479 . . . . 5 (𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin) ↔ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin))
16 fvex 6392 . . . . . . . . 9 ((coe1𝑏)‘𝑋) ∈ V
17 eqid 2765 . . . . . . . . 9 (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋))
1816, 17fnmpti 6202 . . . . . . . 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 787 . . . . . . . 8 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ((𝑆𝐼)‘𝑋))
21 eqid 2765 . . . . . . . . . . . 12 ( deg1𝑅) = ( deg1𝑅)
226, 7, 8, 21hbtlem1 38394 . . . . . . . . . . 11 ((𝑅 ∈ LNoeR ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))})
231, 4, 5, 22syl3anc 1490 . . . . . . . . . 10 (𝜑 → ((𝑆𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))})
2417rnmpt 5542 . . . . . . . . . . 11 ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑑 ∣ ∃𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1𝑏)‘𝑋)}
25 fveq2 6379 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → (( deg1𝑅)‘𝑐) = (( deg1𝑅)‘𝑏))
2625breq1d 4821 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → ((( deg1𝑅)‘𝑐) ≤ 𝑋 ↔ (( deg1𝑅)‘𝑏) ≤ 𝑋))
2726rexrab 3529 . . . . . . . . . . . 12 (∃𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1𝑏)‘𝑋) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋)))
2827abbii 2882 . . . . . . . . . . 11 {𝑑 ∣ ∃𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1𝑏)‘𝑋)} = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))}
2924, 28eqtri 2787 . . . . . . . . . 10 ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑑 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))}
3023, 29syl6eqr 2817 . . . . . . . . 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 3803 . . . . . . 7 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ran (𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
33 simprr 789 . . . . . . 7 ((𝜑 ∧ (𝑎 ⊆ ((𝑆𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ∈ Fin)
34 fipreima 8483 . . . . . . 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 1490 . . . . . 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 8479 . . . . . . . . . 10 (𝑘 ∈ (𝒫 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∩ Fin) ↔ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin))
37 ssrab2 3849 . . . . . . . . . . . . . . . . 17 {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼
38 sstr2 3770 . . . . . . . . . . . . . . . . 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 4324 . . . . . . . . . . . . . . 15 (𝑘 ∈ 𝒫 𝐼𝑘𝐼)
4240, 41sylibr 225 . . . . . . . . . . . . . 14 ((𝜑𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}) → 𝑘 ∈ 𝒫 𝐼)
4342adantrr 708 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ 𝒫 𝐼)
44 simprr 789 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ Fin)
4543, 44elind 3962 . . . . . . . . . . . 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 19905 . . . . . . . . . . . . . . . . 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 787 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋})
5150, 37syl6ss 3775 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘𝐼)
52 eqid 2765 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑃) = (Base‘𝑃)
5352, 7lidlss 19498 . . . . . . . . . . . . . . . . . 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 3773 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ (Base‘𝑃))
57 hbtlem6.n . . . . . . . . . . . . . . . 16 𝑁 = (RSpan‘𝑃)
5857, 52, 7rspcl 19510 . . . . . . . . . . . . . . 15 ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → (𝑁𝑘) ∈ 𝑈)
5949, 56, 58syl2anc 579 . . . . . . . . . . . . . 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 38395 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑁𝑘) ∈ 𝑈𝑋 ∈ ℕ0) → ((𝑆‘(𝑁𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
6246, 59, 60, 61syl3anc 1490 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
63 df-ima 5292 . . . . . . . . . . . . . . 15 ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = ran ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘)
6457, 52rspssid 19511 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → 𝑘 ⊆ (𝑁𝑘))
6549, 56, 64syl2anc 579 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ (𝑁𝑘))
66 ssrab 3842 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘𝐼 ∧ ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋))
6766simprbi 490 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋)
6867ad2antrl 719 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋)
69 ssrab 3842 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ⊆ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘 ⊆ (𝑁𝑘) ∧ ∀𝑐𝑘 (( deg1𝑅)‘𝑐) ≤ 𝑋))
7065, 68, 69sylanbrc 578 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋})
7170resmptd 5631 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) = (𝑏𝑘 ↦ ((coe1𝑏)‘𝑋)))
72 resmpt 5628 . . . . . . . . . . . . . . . . . . 19 (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) = (𝑏𝑘 ↦ ((coe1𝑏)‘𝑋)))
7372ad2antrl 719 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) = (𝑏𝑘 ↦ ((coe1𝑏)‘𝑋)))
7471, 73eqtr4d 2802 . . . . . . . . . . . . . . . . 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 5599 . . . . . . . . . . . . . . . . 17 ((𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) ↾ 𝑘) ⊆ (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋))
7674, 75syl6eqssr 3818 . . . . . . . . . . . . . . . 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 5524 . . . . . . . . . . . . . . . 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 3811 . . . . . . . . . . . . . 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 38394 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ (𝑁𝑘) ∈ 𝑈𝑋 ∈ ℕ0) → ((𝑆‘(𝑁𝑘))‘𝑋) = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))})
8146, 59, 60, 80syl3anc 1490 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁𝑘))‘𝑋) = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))})
82 eqid 2765 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋))
8382rnmpt 5542 . . . . . . . . . . . . . . . 16 ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑒 ∣ ∃𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1𝑏)‘𝑋)}
8426rexrab 3529 . . . . . . . . . . . . . . . . 17 (∃𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1𝑏)‘𝑋) ↔ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋)))
8584abbii 2882 . . . . . . . . . . . . . . . 16 {𝑒 ∣ ∃𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1𝑏)‘𝑋)} = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))}
8683, 85eqtri 2787 . . . . . . . . . . . . . . 15 ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) = {𝑒 ∣ ∃𝑏 ∈ (𝑁𝑘)((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))}
8781, 86syl6eqr 2817 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁𝑘))‘𝑋) = ran (𝑏 ∈ {𝑐 ∈ (𝑁𝑘) ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)))
8879, 87sseqtr4d 3804 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
8912, 9rspssp 19514 . . . . . . . . . . . . 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 1490 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ⊆ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘)) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
9145, 90jca 507 . . . . . . . . . . 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 6379 . . . . . . . . . . . . 13 (((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘) = 𝑎 → ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐𝐼 ∣ (( deg1𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1𝑏)‘𝑋)) “ 𝑘)) = ((RSpan‘𝑅)‘𝑎))
9392sseq1d 3794 . . . . . . . . . . . 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 622 . . . . . . . . . . 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 236 . . . . . . . . . 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 587 . . . . . . . . 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 445 . . . . . . . 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 3160 . . . . . 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 587 . . . 4 ((𝜑𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))
102 sseq1 3788 . . . . 5 (((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → (((𝑆𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋) ↔ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)))
103102rexbidv 3199 . . . 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 238 . . 3 ((𝜑𝑎 ∈ (𝒫 ((𝑆𝐼)‘𝑋) ∩ Fin)) → (((𝑆𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋)))
105104rexlimdva 3178 . 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 1652  wcel 2155  {cab 2751  wral 3055  wrex 3056  {crab 3059  cin 3733  wss 3734  𝒫 cpw 4317   class class class wbr 4811  cmpt 4890  ran crn 5280  cres 5281  cima 5282   Fn wfn 6065  cfv 6070  Fincfn 8164  cle 10333  0cn0 11542  Basecbs 16144  Ringcrg 18828  LIdealclidl 19458  RSpancrsp 19459  Poly1cpl1 19834  coe1cco1 19835   deg1 cdg1 24119  LNoeRclnr 38380  ldgIdlSeqcldgis 38392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271  ax-addf 10272  ax-mulf 10273
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-ofr 7100  df-om 7268  df-1st 7370  df-2nd 7371  df-supp 7502  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-er 7951  df-map 8066  df-pm 8067  df-ixp 8118  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-fsupp 8487  df-sup 8559  df-oi 8626  df-card 9020  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-nn 11279  df-2 11339  df-3 11340  df-4 11341  df-5 11342  df-6 11343  df-7 11344  df-8 11345  df-9 11346  df-n0 11543  df-z 11629  df-dec 11746  df-uz 11892  df-fz 12539  df-fzo 12679  df-seq 13014  df-hash 13327  df-struct 16146  df-ndx 16147  df-slot 16148  df-base 16150  df-sets 16151  df-ress 16152  df-plusg 16241  df-mulr 16242  df-starv 16243  df-sca 16244  df-vsca 16245  df-ip 16246  df-tset 16247  df-ple 16248  df-ds 16250  df-unif 16251  df-0g 16382  df-gsum 16383  df-mre 16526  df-mrc 16527  df-acs 16529  df-mgm 17522  df-sgrp 17564  df-mnd 17575  df-mhm 17615  df-submnd 17616  df-grp 17706  df-minusg 17707  df-sbg 17708  df-mulg 17822  df-subg 17869  df-ghm 17936  df-cntz 18027  df-cmn 18475  df-abl 18476  df-mgp 18771  df-ur 18783  df-ring 18830  df-cring 18831  df-subrg 19061  df-lmod 19148  df-lss 19216  df-lsp 19258  df-sra 19460  df-rgmod 19461  df-lidl 19462  df-rsp 19463  df-ascl 19602  df-psr 19644  df-mvr 19645  df-mpl 19646  df-opsr 19648  df-psr1 19837  df-vr1 19838  df-ply1 19839  df-coe1 19840  df-cnfld 20034  df-mdeg 24120  df-deg1 24121  df-lfig 38339  df-lnm 38347  df-lnr 38381  df-ldgis 38393
This theorem is referenced by:  hbt  38401
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