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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hbtlem3 | Structured version Visualization version GIF version |
Description: The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Ref | Expression |
---|---|
hbtlem.p | ⊢ 𝑃 = (Poly1‘𝑅) |
hbtlem.u | ⊢ 𝑈 = (LIdeal‘𝑃) |
hbtlem.s | ⊢ 𝑆 = (ldgIdlSeq‘𝑅) |
hbtlem3.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
hbtlem3.i | ⊢ (𝜑 → 𝐼 ∈ 𝑈) |
hbtlem3.j | ⊢ (𝜑 → 𝐽 ∈ 𝑈) |
hbtlem3.ij | ⊢ (𝜑 → 𝐼 ⊆ 𝐽) |
hbtlem3.x | ⊢ (𝜑 → 𝑋 ∈ ℕ0) |
Ref | Expression |
---|---|
hbtlem3 | ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘𝐽)‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbtlem3.ij | . . . 4 ⊢ (𝜑 → 𝐼 ⊆ 𝐽) | |
2 | ssrexv 4052 | . . . 4 ⊢ (𝐼 ⊆ 𝐽 → (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) → ∃𝑏 ∈ 𝐽 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) → ∃𝑏 ∈ 𝐽 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)))) |
4 | 3 | ss2abdv 4065 | . 2 ⊢ (𝜑 → {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ⊆ {𝑎 ∣ ∃𝑏 ∈ 𝐽 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
5 | hbtlem3.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
6 | hbtlem3.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑈) | |
7 | hbtlem3.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℕ0) | |
8 | hbtlem.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
9 | hbtlem.u | . . . 4 ⊢ 𝑈 = (LIdeal‘𝑃) | |
10 | hbtlem.s | . . . 4 ⊢ 𝑆 = (ldgIdlSeq‘𝑅) | |
11 | eqid 2736 | . . . 4 ⊢ (deg1‘𝑅) = (deg1‘𝑅) | |
12 | 8, 9, 10, 11 | hbtlem1 43113 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
13 | 5, 6, 7, 12 | syl3anc 1373 | . 2 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
14 | hbtlem3.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑈) | |
15 | 8, 9, 10, 11 | hbtlem1 43113 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐽 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐽)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐽 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
16 | 5, 14, 7, 15 | syl3anc 1373 | . 2 ⊢ (𝜑 → ((𝑆‘𝐽)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐽 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
17 | 4, 13, 16 | 3sstr4d 4038 | 1 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘𝐽)‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2713 ∃wrex 3069 ⊆ wss 3950 class class class wbr 5141 ‘cfv 6559 ≤ cle 11292 ℕ0cn0 12522 Ringcrg 20226 LIdealclidl 21208 Poly1cpl1 22168 coe1cco1 22169 deg1cdg1 26083 ldgIdlSeqcldgis 43111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-1cn 11209 ax-addcl 11211 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-ov 7432 df-om 7884 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-nn 12263 df-n0 12523 df-ldgis 43112 |
This theorem is referenced by: hbt 43120 |
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