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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 3982 | . . . 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 3991 | . 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 2798 | . . . 4 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
12 | 8, 9, 10, 11 | hbtlem1 40067 | . . 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 1368 | . 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 40067 | . . 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 1368 | . 2 ⊢ (𝜑 → ((𝑆‘𝐽)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
17 | 4, 13, 16 | 3sstr4d 3962 | 1 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘𝐽)‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2776 ∃wrex 3107 ⊆ wss 3881 class class class wbr 5030 ‘cfv 6324 ≤ cle 10665 ℕ0cn0 11885 Ringcrg 19290 LIdealclidl 19935 Poly1cpl1 20806 coe1cco1 20807 deg1 cdg1 24655 ldgIdlSeqcldgis 40065 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
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 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 df-n0 11886 df-ldgis 40066 |
This theorem is referenced by: hbt 40074 |
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