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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 4065 | . . . 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 4076 | . 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 2733 | . . . 4 ⊢ (deg1‘𝑅) = (deg1‘𝑅) | |
12 | 8, 9, 10, 11 | hbtlem1 43066 | . . 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 1369 | . 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 43066 | . . 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 1369 | . 2 ⊢ (𝜑 → ((𝑆‘𝐽)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐽 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}) |
17 | 4, 13, 16 | 3sstr4d 4043 | 1 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘𝐽)‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1535 ∈ wcel 2104 {cab 2710 ∃wrex 3066 ⊆ wss 3963 class class class wbr 5149 ‘cfv 6558 ≤ cle 11287 ℕ0cn0 12517 Ringcrg 20236 LIdealclidl 21215 Poly1cpl1 22175 coe1cco1 22176 deg1cdg1 26089 ldgIdlSeqcldgis 43064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5430 ax-un 7747 ax-cnex 11202 ax-1cn 11204 ax-addcl 11206 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-ov 7428 df-om 7881 df-2nd 8008 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-nn 12258 df-n0 12518 df-ldgis 43065 |
This theorem is referenced by: hbt 43073 |
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