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Mirrors > Home > MPE Home > Th. List > mdegpropd | Structured version Visualization version GIF version |
Description: Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
Ref | Expression |
---|---|
mdegpropd.b1 | ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
mdegpropd.b2 | ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) |
mdegpropd.p | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) |
Ref | Expression |
---|---|
mdegpropd | ⊢ (𝜑 → (𝐼 mDeg 𝑅) = (𝐼 mDeg 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdegpropd.b1 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) | |
2 | mdegpropd.b2 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) | |
3 | mdegpropd.p | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) | |
4 | 1, 2, 3 | mplbaspropd 21130 | . . 3 ⊢ (𝜑 → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
5 | 1, 2, 3 | grpidpropd 18106 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝑆)) |
6 | 5 | oveq2d 7218 | . . . . 5 ⊢ (𝜑 → (𝑐 supp (0g‘𝑅)) = (𝑐 supp (0g‘𝑆))) |
7 | 6 | imaeq2d 5918 | . . . 4 ⊢ (𝜑 → ((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑅))) = ((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑆)))) |
8 | 7 | supeq1d 9051 | . . 3 ⊢ (𝜑 → sup(((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑅))), ℝ*, < ) = sup(((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑆))), ℝ*, < )) |
9 | 4, 8 | mpteq12dv 5129 | . 2 ⊢ (𝜑 → (𝑐 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ sup(((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑅))), ℝ*, < )) = (𝑐 ∈ (Base‘(𝐼 mPoly 𝑆)) ↦ sup(((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑆))), ℝ*, < ))) |
10 | eqid 2734 | . . 3 ⊢ (𝐼 mDeg 𝑅) = (𝐼 mDeg 𝑅) | |
11 | eqid 2734 | . . 3 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
12 | eqid 2734 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅)) | |
13 | eqid 2734 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
14 | eqid 2734 | . . 3 ⊢ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} = {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} | |
15 | eqid 2734 | . . 3 ⊢ (𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) = (𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) | |
16 | 10, 11, 12, 13, 14, 15 | mdegfval 24932 | . 2 ⊢ (𝐼 mDeg 𝑅) = (𝑐 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ sup(((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑅))), ℝ*, < )) |
17 | eqid 2734 | . . 3 ⊢ (𝐼 mDeg 𝑆) = (𝐼 mDeg 𝑆) | |
18 | eqid 2734 | . . 3 ⊢ (𝐼 mPoly 𝑆) = (𝐼 mPoly 𝑆) | |
19 | eqid 2734 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑆)) = (Base‘(𝐼 mPoly 𝑆)) | |
20 | eqid 2734 | . . 3 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
21 | 17, 18, 19, 20, 14, 15 | mdegfval 24932 | . 2 ⊢ (𝐼 mDeg 𝑆) = (𝑐 ∈ (Base‘(𝐼 mPoly 𝑆)) ↦ sup(((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑆))), ℝ*, < )) |
22 | 9, 16, 21 | 3eqtr4g 2799 | 1 ⊢ (𝜑 → (𝐼 mDeg 𝑅) = (𝐼 mDeg 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {crab 3058 ↦ cmpt 5124 ◡ccnv 5539 “ cima 5543 ‘cfv 6369 (class class class)co 7202 supp csupp 7892 ↑m cmap 8497 Fincfn 8615 supcsup 9045 ℝ*cxr 10849 < clt 10850 ℕcn 11813 ℕ0cn0 12073 Basecbs 16684 +gcplusg 16767 0gc0g 16916 Σg cgsu 16917 ℂfldccnfld 20335 mPoly cmpl 20837 mDeg cmdg 24920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-sup 9047 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-sca 16783 df-vsca 16784 df-tset 16786 df-0g 16918 df-psr 20840 df-mpl 20842 df-mdeg 24922 |
This theorem is referenced by: deg1propd 24956 |
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