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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 19428 | . . 3 ⊢ (𝜑 → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
5 | 1, 2, 3 | grpidpropd 17084 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝑆)) |
6 | 5 | oveq2d 6565 | . . . . 5 ⊢ (𝜑 → (𝑐 supp (0g‘𝑅)) = (𝑐 supp (0g‘𝑆))) |
7 | 6 | imaeq2d 5385 | . . . 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 8235 | . . 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 4663 | . 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 2610 | . . 3 ⊢ (𝐼 mDeg 𝑅) = (𝐼 mDeg 𝑅) | |
11 | eqid 2610 | . . 3 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
12 | eqid 2610 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅)) | |
13 | eqid 2610 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
14 | eqid 2610 | . . 3 ⊢ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} = {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} | |
15 | eqid 2610 | . . 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 23626 | . 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 2610 | . . 3 ⊢ (𝐼 mDeg 𝑆) = (𝐼 mDeg 𝑆) | |
18 | eqid 2610 | . . 3 ⊢ (𝐼 mPoly 𝑆) = (𝐼 mPoly 𝑆) | |
19 | eqid 2610 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑆)) = (Base‘(𝐼 mPoly 𝑆)) | |
20 | eqid 2610 | . . 3 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
21 | 17, 18, 19, 20, 14, 15 | mdegfval 23626 | . 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 2669 | 1 ⊢ (𝜑 → (𝐼 mDeg 𝑅) = (𝐼 mDeg 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ↦ cmpt 4643 ◡ccnv 5037 “ cima 5041 ‘cfv 5804 (class class class)co 6549 supp csupp 7182 ↑𝑚 cmap 7744 Fincfn 7841 supcsup 8229 ℝ*cxr 9952 < clt 9953 ℕcn 10897 ℕ0cn0 11169 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Σg cgsu 15924 mPoly cmpl 19174 ℂfldccnfld 19567 mDeg cmdg 23617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-0g 15925 df-psr 19177 df-mpl 19179 df-mdeg 23619 |
This theorem is referenced by: deg1propd 23650 |
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