Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mdegxrf | Structured version Visualization version GIF version |
Description: Functionality of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
Ref | Expression |
---|---|
mdegxrcl.d | ⊢ 𝐷 = (𝐼 mDeg 𝑅) |
mdegxrcl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mdegxrcl.b | ⊢ 𝐵 = (Base‘𝑃) |
Ref | Expression |
---|---|
mdegxrf | ⊢ 𝐷:𝐵⟶ℝ* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltso 12537 | . . . 4 ⊢ < Or ℝ* | |
2 | 1 | supex 8930 | . . 3 ⊢ sup(((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ (𝑧 supp (0g‘𝑅))), ℝ*, < ) ∈ V |
3 | mdegxrcl.d | . . . 4 ⊢ 𝐷 = (𝐼 mDeg 𝑅) | |
4 | mdegxrcl.p | . . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
5 | mdegxrcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
6 | eqid 2824 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | eqid 2824 | . . . 4 ⊢ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} = {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} | |
8 | eqid 2824 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) = (𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) | |
9 | 3, 4, 5, 6, 7, 8 | mdegfval 24659 | . . 3 ⊢ 𝐷 = (𝑧 ∈ 𝐵 ↦ sup(((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ (𝑧 supp (0g‘𝑅))), ℝ*, < )) |
10 | 2, 9 | fnmpti 6494 | . 2 ⊢ 𝐷 Fn 𝐵 |
11 | 3, 4, 5 | mdegxrcl 24664 | . . 3 ⊢ (𝑓 ∈ 𝐵 → (𝐷‘𝑓) ∈ ℝ*) |
12 | 11 | rgen 3151 | . 2 ⊢ ∀𝑓 ∈ 𝐵 (𝐷‘𝑓) ∈ ℝ* |
13 | ffnfv 6885 | . 2 ⊢ (𝐷:𝐵⟶ℝ* ↔ (𝐷 Fn 𝐵 ∧ ∀𝑓 ∈ 𝐵 (𝐷‘𝑓) ∈ ℝ*)) | |
14 | 10, 12, 13 | mpbir2an 709 | 1 ⊢ 𝐷:𝐵⟶ℝ* |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 ∀wral 3141 {crab 3145 ↦ cmpt 5149 ◡ccnv 5557 “ cima 5561 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 supp csupp 7833 ↑m cmap 8409 Fincfn 8512 supcsup 8907 ℝ*cxr 10677 < clt 10678 ℕcn 11641 ℕ0cn0 11900 Basecbs 16486 0gc0g 16716 Σg cgsu 16717 mPoly cmpl 20136 ℂfldccnfld 20548 mDeg cmdg 24650 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-0g 16718 df-gsum 16719 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-cntz 18450 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-psr 20139 df-mpl 20141 df-cnfld 20549 df-mdeg 24652 |
This theorem is referenced by: deg1xrf 24678 |
Copyright terms: Public domain | W3C validator |