mdegxrf - Metamath Proof Explorer

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Theorem mdegxrf 25138

Description: Functionality of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)

**Hypotheses**
Ref	Expression
mdegxrcl.d	⊢ 𝐷 = (𝐼 mDeg 𝑅)
mdegxrcl.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mdegxrcl.b	⊢ 𝐵 = (Base‘𝑃)

**Assertion**
Ref	Expression
mdegxrf	⊢ 𝐷:𝐵⟶ℝ^*

Proof of Theorem mdegxrf Dummy variables 𝑥 𝑦 𝑓 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrltso 12804	. . . 4 ⊢ < Or ℝ^*
2	1	supex 9152	. . 3 ⊢ sup(((𝑦 ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑦)) “ (𝑧 supp (0_g‘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 2738	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
7		eqid 2738	. . . 4 ⊢ {𝑥 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑥 “ ℕ) ∈ Fin} = {𝑥 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑥 “ ℕ) ∈ Fin}
8		eqid 2738	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑦)) = (𝑦 ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑦))
9	3, 4, 5, 6, 7, 8	mdegfval 25132	. . 3 ⊢ 𝐷 = (𝑧 ∈ 𝐵 ↦ sup(((𝑦 ∈ {𝑥 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂ_fld Σ_g 𝑦)) “ (𝑧 supp (0_g‘𝑅))), ℝ^*, < ))
10	2, 9	fnmpti 6560	. 2 ⊢ 𝐷 Fn 𝐵
11	3, 4, 5	mdegxrcl 25137	. . 3 ⊢ (𝑓 ∈ 𝐵 → (𝐷‘𝑓) ∈ ℝ^*)
12	11	rgen 3073	. 2 ⊢ ∀𝑓 ∈ 𝐵 (𝐷‘𝑓) ∈ ℝ^*
13		ffnfv 6974	. 2 ⊢ (𝐷:𝐵⟶ℝ^* ↔ (𝐷 Fn 𝐵 ∧ ∀𝑓 ∈ 𝐵 (𝐷‘𝑓) ∈ ℝ^*))
14	10, 12, 13	mpbir2an 707	1 ⊢ 𝐷:𝐵⟶ℝ^*

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2108 ∀wral 3063 {crab 3067 ↦ cmpt 5153 ^◡ccnv 5579 “ cima 5583 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 supp csupp 7948 ↑_m cmap 8573 Fincfn 8691 supcsup 9129 ℝ^*cxr 10939 < clt 10940 ℕcn 11903 ℕ₀cn0 12163 Basecbs 16840 0_gc0g 17067 Σ_g cgsu 17068 ℂ_fldccnfld 20510 mPoly cmpl 21019 mDeg cmdg 25120

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-fzo 13312 df-seq 13650 df-hash 13973 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-0g 17069 df-gsum 17070 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-cnfld 20511 df-psr 21022 df-mpl 21024 df-mdeg 25122

This theorem is referenced by: deg1xrf 25151

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