mdegxrf - Metamath Proof Explorer

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Theorem mdegxrf 26022

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 13158	. . . 4 ⊢ < Or ℝ^*
2	1	supex 9492	. . 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 2727	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
7		eqid 2727	. . . 4 ⊢ {𝑥 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑥 “ ℕ) ∈ Fin} = {𝑥 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑥 “ ℕ) ∈ Fin}
8		eqid 2727	. . . 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 26016	. . 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 6701	. 2 ⊢ 𝐷 Fn 𝐵
11	3, 4, 5	mdegxrcl 26021	. . 3 ⊢ (𝑓 ∈ 𝐵 → (𝐷‘𝑓) ∈ ℝ^*)
12	11	rgen 3059	. 2 ⊢ ∀𝑓 ∈ 𝐵 (𝐷‘𝑓) ∈ ℝ^*
13		ffnfv 7132	. 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 1533 ∈ wcel 2098 ∀wral 3057 {crab 3428 ↦ cmpt 5233 ^◡ccnv 5679 “ cima 5683 Fn wfn 6546 ⟶wf 6547 ‘cfv 6551 (class class class)co 7424 supp csupp 8169 ↑_m cmap 8849 Fincfn 8968 supcsup 9469 ℝ^*cxr 11283 < clt 11284 ℕcn 12248 ℕ₀cn0 12508 Basecbs 17185 0_gc0g 17426 Σ_g cgsu 17427 ℂ_fldccnfld 21284 mPoly cmpl 21844 mDeg cmdg 26004

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 ax-addf 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-sup 9471 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-fz 13523 df-fzo 13666 df-seq 14005 df-hash 14328 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-starv 17253 df-sca 17254 df-vsca 17255 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-0g 17428 df-gsum 17429 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-grp 18898 df-minusg 18899 df-cntz 19273 df-cmn 19742 df-abl 19743 df-mgp 20080 df-ur 20127 df-ring 20180 df-cring 20181 df-cnfld 21285 df-psr 21847 df-mpl 21849 df-mdeg 26006

This theorem is referenced by: deg1xrf 26035

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