mdegxrf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mdegxrf		Structured version Visualization version GIF version

Theorem mdegxrf 25585

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 13119	. . . 4 ⊢ < Or ℝ^*
2	1	supex 9457	. . 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 2732	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
7		eqid 2732	. . . 4 ⊢ {𝑥 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑥 “ ℕ) ∈ Fin} = {𝑥 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑥 “ ℕ) ∈ Fin}
8		eqid 2732	. . . 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 25579	. . 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 6693	. 2 ⊢ 𝐷 Fn 𝐵
11	3, 4, 5	mdegxrcl 25584	. . 3 ⊢ (𝑓 ∈ 𝐵 → (𝐷‘𝑓) ∈ ℝ^*)
12	11	rgen 3063	. 2 ⊢ ∀𝑓 ∈ 𝐵 (𝐷‘𝑓) ∈ ℝ^*
13		ffnfv 7117	. 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 1541 ∈ wcel 2106 ∀wral 3061 {crab 3432 ↦ cmpt 5231 ^◡ccnv 5675 “ cima 5679 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 supp csupp 8145 ↑_m cmap 8819 Fincfn 8938 supcsup 9434 ℝ^*cxr 11246 < clt 11247 ℕcn 12211 ℕ₀cn0 12471 Basecbs 17143 0_gc0g 17384 Σ_g cgsu 17385 ℂ_fldccnfld 20943 mPoly cmpl 21458 mDeg cmdg 25567

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-0g 17386 df-gsum 17387 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-grp 18821 df-minusg 18822 df-cntz 19180 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-cnfld 20944 df-psr 21461 df-mpl 21463 df-mdeg 25569

This theorem is referenced by: deg1xrf 25598

W3C validator