mdegxrf - Metamath Proof Explorer

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Theorem mdegxrf 25955

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 13123	. . . 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 2726	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
7		eqid 2726	. . . 4 ⊢ {𝑥 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑥 “ ℕ) ∈ Fin} = {𝑥 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑥 “ ℕ) ∈ Fin}
8		eqid 2726	. . . 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 25949	. . 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 6686	. 2 ⊢ 𝐷 Fn 𝐵
11	3, 4, 5	mdegxrcl 25954	. . 3 ⊢ (𝑓 ∈ 𝐵 → (𝐷‘𝑓) ∈ ℝ^*)
12	11	rgen 3057	. 2 ⊢ ∀𝑓 ∈ 𝐵 (𝐷‘𝑓) ∈ ℝ^*
13		ffnfv 7113	. 2 ⊢ (𝐷:𝐵⟶ℝ^* ↔ (𝐷 Fn 𝐵 ∧ ∀𝑓 ∈ 𝐵 (𝐷‘𝑓) ∈ ℝ^*))
14	10, 12, 13	mpbir2an 708	1 ⊢ 𝐷:𝐵⟶ℝ^*

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 ∀wral 3055 {crab 3426 ↦ cmpt 5224 ^◡ccnv 5668 “ cima 5672 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 supp csupp 8143 ↑_m cmap 8819 Fincfn 8938 supcsup 9434 ℝ^*cxr 11248 < clt 11249 ℕcn 12213 ℕ₀cn0 12473 Basecbs 17151 0_gc0g 17392 Σ_g cgsu 17393 ℂ_fldccnfld 21236 mPoly cmpl 21796 mDeg cmdg 25937

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 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 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14294 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-0g 17394 df-gsum 17395 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-cntz 19231 df-cmn 19700 df-abl 19701 df-mgp 20038 df-ur 20085 df-ring 20138 df-cring 20139 df-cnfld 21237 df-psr 21799 df-mpl 21801 df-mdeg 25939

This theorem is referenced by: deg1xrf 25968

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