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Theorem mdeglt 24225

Description: If there is an upper limit on the degree of a polynomial that is lower than the degree of some exponent bag, then that exponent bag is unrepresented in the polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)

**Hypotheses**
Ref	Expression
mdegval.d	⊢ 𝐷 = (𝐼 mDeg 𝑅)
mdegval.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mdegval.b	⊢ 𝐵 = (Base‘𝑃)
mdegval.z	⊢ 0 = (0_g‘𝑅)
mdegval.a	⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin}
mdegval.h	⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ))
mdeglt.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
medglt.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
mdeglt.lt	⊢ (𝜑 → (𝐷‘𝐹) < (𝐻‘𝑋))

**Assertion**
Ref	Expression
mdeglt	⊢ (𝜑 → (𝐹‘𝑋) = 0 )

Distinct variable groups: 𝐴,ℎ 𝑚,𝐼 0 ,ℎ ℎ,𝐼,𝑚 Allowed substitution hints: 𝜑(ℎ,𝑚) 𝐴(𝑚) 𝐵(ℎ,𝑚) 𝐷(ℎ,𝑚) 𝑃(ℎ,𝑚) 𝑅(ℎ,𝑚) 𝐹(ℎ,𝑚) 𝐻(ℎ,𝑚) 𝑋(ℎ,𝑚) 0 (𝑚)

Proof of Theorem mdeglt Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdeglt.lt	. 2 ⊢ (𝜑 → (𝐷‘𝐹) < (𝐻‘𝑋))
2		fveq2 6434	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐻‘𝑥) = (𝐻‘𝑋))
3	2	breq2d 4886	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝐹) < (𝐻‘𝑥) ↔ (𝐷‘𝐹) < (𝐻‘𝑋)))
4		fveqeq2 6443	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 0 ↔ (𝐹‘𝑋) = 0 ))
5	3, 4	imbi12d 336	. . 3 ⊢ (𝑥 = 𝑋 → (((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 ) ↔ ((𝐷‘𝐹) < (𝐻‘𝑋) → (𝐹‘𝑋) = 0 )))
6		mdeglt.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
7		mdegval.d	. . . . . . . 8 ⊢ 𝐷 = (𝐼 mDeg 𝑅)
8		mdegval.p	. . . . . . . 8 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
9		mdegval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑃)
10		mdegval.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑅)
11		mdegval.a	. . . . . . . 8 ⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin}
12		mdegval.h	. . . . . . . 8 ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ))
13	7, 8, 9, 10, 11, 12	mdegval 24223	. . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ))
14	6, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ))
15		imassrn 5719	. . . . . . . 8 ⊢ (𝐻 “ (𝐹 supp 0 )) ⊆ ran 𝐻
16	8, 9	mplrcl 19851	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V)
17	11, 12	tdeglem1 24218	. . . . . . . . . 10 ⊢ (𝐼 ∈ V → 𝐻:𝐴⟶ℕ₀)
18		frn 6285	. . . . . . . . . 10 ⊢ (𝐻:𝐴⟶ℕ₀ → ran 𝐻 ⊆ ℕ₀)
19	6, 16, 17, 18	4syl 19	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ⊆ ℕ₀)
20		nn0ssre 11623	. . . . . . . . . 10 ⊢ ℕ₀ ⊆ ℝ
21		ressxr 10401	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ^*
22	20, 21	sstri 3837	. . . . . . . . 9 ⊢ ℕ₀ ⊆ ℝ^*
23	19, 22	syl6ss 3840	. . . . . . . 8 ⊢ (𝜑 → ran 𝐻 ⊆ ℝ^*)
24	15, 23	syl5ss 3839	. . . . . . 7 ⊢ (𝜑 → (𝐻 “ (𝐹 supp 0 )) ⊆ ℝ^*)
25		supxrcl 12434	. . . . . . 7 ⊢ ((𝐻 “ (𝐹 supp 0 )) ⊆ ℝ^* → sup((𝐻 “ (𝐹 supp 0 )), ℝ^, < ) ∈ ℝ^)
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → sup((𝐻 “ (𝐹 supp 0 )), ℝ^, < ) ∈ ℝ^)
27	14, 26	eqeltrd 2907	. . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ^*)
28	27	xrleidd 12272	. . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ≤ (𝐷‘𝐹))
29	7, 8, 9, 10, 11, 12	mdegleb 24224	. . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ (𝐷‘𝐹) ∈ ℝ^*) → ((𝐷‘𝐹) ≤ (𝐷‘𝐹) ↔ ∀𝑥 ∈ 𝐴 ((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 )))
30	6, 27, 29	syl2anc 581	. . . 4 ⊢ (𝜑 → ((𝐷‘𝐹) ≤ (𝐷‘𝐹) ↔ ∀𝑥 ∈ 𝐴 ((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 )))
31	28, 30	mpbid 224	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 ))
32		medglt.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
33	5, 31, 32	rspcdva 3533	. 2 ⊢ (𝜑 → ((𝐷‘𝐹) < (𝐻‘𝑋) → (𝐹‘𝑋) = 0 ))
34	1, 33	mpd 15	1 ⊢ (𝜑 → (𝐹‘𝑋) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 = wceq 1658 ∈ wcel 2166 ∀wral 3118 {crab 3122 Vcvv 3415 ⊆ wss 3799 class class class wbr 4874 ↦ cmpt 4953 ^◡ccnv 5342 ran crn 5344 “ cima 5346 ⟶wf 6120 ‘cfv 6124 (class class class)co 6906 supp csupp 7560 ↑_𝑚 cmap 8123 Fincfn 8223 supcsup 8616 ℝcr 10252 ℝ^*cxr 10391 < clt 10392 ≤ cle 10393 ℕcn 11351 ℕ₀cn0 11619 Basecbs 16223 0_gc0g 16454 Σ_g cgsu 16455 mPoly cmpl 19715 ℂ_fldccnfld 20107 mDeg cmdg 24213

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 ax-addf 10332 ax-mulf 10333

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-of 7158 df-om 7328 df-1st 7429 df-2nd 7430 df-supp 7561 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-fsupp 8546 df-sup 8618 df-oi 8685 df-card 9079 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-uz 11970 df-fz 12621 df-fzo 12762 df-seq 13097 df-hash 13412 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-starv 16321 df-sca 16322 df-vsca 16323 df-tset 16325 df-ple 16326 df-ds 16328 df-unif 16329 df-0g 16456 df-gsum 16457 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-submnd 17690 df-grp 17780 df-minusg 17781 df-cntz 18101 df-cmn 18549 df-abl 18550 df-mgp 18845 df-ur 18857 df-ring 18904 df-cring 18905 df-psr 19718 df-mpl 19720 df-cnfld 20108 df-mdeg 24215

This theorem is referenced by: mdegaddle 24234 mdegvscale 24235 mdegmullem 24238

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