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Theorem mdegval 25968

Description: Value of the multivariate degree function at some particular polynomial. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-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))

**Assertion**
Ref	Expression
mdegval	⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ))

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

Proof of Theorem mdegval Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7394	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 supp 0 ) = (𝐹 supp 0 ))
2	1	imaeq2d 6031	. . 3 ⊢ (𝑓 = 𝐹 → (𝐻 “ (𝑓 supp 0 )) = (𝐻 “ (𝐹 supp 0 )))
3	2	supeq1d 9397	. 2 ⊢ (𝑓 = 𝐹 → sup((𝐻 “ (𝑓 supp 0 )), ℝ^, < ) = sup((𝐻 “ (𝐹 supp 0 )), ℝ^, < ))
4		mdegval.d	. . 3 ⊢ 𝐷 = (𝐼 mDeg 𝑅)
5		mdegval.p	. . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
6		mdegval.b	. . 3 ⊢ 𝐵 = (Base‘𝑃)
7		mdegval.z	. . 3 ⊢ 0 = (0_g‘𝑅)
8		mdegval.a	. . 3 ⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin}
9		mdegval.h	. . 3 ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ))
10	4, 5, 6, 7, 8, 9	mdegfval 25967	. 2 ⊢ 𝐷 = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ^*, < ))
11		xrltso 13101	. . 3 ⊢ < Or ℝ^*
12	11	supex 9415	. 2 ⊢ sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ) ∈ V
13	3, 10, 12	fvmpt 6968	1 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 {crab 3405 ↦ cmpt 5188 ^◡ccnv 5637 “ cima 5641 ‘cfv 6511 (class class class)co 7387 supp csupp 8139 ↑_m cmap 8799 Fincfn 8918 supcsup 9391 ℝ^*cxr 11207 < clt 11208 ℕcn 12186 ℕ₀cn0 12442 Basecbs 17179 0_gc0g 17402 Σ_g cgsu 17403 ℂ_fldccnfld 21264 mPoly cmpl 21815 mDeg cmdg 25958

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-tset 17239 df-psr 21818 df-mpl 21820 df-mdeg 25960

This theorem is referenced by: mdegleb 25969 mdeglt 25970 mdegldg 25971 mdegxrcl 25972 mdegcl 25974 mdeg0 25975 mdegvsca 25981 deg1val 26001

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