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Theorem mdegval 26053

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 7370	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 supp 0 ) = (𝐹 supp 0 ))
2	1	imaeq2d 6019	. . 3 ⊢ (𝑓 = 𝐹 → (𝐻 “ (𝑓 supp 0 )) = (𝐻 “ (𝐹 supp 0 )))
3	2	supeq1d 9356	. 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 26052	. 2 ⊢ 𝐷 = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ^*, < ))
11		xrltso 13090	. . 3 ⊢ < Or ℝ^*
12	11	supex 9374	. 2 ⊢ sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ) ∈ V
13	3, 10, 12	fvmpt 6942	1 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 {crab 3392 ↦ cmpt 5160 ^◡ccnv 5624 “ cima 5628 ‘cfv 6492 (class class class)co 7363 supp csupp 8107 ↑_m cmap 8770 Fincfn 8890 supcsup 9350 ℝ^*cxr 11176 < clt 11177 ℕcn 12172 ℕ₀cn0 12435 Basecbs 17177 0_gc0g 17400 Σ_g cgsu 17401 ℂ_fldccnfld 21354 mPoly cmpl 21888 mDeg cmdg 26043

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-sup 9352 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-tset 17237 df-psr 21891 df-mpl 21893 df-mdeg 26045

This theorem is referenced by: mdegleb 26054 mdeglt 26055 mdegldg 26056 mdegxrcl 26057 mdegcl 26059 mdeg0 26060 mdegvsca 26066 deg1val 26086

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