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Theorem mdegval 26116

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 7437	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 supp 0 ) = (𝐹 supp 0 ))
2	1	imaeq2d 6079	. . 3 ⊢ (𝑓 = 𝐹 → (𝐻 “ (𝑓 supp 0 )) = (𝐻 “ (𝐹 supp 0 )))
3	2	supeq1d 9483	. 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 26115	. 2 ⊢ 𝐷 = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ^*, < ))
11		xrltso 13179	. . 3 ⊢ < Or ℝ^*
12	11	supex 9500	. 2 ⊢ sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ) ∈ V
13	3, 10, 12	fvmpt 7015	1 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 {crab 3432 ↦ cmpt 5230 ^◡ccnv 5687 “ cima 5691 ‘cfv 6562 (class class class)co 7430 supp csupp 8183 ↑_m cmap 8864 Fincfn 8983 supcsup 9477 ℝ^*cxr 11291 < clt 11292 ℕcn 12263 ℕ₀cn0 12523 Basecbs 17244 0_gc0g 17485 Σ_g cgsu 17486 ℂ_fldccnfld 21381 mPoly cmpl 21943 mDeg cmdg 26106

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-psr 21946 df-mpl 21948 df-mdeg 26108

This theorem is referenced by: mdegleb 26117 mdeglt 26118 mdegldg 26119 mdegxrcl 26120 mdegcl 26122 mdeg0 26123 mdegvsca 26129 deg1val 26149

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