mdegval - Metamath Proof Explorer

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Theorem mdegval 26257

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 7430	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 supp 0 ) = (𝐹 supp 0 ))
2	1	imaeq2d 6067	. . 3 ⊢ (𝑓 = 𝐹 → (𝐻 “ (𝑓 supp 0 )) = (𝐻 “ (𝐹 supp 0 )))
3	2	supeq1d 9416	. 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 26256	. 2 ⊢ 𝐷 = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ^*, < ))
11		xrltso 13184	. . 3 ⊢ < Or ℝ^*
12	11	supex 9434	. 2 ⊢ sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ) ∈ V
13	3, 10, 12	fvmpt 6996	1 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 {crab 3419 ↦ cmpt 5197 ^◡ccnv 5665 “ cima 5669 ‘cfv 6543 (class class class)co 7423 supp csupp 8165 ↑_m cmap 8833 Fincfn 8952 supcsup 9410 ℝ^*cxr 11260 < clt 11261 ℕcn 12251 ℕ₀cn0 12522 Basecbs 17294 0_gc0g 17517 Σ_g cgsu 17518 ℂ_fldccnfld 21559 mPoly cmpl 22093 mDeg cmdg 26247

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-sup 9412 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-uz 12881 df-fz 13554 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-sca 17351 df-vsca 17352 df-tset 17354 df-psr 22096 df-mpl 22098 df-mdeg 26249

This theorem is used by: mdegleb 26258 mdeglt 26259 mdegldg 26260 mdegxrcl 26261 mdegcl 26263 mdeg0 26264 mdegvsca 26270 deg1val 26290

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