evlsvval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > evlsvval		Structured version Visualization version GIF version

Theorem evlsvval 22057

Description: Give a formula for the evaluation of a polynomial. (Contributed by SN, 9-Feb-2025.)

**Hypotheses**
Ref	Expression
evlsvval.q	⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
evlsvval.p	⊢ 𝑃 = (𝐼 mPoly 𝑈)
evlsvval.b	⊢ 𝐵 = (Base‘𝑃)
evlsvval.d	⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
evlsvval.k	⊢ 𝐾 = (Base‘𝑆)
evlsvval.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
evlsvval.t	⊢ 𝑇 = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
evlsvval.m	⊢ 𝑀 = (mulGrp‘𝑇)
evlsvval.w	⊢ ↑ = (._g‘𝑀)
evlsvval.x	⊢ · = (._r‘𝑇)
evlsvval.f	⊢ 𝐹 = (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))
evlsvval.g	⊢ 𝐺 = (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))
evlsvval.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
evlsvval.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evlsvval.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evlsvval.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)

**Assertion**
Ref	Expression
evlsvval	⊢ (𝜑 → (𝑄‘𝐴) = (𝑇 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σ_g (𝑏 ∘_f ↑ 𝐺))))))

Distinct variable groups: 𝜑,𝑎,𝑥 𝜑,𝑏 𝑃,𝑏 𝐵,𝑏 𝐷,𝑏 𝐾,𝑎,𝑥 𝑈,𝑏,ℎ 𝑇,𝑏 𝑥,𝑇 𝑀,𝑏 ↑ ,𝑏 · ,𝑏 𝐹,𝑏 𝐺,𝑏 𝐼,𝑎,𝑥 𝐼,𝑏,ℎ 𝑆,𝑎,𝑥 𝑥,𝑅 𝐴,𝑏 Allowed substitution hints: 𝜑(ℎ) 𝐴(𝑥,ℎ,𝑎) 𝐵(𝑥,ℎ,𝑎) 𝐷(𝑥,ℎ,𝑎) 𝑃(𝑥,ℎ,𝑎) 𝑄(𝑥,ℎ,𝑎,𝑏) 𝑅(ℎ,𝑎,𝑏) 𝑆(ℎ,𝑏) 𝑇(ℎ,𝑎) · (𝑥,ℎ,𝑎) 𝑈(𝑥,𝑎) ↑ (𝑥,ℎ,𝑎) 𝐹(𝑥,ℎ,𝑎) 𝐺(𝑥,ℎ,𝑎) 𝐾(ℎ,𝑏) 𝑀(𝑥,ℎ,𝑎) 𝑉(𝑥,ℎ,𝑎,𝑏)

Proof of Theorem evlsvval Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6841	. . . . . 6 ⊢ (𝑝 = 𝐴 → (𝑝‘𝑏) = (𝐴‘𝑏))
2	1	fveq2d 6846	. . . . 5 ⊢ (𝑝 = 𝐴 → (𝐹‘(𝑝‘𝑏)) = (𝐹‘(𝐴‘𝑏)))
3	2	oveq1d 7383	. . . 4 ⊢ (𝑝 = 𝐴 → ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σ_g (𝑏 ∘_f ↑ 𝐺))) = ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σ_g (𝑏 ∘_f ↑ 𝐺))))
4	3	mpteq2dv 5194	. . 3 ⊢ (𝑝 = 𝐴 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σ_g (𝑏 ∘_f ↑ 𝐺)))) = (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σ_g (𝑏 ∘_f ↑ 𝐺)))))
5	4	oveq2d 7384	. 2 ⊢ (𝑝 = 𝐴 → (𝑇 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σ_g (𝑏 ∘_f ↑ 𝐺))))) = (𝑇 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σ_g (𝑏 ∘_f ↑ 𝐺))))))
6		evlsvval.q	. . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
7		evlsvval.p	. . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑈)
8		evlsvval.b	. . 3 ⊢ 𝐵 = (Base‘𝑃)
9		evlsvval.d	. . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
10		evlsvval.k	. . 3 ⊢ 𝐾 = (Base‘𝑆)
11		evlsvval.u	. . 3 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
12		evlsvval.t	. . 3 ⊢ 𝑇 = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
13		evlsvval.m	. . 3 ⊢ 𝑀 = (mulGrp‘𝑇)
14		evlsvval.w	. . 3 ⊢ ↑ = (._g‘𝑀)
15		evlsvval.x	. . 3 ⊢ · = (._r‘𝑇)
16		eqid 2737	. . 3 ⊢ (𝑝 ∈ 𝐵 ↦ (𝑇 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σ_g (𝑏 ∘_f ↑ 𝐺)))))) = (𝑝 ∈ 𝐵 ↦ (𝑇 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σ_g (𝑏 ∘_f ↑ 𝐺))))))
17		evlsvval.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))
18		evlsvval.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))
19		evlsvval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
20		evlsvval.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ CRing)
21		evlsvval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
22	6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21	evlsval3 22056	. 2 ⊢ (𝜑 → 𝑄 = (𝑝 ∈ 𝐵 ↦ (𝑇 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σ_g (𝑏 ∘_f ↑ 𝐺)))))))
23		evlsvval.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
24		ovexd 7403	. 2 ⊢ (𝜑 → (𝑇 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σ_g (𝑏 ∘_f ↑ 𝐺))))) ∈ V)
25	5, 22, 23, 24	fvmptd4 6974	1 ⊢ (𝜑 → (𝑄‘𝐴) = (𝑇 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σ_g (𝑏 ∘_f ↑ 𝐺))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 {crab 3401 Vcvv 3442 {csn 4582 ↦ cmpt 5181 × cxp 5630 ^◡ccnv 5631 “ cima 5635 ‘cfv 6500 (class class class)co 7368 ∘_f cof 7630 ↑_m cmap 8775 Fincfn 8895 ℕcn 12157 ℕ₀cn0 12413 Basecbs 17148 ↾_s cress 17169 ._rcmulr 17190 Σ_g cgsu 17372 ↑_s cpws 17378 ._gcmg 19009 mulGrpcmgp 20087 CRingccrg 20181 SubRingcsubrg 20514 mPoly cmpl 21874 evalSub ces 22039

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-ghm 19154 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-rhm 20420 df-subrng 20491 df-subrg 20515 df-lmod 20825 df-lss 20895 df-lsp 20935 df-assa 21820 df-asp 21821 df-ascl 21822 df-psr 21877 df-mvr 21878 df-mpl 21879 df-evls 22041

This theorem is referenced by: evlsvvval 22060 evlsevl 42926

W3C validator