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Mirrors > Home > MPE Home > Th. List > Mathboxes > evlsvval | Structured version Visualization version GIF version |
Description: Give a formula for the evaluation of a polynomial. (Contributed by SN, 9-Feb-2025.) |
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) |
Ref | Expression |
---|---|
evlsvval | ⊢ (𝜑 → (𝑄‘𝐴) = (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6906 | . . . . . 6 ⊢ (𝑝 = 𝐴 → (𝑝‘𝑏) = (𝐴‘𝑏)) | |
2 | 1 | fveq2d 6911 | . . . . 5 ⊢ (𝑝 = 𝐴 → (𝐹‘(𝑝‘𝑏)) = (𝐹‘(𝐴‘𝑏))) |
3 | 2 | oveq1d 7446 | . . . 4 ⊢ (𝑝 = 𝐴 → ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺))) = ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺)))) |
4 | 3 | mpteq2dv 5250 | . . 3 ⊢ (𝑝 = 𝐴 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺)))) = (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺))))) |
5 | 4 | oveq2d 7447 | . 2 ⊢ (𝑝 = 𝐴 → (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺))))) = (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σg (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 2735 | . . 3 ⊢ (𝑝 ∈ 𝐵 ↦ (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺)))))) = (𝑝 ∈ 𝐵 ↦ (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σg (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 42546 | . 2 ⊢ (𝜑 → 𝑄 = (𝑝 ∈ 𝐵 ↦ (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺))))))) |
23 | evlsvval.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
24 | ovexd 7466 | . 2 ⊢ (𝜑 → (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺))))) ∈ V) | |
25 | 5, 22, 23, 24 | fvmptd4 7040 | 1 ⊢ (𝜑 → (𝑄‘𝐴) = (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 {csn 4631 ↦ cmpt 5231 × cxp 5687 ◡ccnv 5688 “ cima 5692 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 ↑m cmap 8865 Fincfn 8984 ℕcn 12264 ℕ0cn0 12524 Basecbs 17245 ↾s cress 17274 .rcmulr 17299 Σg cgsu 17487 ↑s cpws 17493 .gcmg 19098 mulGrpcmgp 20152 CRingccrg 20252 SubRingcsubrg 20586 mPoly cmpl 21944 evalSub ces 22114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-evls 22116 |
This theorem is referenced by: evlsvvval 42550 evlsevl 42558 |
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