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Mirrors > Home > MPE Home > Th. List > fveval1fvcl | Structured version Visualization version GIF version |
Description: The function value of the evaluation function of a polynomial is an element of the underlying ring. (Contributed by AV, 17-Sep-2019.) |
Ref | Expression |
---|---|
fveval1fvcl.q | ⊢ 𝑂 = (eval1‘𝑅) |
fveval1fvcl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
fveval1fvcl.b | ⊢ 𝐵 = (Base‘𝑅) |
fveval1fvcl.u | ⊢ 𝑈 = (Base‘𝑃) |
fveval1fvcl.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
fveval1fvcl.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
fveval1fvcl.m | ⊢ (𝜑 → 𝑀 ∈ 𝑈) |
Ref | Expression |
---|---|
fveval1fvcl | ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
2 | fveval1fvcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | eqid 2821 | . . 3 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
4 | fveval1fvcl.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
5 | 2 | fvexi 6683 | . . . 4 ⊢ 𝐵 ∈ V |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
7 | fveval1fvcl.q | . . . . . 6 ⊢ 𝑂 = (eval1‘𝑅) | |
8 | fveval1fvcl.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
9 | 7, 8, 1, 2 | evl1rhm 20494 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
10 | fveval1fvcl.u | . . . . . 6 ⊢ 𝑈 = (Base‘𝑃) | |
11 | 10, 3 | rhmf 19477 | . . . . 5 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵)) → 𝑂:𝑈⟶(Base‘(𝑅 ↑s 𝐵))) |
12 | 4, 9, 11 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑂:𝑈⟶(Base‘(𝑅 ↑s 𝐵))) |
13 | fveval1fvcl.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑈) | |
14 | 12, 13 | ffvelrnd 6851 | . . 3 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑s 𝐵))) |
15 | 1, 2, 3, 4, 6, 14 | pwselbas 16761 | . 2 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵) |
16 | fveval1fvcl.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
17 | 15, 16 | ffvelrnd 6851 | 1 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 ↑s cpws 16719 CRingccrg 19297 RingHom crh 19463 Poly1cpl1 20344 eval1ce1 20476 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-ofr 7409 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-hom 16588 df-cco 16589 df-0g 16714 df-gsum 16715 df-prds 16720 df-pws 16722 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-ghm 18355 df-cntz 18446 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-srg 19255 df-ring 19298 df-cring 19299 df-rnghom 19466 df-subrg 19532 df-lmod 19635 df-lss 19703 df-lsp 19743 df-assa 20084 df-asp 20085 df-ascl 20086 df-psr 20135 df-mvr 20136 df-mpl 20137 df-opsr 20139 df-evls 20285 df-evl 20286 df-psr1 20347 df-ply1 20349 df-evl1 20478 |
This theorem is referenced by: evl1gsumdlem 20518 facth1 24757 |
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