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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 2735 | . . 3 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
2 | fveval1fvcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | eqid 2735 | . . 3 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
4 | fveval1fvcl.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
5 | 2 | fvexi 6921 | . . . 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 22352 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
10 | fveval1fvcl.u | . . . . . 6 ⊢ 𝑈 = (Base‘𝑃) | |
11 | 10, 3 | rhmf 20502 | . . . . 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 | ffvelcdmd 7105 | . . 3 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑s 𝐵))) |
15 | 1, 2, 3, 4, 6, 14 | pwselbas 17536 | . 2 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵) |
16 | fveval1fvcl.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
17 | 15, 16 | ffvelcdmd 7105 | 1 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↑s cpws 17493 CRingccrg 20252 RingHom crh 20486 Poly1cpl1 22194 eval1ce1 22334 |
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-opsr 21951 df-evls 22116 df-evl 22117 df-psr1 22197 df-ply1 22199 df-evl1 22336 |
This theorem is referenced by: evl1gsumdlem 22376 evls1fvcl 22395 evls1maprhm 22396 facth1 26221 ply1mulrtss 33586 aks6d1c1p4 42093 aks6d1c1p5 42094 aks6d1c1p8 42097 evl1gprodd 42099 aks6d1c2lem4 42109 aks6d1c5lem2 42120 aks6d1c6lem2 42153 aks6d1c6lem3 42154 |
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