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Mirrors > Home > MPE Home > Th. List > evl1scad | Structured version Visualization version GIF version |
Description: Polynomial evaluation builder for scalars. (Contributed by Mario Carneiro, 4-Jul-2015.) |
Ref | Expression |
---|---|
evl1sca.o | ⊢ 𝑂 = (eval1‘𝑅) |
evl1sca.p | ⊢ 𝑃 = (Poly1‘𝑅) |
evl1sca.b | ⊢ 𝐵 = (Base‘𝑅) |
evl1sca.a | ⊢ 𝐴 = (algSc‘𝑃) |
evl1scad.u | ⊢ 𝑈 = (Base‘𝑃) |
evl1scad.1 | ⊢ (𝜑 → 𝑅 ∈ CRing) |
evl1scad.2 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
evl1scad.3 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
evl1scad | ⊢ (𝜑 → ((𝐴‘𝑋) ∈ 𝑈 ∧ ((𝑂‘(𝐴‘𝑋))‘𝑌) = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1scad.1 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
2 | crngring 19003 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
3 | evl1sca.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | evl1sca.a | . . . . 5 ⊢ 𝐴 = (algSc‘𝑃) | |
5 | evl1sca.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
6 | evl1scad.u | . . . . 5 ⊢ 𝑈 = (Base‘𝑃) | |
7 | 3, 4, 5, 6 | ply1sclf 20141 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝐴:𝐵⟶𝑈) |
8 | 1, 2, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐴:𝐵⟶𝑈) |
9 | evl1scad.2 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
10 | 8, 9 | ffvelrnd 6722 | . 2 ⊢ (𝜑 → (𝐴‘𝑋) ∈ 𝑈) |
11 | evl1sca.o | . . . . . 6 ⊢ 𝑂 = (eval1‘𝑅) | |
12 | 11, 3, 5, 4 | evl1sca 20184 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (𝐵 × {𝑋})) |
13 | 1, 9, 12 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑂‘(𝐴‘𝑋)) = (𝐵 × {𝑋})) |
14 | 13 | fveq1d 6545 | . . 3 ⊢ (𝜑 → ((𝑂‘(𝐴‘𝑋))‘𝑌) = ((𝐵 × {𝑋})‘𝑌)) |
15 | evl1scad.3 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
16 | fvconst2g 6836 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐵 × {𝑋})‘𝑌) = 𝑋) | |
17 | 9, 15, 16 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((𝐵 × {𝑋})‘𝑌) = 𝑋) |
18 | 14, 17 | eqtrd 2831 | . 2 ⊢ (𝜑 → ((𝑂‘(𝐴‘𝑋))‘𝑌) = 𝑋) |
19 | 10, 18 | jca 512 | 1 ⊢ (𝜑 → ((𝐴‘𝑋) ∈ 𝑈 ∧ ((𝑂‘(𝐴‘𝑋))‘𝑌) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 {csn 4476 × cxp 5446 ⟶wf 6226 ‘cfv 6230 Basecbs 16317 Ringcrg 18992 CRingccrg 18993 algSccascl 19778 Poly1cpl1 20033 eval1ce1 20165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-iin 4832 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-se 5408 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-isom 6239 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-of 7272 df-ofr 7273 df-om 7442 df-1st 7550 df-2nd 7551 df-supp 7687 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-2o 7959 df-oadd 7962 df-er 8144 df-map 8263 df-pm 8264 df-ixp 8316 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-fsupp 8685 df-sup 8757 df-oi 8825 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-dec 11953 df-uz 12099 df-fz 12748 df-fzo 12889 df-seq 13225 df-hash 13546 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-mulr 16413 df-sca 16415 df-vsca 16416 df-ip 16417 df-tset 16418 df-ple 16419 df-ds 16421 df-hom 16423 df-cco 16424 df-0g 16549 df-gsum 16550 df-prds 16555 df-pws 16557 df-mre 16691 df-mrc 16692 df-acs 16694 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-mhm 17779 df-submnd 17780 df-grp 17869 df-minusg 17870 df-sbg 17871 df-mulg 17987 df-subg 18035 df-ghm 18102 df-cntz 18193 df-cmn 18640 df-abl 18641 df-mgp 18935 df-ur 18947 df-srg 18951 df-ring 18994 df-cring 18995 df-rnghom 19162 df-subrg 19228 df-lmod 19331 df-lss 19399 df-lsp 19439 df-assa 19779 df-asp 19780 df-ascl 19781 df-psr 19829 df-mvr 19830 df-mpl 19831 df-opsr 19833 df-evls 19978 df-evl 19979 df-psr1 20036 df-ply1 20038 df-evl1 20167 |
This theorem is referenced by: evl1vsd 20194 evl1gsumd 20207 ply1remlem 24444 lgsqrlem1 25609 idomrootle 39306 evl1at0 43952 evl1at1 43953 lineval 43955 |
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