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| Mirrors > Home > MPE Home > Th. List > evl1sca | Structured version Visualization version GIF version | ||
| Description: Polynomial evaluation maps scalars to constant functions. (Contributed by Mario Carneiro, 12-Jun-2015.) |
| Ref | Expression |
|---|---|
| evl1sca.o | ⊢ 𝑂 = (eval1‘𝑅) |
| evl1sca.p | ⊢ 𝑃 = (Poly1‘𝑅) |
| evl1sca.b | ⊢ 𝐵 = (Base‘𝑅) |
| evl1sca.a | ⊢ 𝐴 = (algSc‘𝑃) |
| Ref | Expression |
|---|---|
| evl1sca | ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (𝐵 × {𝑋})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | crngring 20242 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 2 | 1 | adantr 480 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Ring) |
| 3 | evl1sca.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 4 | evl1sca.a | . . . . . 6 ⊢ 𝐴 = (algSc‘𝑃) | |
| 5 | evl1sca.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
| 6 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
| 7 | 3, 4, 5, 6 | ply1sclf 22288 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝐴:𝐵⟶(Base‘𝑃)) |
| 8 | 2, 7 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴:𝐵⟶(Base‘𝑃)) |
| 9 | ffvelcdm 7101 | . . . 4 ⊢ ((𝐴:𝐵⟶(Base‘𝑃) ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ (Base‘𝑃)) | |
| 10 | 8, 9 | sylancom 588 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ (Base‘𝑃)) |
| 11 | evl1sca.o | . . . 4 ⊢ 𝑂 = (eval1‘𝑅) | |
| 12 | eqid 2737 | . . . 4 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
| 13 | eqid 2737 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
| 14 | 3, 6 | ply1bas 22196 | . . . 4 ⊢ (Base‘𝑃) = (Base‘(1o mPoly 𝑅)) |
| 15 | 11, 12, 5, 13, 14 | evl1val 22333 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐴‘𝑋) ∈ (Base‘𝑃)) → (𝑂‘(𝐴‘𝑋)) = (((1o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
| 16 | 10, 15 | syldan 591 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (((1o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
| 17 | 3, 4 | ply1ascl 22261 | . . . . . . 7 ⊢ 𝐴 = (algSc‘(1o mPoly 𝑅)) |
| 18 | 5 | ressid 17290 | . . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅) |
| 19 | 18 | adantr 480 | . . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑅 ↾s 𝐵) = 𝑅) |
| 20 | 19 | oveq2d 7447 | . . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly 𝑅)) |
| 21 | 20 | fveq2d 6910 | . . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(1o mPoly 𝑅))) |
| 22 | 17, 21 | eqtr4id 2796 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴 = (algSc‘(1o mPoly (𝑅 ↾s 𝐵)))) |
| 23 | 22 | fveq1d 6908 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) = ((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑋)) |
| 24 | 23 | fveq2d 6910 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1o eval 𝑅)‘(𝐴‘𝑋)) = ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑋))) |
| 25 | 12, 5 | evlval 22119 | . . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵) |
| 26 | eqid 2737 | . . . . 5 ⊢ (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly (𝑅 ↾s 𝐵)) | |
| 27 | eqid 2737 | . . . . 5 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵) | |
| 28 | eqid 2737 | . . . . 5 ⊢ (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) | |
| 29 | 1on 8518 | . . . . . 6 ⊢ 1o ∈ On | |
| 30 | 29 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 1o ∈ On) |
| 31 | simpl 482 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ CRing) | |
| 32 | 5 | subrgid 20573 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) |
| 33 | 2, 32 | syl 17 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ (SubRing‘𝑅)) |
| 34 | simpr 484 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
| 35 | 25, 26, 27, 5, 28, 30, 31, 33, 34 | evlssca 22113 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑋)) = ((𝐵 ↑m 1o) × {𝑋})) |
| 36 | 24, 35 | eqtrd 2777 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1o eval 𝑅)‘(𝐴‘𝑋)) = ((𝐵 ↑m 1o) × {𝑋})) |
| 37 | 36 | coeq1d 5872 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((1o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (((𝐵 ↑m 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
| 38 | df1o2 8513 | . . . . . . 7 ⊢ 1o = {∅} | |
| 39 | 5 | fvexi 6920 | . . . . . . 7 ⊢ 𝐵 ∈ V |
| 40 | 0ex 5307 | . . . . . . 7 ⊢ ∅ ∈ V | |
| 41 | eqid 2737 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) | |
| 42 | 38, 39, 40, 41 | mapsnf1o3 8935 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑m 1o) |
| 43 | f1of 6848 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑m 1o) → (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵⟶(𝐵 ↑m 1o)) | |
| 44 | 42, 43 | mp1i 13 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵⟶(𝐵 ↑m 1o)) |
| 45 | 41 | fmpt 7130 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 (1o × {𝑦}) ∈ (𝐵 ↑m 1o) ↔ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵⟶(𝐵 ↑m 1o)) |
| 46 | 44, 45 | sylibr 234 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 (1o × {𝑦}) ∈ (𝐵 ↑m 1o)) |
| 47 | eqidd 2738 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) | |
| 48 | fconstmpt 5747 | . . . . 5 ⊢ ((𝐵 ↑m 1o) × {𝑋}) = (𝑥 ∈ (𝐵 ↑m 1o) ↦ 𝑋) | |
| 49 | 48 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((𝐵 ↑m 1o) × {𝑋}) = (𝑥 ∈ (𝐵 ↑m 1o) ↦ 𝑋)) |
| 50 | eqidd 2738 | . . . 4 ⊢ (𝑥 = (1o × {𝑦}) → 𝑋 = 𝑋) | |
| 51 | 46, 47, 49, 50 | fmptcof 7150 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑m 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (𝑦 ∈ 𝐵 ↦ 𝑋)) |
| 52 | fconstmpt 5747 | . . 3 ⊢ (𝐵 × {𝑋}) = (𝑦 ∈ 𝐵 ↦ 𝑋) | |
| 53 | 51, 52 | eqtr4di 2795 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑m 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (𝐵 × {𝑋})) |
| 54 | 16, 37, 53 | 3eqtrd 2781 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (𝐵 × {𝑋})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∅c0 4333 {csn 4626 ↦ cmpt 5225 × cxp 5683 ∘ ccom 5689 Oncon0 6384 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 1oc1o 8499 ↑m cmap 8866 Basecbs 17247 ↾s cress 17274 Ringcrg 20230 CRingccrg 20231 SubRingcsubrg 20569 algSccascl 21872 mPoly cmpl 21926 eval cevl 22097 Poly1cpl1 22178 eval1ce1 22318 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-srg 20184 df-ring 20232 df-cring 20233 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-lsp 20970 df-assa 21873 df-asp 21874 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-evls 22098 df-evl 22099 df-psr1 22181 df-ply1 22183 df-evl1 22320 |
| This theorem is referenced by: evl1scad 22339 pf1const 22350 pf1ind 22359 evl1scvarpw 22367 ply1rem 26205 fta1g 26209 fta1blem 26210 plypf1 26251 |
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