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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 19302 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | 1 | adantr 484 | . . . . 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 2798 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
7 | 3, 4, 5, 6 | ply1sclf 20914 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝐴:𝐵⟶(Base‘𝑃)) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴:𝐵⟶(Base‘𝑃)) |
9 | ffvelrn 6826 | . . . 4 ⊢ ((𝐴:𝐵⟶(Base‘𝑃) ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ (Base‘𝑃)) | |
10 | 8, 9 | sylancom 591 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ (Base‘𝑃)) |
11 | evl1sca.o | . . . 4 ⊢ 𝑂 = (eval1‘𝑅) | |
12 | eqid 2798 | . . . 4 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
13 | eqid 2798 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
14 | eqid 2798 | . . . . 5 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
15 | 3, 14, 6 | ply1bas 20824 | . . . 4 ⊢ (Base‘𝑃) = (Base‘(1o mPoly 𝑅)) |
16 | 11, 12, 5, 13, 15 | evl1val 20953 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐴‘𝑋) ∈ (Base‘𝑃)) → (𝑂‘(𝐴‘𝑋)) = (((1o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
17 | 10, 16 | syldan 594 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (((1o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
18 | 3, 4 | ply1ascl 20887 | . . . . . . 7 ⊢ 𝐴 = (algSc‘(1o mPoly 𝑅)) |
19 | 5 | ressid 16551 | . . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅) |
20 | 19 | adantr 484 | . . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑅 ↾s 𝐵) = 𝑅) |
21 | 20 | oveq2d 7151 | . . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly 𝑅)) |
22 | 21 | fveq2d 6649 | . . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(1o mPoly 𝑅))) |
23 | 18, 22 | eqtr4id 2852 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴 = (algSc‘(1o mPoly (𝑅 ↾s 𝐵)))) |
24 | 23 | fveq1d 6647 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) = ((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑋)) |
25 | 24 | fveq2d 6649 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1o eval 𝑅)‘(𝐴‘𝑋)) = ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑋))) |
26 | 12, 5 | evlval 20767 | . . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵) |
27 | eqid 2798 | . . . . 5 ⊢ (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly (𝑅 ↾s 𝐵)) | |
28 | eqid 2798 | . . . . 5 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵) | |
29 | eqid 2798 | . . . . 5 ⊢ (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) | |
30 | 1on 8092 | . . . . . 6 ⊢ 1o ∈ On | |
31 | 30 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 1o ∈ On) |
32 | simpl 486 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ CRing) | |
33 | 5 | subrgid 19530 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) |
34 | 2, 33 | syl 17 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ (SubRing‘𝑅)) |
35 | simpr 488 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
36 | 26, 27, 28, 5, 29, 31, 32, 34, 35 | evlssca 20761 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑋)) = ((𝐵 ↑m 1o) × {𝑋})) |
37 | 25, 36 | eqtrd 2833 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1o eval 𝑅)‘(𝐴‘𝑋)) = ((𝐵 ↑m 1o) × {𝑋})) |
38 | 37 | coeq1d 5696 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((1o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (((𝐵 ↑m 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
39 | df1o2 8099 | . . . . . . 7 ⊢ 1o = {∅} | |
40 | 5 | fvexi 6659 | . . . . . . 7 ⊢ 𝐵 ∈ V |
41 | 0ex 5175 | . . . . . . 7 ⊢ ∅ ∈ V | |
42 | eqid 2798 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) | |
43 | 39, 40, 41, 42 | mapsnf1o3 8442 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑m 1o) |
44 | f1of 6590 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑m 1o) → (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵⟶(𝐵 ↑m 1o)) | |
45 | 43, 44 | mp1i 13 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵⟶(𝐵 ↑m 1o)) |
46 | 42 | fmpt 6851 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 (1o × {𝑦}) ∈ (𝐵 ↑m 1o) ↔ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵⟶(𝐵 ↑m 1o)) |
47 | 45, 46 | sylibr 237 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 (1o × {𝑦}) ∈ (𝐵 ↑m 1o)) |
48 | eqidd 2799 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) | |
49 | fconstmpt 5578 | . . . . 5 ⊢ ((𝐵 ↑m 1o) × {𝑋}) = (𝑥 ∈ (𝐵 ↑m 1o) ↦ 𝑋) | |
50 | 49 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((𝐵 ↑m 1o) × {𝑋}) = (𝑥 ∈ (𝐵 ↑m 1o) ↦ 𝑋)) |
51 | eqidd 2799 | . . . 4 ⊢ (𝑥 = (1o × {𝑦}) → 𝑋 = 𝑋) | |
52 | 47, 48, 50, 51 | fmptcof 6869 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑m 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (𝑦 ∈ 𝐵 ↦ 𝑋)) |
53 | fconstmpt 5578 | . . 3 ⊢ (𝐵 × {𝑋}) = (𝑦 ∈ 𝐵 ↦ 𝑋) | |
54 | 52, 53 | eqtr4di 2851 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑m 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (𝐵 × {𝑋})) |
55 | 17, 38, 54 | 3eqtrd 2837 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (𝐵 × {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∅c0 4243 {csn 4525 ↦ cmpt 5110 × cxp 5517 ∘ ccom 5523 Oncon0 6159 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 ↑m cmap 8389 Basecbs 16475 ↾s cress 16476 Ringcrg 19290 CRingccrg 19291 SubRingcsubrg 19524 algSccascl 20541 mPoly cmpl 20591 eval cevl 20744 PwSer1cps1 20804 Poly1cpl1 20806 eval1ce1 20938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-srg 19249 df-ring 19292 df-cring 19293 df-rnghom 19463 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-assa 20542 df-asp 20543 df-ascl 20544 df-psr 20594 df-mvr 20595 df-mpl 20596 df-opsr 20598 df-evls 20745 df-evl 20746 df-psr1 20809 df-ply1 20811 df-evl1 20940 |
This theorem is referenced by: evl1scad 20959 pf1const 20970 pf1ind 20979 evl1scvarpw 20987 ply1rem 24764 fta1g 24768 fta1blem 24769 plypf1 24809 |
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