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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 18998 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | 1 | adantr 481 | . . . . 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 2795 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
7 | 3, 4, 5, 6 | ply1sclf 20136 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝐴:𝐵⟶(Base‘𝑃)) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴:𝐵⟶(Base‘𝑃)) |
9 | ffvelrn 6714 | . . . 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 2795 | . . . 4 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
13 | eqid 2795 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
14 | eqid 2795 | . . . . 5 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
15 | 3, 14, 6 | ply1bas 20046 | . . . 4 ⊢ (Base‘𝑃) = (Base‘(1o mPoly 𝑅)) |
16 | 11, 12, 5, 13, 15 | evl1val 20174 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐴‘𝑋) ∈ (Base‘𝑃)) → (𝑂‘(𝐴‘𝑋)) = (((1o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
17 | 10, 16 | syldan 591 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (((1o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
18 | 5 | ressid 16388 | . . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅) |
19 | 18 | adantr 481 | . . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑅 ↾s 𝐵) = 𝑅) |
20 | 19 | oveq2d 7032 | . . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly 𝑅)) |
21 | 20 | fveq2d 6542 | . . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(1o mPoly 𝑅))) |
22 | 3, 4 | ply1ascl 20109 | . . . . . . 7 ⊢ 𝐴 = (algSc‘(1o mPoly 𝑅)) |
23 | 21, 22 | syl6reqr 2850 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴 = (algSc‘(1o mPoly (𝑅 ↾s 𝐵)))) |
24 | 23 | fveq1d 6540 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) = ((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑋)) |
25 | 24 | fveq2d 6542 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1o eval 𝑅)‘(𝐴‘𝑋)) = ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑋))) |
26 | 12, 5 | evlval 19991 | . . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵) |
27 | eqid 2795 | . . . . 5 ⊢ (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly (𝑅 ↾s 𝐵)) | |
28 | eqid 2795 | . . . . 5 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵) | |
29 | eqid 2795 | . . . . 5 ⊢ (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) | |
30 | 1on 7960 | . . . . . 6 ⊢ 1o ∈ On | |
31 | 30 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 1o ∈ On) |
32 | simpl 483 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ CRing) | |
33 | 5 | subrgid 19227 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) |
34 | 2, 33 | syl 17 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ (SubRing‘𝑅)) |
35 | simpr 485 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
36 | 26, 27, 28, 5, 29, 31, 32, 34, 35 | evlssca 19989 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑋)) = ((𝐵 ↑𝑚 1o) × {𝑋})) |
37 | 25, 36 | eqtrd 2831 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1o eval 𝑅)‘(𝐴‘𝑋)) = ((𝐵 ↑𝑚 1o) × {𝑋})) |
38 | 37 | coeq1d 5618 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((1o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (((𝐵 ↑𝑚 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
39 | df1o2 7967 | . . . . . . 7 ⊢ 1o = {∅} | |
40 | 5 | fvexi 6552 | . . . . . . 7 ⊢ 𝐵 ∈ V |
41 | 0ex 5102 | . . . . . . 7 ⊢ ∅ ∈ V | |
42 | eqid 2795 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) | |
43 | 39, 40, 41, 42 | mapsnf1o3 8308 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑𝑚 1o) |
44 | f1of 6483 | . . . . . 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 6737 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 (1o × {𝑦}) ∈ (𝐵 ↑𝑚 1o) ↔ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵⟶(𝐵 ↑𝑚 1o)) |
47 | 45, 46 | sylibr 235 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 (1o × {𝑦}) ∈ (𝐵 ↑𝑚 1o)) |
48 | eqidd 2796 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) | |
49 | fconstmpt 5500 | . . . . 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 2796 | . . . 4 ⊢ (𝑥 = (1o × {𝑦}) → 𝑋 = 𝑋) | |
52 | 47, 48, 50, 51 | fmptcof 6755 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑𝑚 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (𝑦 ∈ 𝐵 ↦ 𝑋)) |
53 | fconstmpt 5500 | . . 3 ⊢ (𝐵 × {𝑋}) = (𝑦 ∈ 𝐵 ↦ 𝑋) | |
54 | 52, 53 | syl6eqr 2849 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑𝑚 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (𝐵 × {𝑋})) |
55 | 17, 38, 54 | 3eqtrd 2835 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (𝐵 × {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∀wral 3105 ∅c0 4211 {csn 4472 ↦ cmpt 5041 × cxp 5441 ∘ ccom 5447 Oncon0 6066 ⟶wf 6221 –1-1-onto→wf1o 6224 ‘cfv 6225 (class class class)co 7016 1oc1o 7946 ↑𝑚 cmap 8256 Basecbs 16312 ↾s cress 16313 Ringcrg 18987 CRingccrg 18988 SubRingcsubrg 19221 algSccascl 19773 mPoly cmpl 19821 eval cevl 19972 PwSer1cps1 20026 Poly1cpl1 20028 eval1ce1 20160 |
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 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
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 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-ofr 7268 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-pm 8259 df-ixp 8311 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-sup 8752 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-fz 12743 df-fzo 12884 df-seq 13220 df-hash 13541 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-hom 16418 df-cco 16419 df-0g 16544 df-gsum 16545 df-prds 16550 df-pws 16552 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-mhm 17774 df-submnd 17775 df-grp 17864 df-minusg 17865 df-sbg 17866 df-mulg 17982 df-subg 18030 df-ghm 18097 df-cntz 18188 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-srg 18946 df-ring 18989 df-cring 18990 df-rnghom 19157 df-subrg 19223 df-lmod 19326 df-lss 19394 df-lsp 19434 df-assa 19774 df-asp 19775 df-ascl 19776 df-psr 19824 df-mvr 19825 df-mpl 19826 df-opsr 19828 df-evls 19973 df-evl 19974 df-psr1 20031 df-ply1 20033 df-evl1 20162 |
This theorem is referenced by: evl1scad 20180 pf1const 20191 pf1ind 20200 evl1scvarpw 20208 ply1rem 24440 fta1g 24444 fta1blem 24445 plypf1 24485 |
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