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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 19310 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | 1 | adantr 483 | . . . . 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 2823 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
7 | 3, 4, 5, 6 | ply1sclf 20455 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝐴:𝐵⟶(Base‘𝑃)) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴:𝐵⟶(Base‘𝑃)) |
9 | ffvelrn 6851 | . . . 4 ⊢ ((𝐴:𝐵⟶(Base‘𝑃) ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ (Base‘𝑃)) | |
10 | 8, 9 | sylancom 590 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ (Base‘𝑃)) |
11 | evl1sca.o | . . . 4 ⊢ 𝑂 = (eval1‘𝑅) | |
12 | eqid 2823 | . . . 4 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
13 | eqid 2823 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
14 | eqid 2823 | . . . . 5 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
15 | 3, 14, 6 | ply1bas 20365 | . . . 4 ⊢ (Base‘𝑃) = (Base‘(1o mPoly 𝑅)) |
16 | 11, 12, 5, 13, 15 | evl1val 20494 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐴‘𝑋) ∈ (Base‘𝑃)) → (𝑂‘(𝐴‘𝑋)) = (((1o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
17 | 10, 16 | syldan 593 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (((1o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
18 | 5 | ressid 16561 | . . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅) |
19 | 18 | adantr 483 | . . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑅 ↾s 𝐵) = 𝑅) |
20 | 19 | oveq2d 7174 | . . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly 𝑅)) |
21 | 20 | fveq2d 6676 | . . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(1o mPoly 𝑅))) |
22 | 3, 4 | ply1ascl 20428 | . . . . . . 7 ⊢ 𝐴 = (algSc‘(1o mPoly 𝑅)) |
23 | 21, 22 | syl6reqr 2877 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴 = (algSc‘(1o mPoly (𝑅 ↾s 𝐵)))) |
24 | 23 | fveq1d 6674 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) = ((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑋)) |
25 | 24 | fveq2d 6676 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1o eval 𝑅)‘(𝐴‘𝑋)) = ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑋))) |
26 | 12, 5 | evlval 20310 | . . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵) |
27 | eqid 2823 | . . . . 5 ⊢ (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly (𝑅 ↾s 𝐵)) | |
28 | eqid 2823 | . . . . 5 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵) | |
29 | eqid 2823 | . . . . 5 ⊢ (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) | |
30 | 1on 8111 | . . . . . 6 ⊢ 1o ∈ On | |
31 | 30 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 1o ∈ On) |
32 | simpl 485 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ CRing) | |
33 | 5 | subrgid 19539 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) |
34 | 2, 33 | syl 17 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ (SubRing‘𝑅)) |
35 | simpr 487 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
36 | 26, 27, 28, 5, 29, 31, 32, 34, 35 | evlssca 20304 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑋)) = ((𝐵 ↑m 1o) × {𝑋})) |
37 | 25, 36 | eqtrd 2858 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1o eval 𝑅)‘(𝐴‘𝑋)) = ((𝐵 ↑m 1o) × {𝑋})) |
38 | 37 | coeq1d 5734 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((1o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (((𝐵 ↑m 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
39 | df1o2 8118 | . . . . . . 7 ⊢ 1o = {∅} | |
40 | 5 | fvexi 6686 | . . . . . . 7 ⊢ 𝐵 ∈ V |
41 | 0ex 5213 | . . . . . . 7 ⊢ ∅ ∈ V | |
42 | eqid 2823 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) | |
43 | 39, 40, 41, 42 | mapsnf1o3 8461 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑m 1o) |
44 | f1of 6617 | . . . . . 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 6876 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 (1o × {𝑦}) ∈ (𝐵 ↑m 1o) ↔ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵⟶(𝐵 ↑m 1o)) |
47 | 45, 46 | sylibr 236 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 (1o × {𝑦}) ∈ (𝐵 ↑m 1o)) |
48 | eqidd 2824 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) | |
49 | fconstmpt 5616 | . . . . 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 2824 | . . . 4 ⊢ (𝑥 = (1o × {𝑦}) → 𝑋 = 𝑋) | |
52 | 47, 48, 50, 51 | fmptcof 6894 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑m 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (𝑦 ∈ 𝐵 ↦ 𝑋)) |
53 | fconstmpt 5616 | . . 3 ⊢ (𝐵 × {𝑋}) = (𝑦 ∈ 𝐵 ↦ 𝑋) | |
54 | 52, 53 | syl6eqr 2876 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑m 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (𝐵 × {𝑋})) |
55 | 17, 38, 54 | 3eqtrd 2862 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (𝐵 × {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∅c0 4293 {csn 4569 ↦ cmpt 5148 × cxp 5555 ∘ ccom 5561 Oncon0 6193 ⟶wf 6353 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 1oc1o 8097 ↑m cmap 8408 Basecbs 16485 ↾s cress 16486 Ringcrg 19299 CRingccrg 19300 SubRingcsubrg 19533 algSccascl 20086 mPoly cmpl 20135 eval cevl 20287 PwSer1cps1 20345 Poly1cpl1 20347 eval1ce1 20479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-0g 16717 df-gsum 16718 df-prds 16723 df-pws 16725 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-srg 19258 df-ring 19301 df-cring 19302 df-rnghom 19469 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-assa 20087 df-asp 20088 df-ascl 20089 df-psr 20138 df-mvr 20139 df-mpl 20140 df-opsr 20142 df-evls 20288 df-evl 20289 df-psr1 20350 df-ply1 20352 df-evl1 20481 |
This theorem is referenced by: evl1scad 20500 pf1const 20511 pf1ind 20520 evl1scvarpw 20528 ply1rem 24759 fta1g 24763 fta1blem 24764 plypf1 24804 |
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