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Mirrors > Home > MPE Home > Th. List > evlssca | Structured version Visualization version GIF version |
Description: Polynomial evaluation maps scalars to constant functions. (Contributed by Stefan O'Rear, 13-Mar-2015.) (Proof shortened by AV, 18-Sep-2021.) |
Ref | Expression |
---|---|
evlssca.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlssca.w | ⊢ 𝑊 = (𝐼 mPoly 𝑈) |
evlssca.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlssca.b | ⊢ 𝐵 = (Base‘𝑆) |
evlssca.a | ⊢ 𝐴 = (algSc‘𝑊) |
evlssca.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
evlssca.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlssca.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlssca.x | ⊢ (𝜑 → 𝑋 ∈ 𝑅) |
Ref | Expression |
---|---|
evlssca | ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑m 𝐼) × {𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlssca.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
2 | evlssca.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
3 | evlssca.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
4 | evlssca.q | . . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
5 | evlssca.w | . . . . . 6 ⊢ 𝑊 = (𝐼 mPoly 𝑈) | |
6 | eqid 2798 | . . . . . 6 ⊢ (𝐼 mVar 𝑈) = (𝐼 mVar 𝑈) | |
7 | evlssca.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
8 | eqid 2798 | . . . . . 6 ⊢ (𝑆 ↑s (𝐵 ↑m 𝐼)) = (𝑆 ↑s (𝐵 ↑m 𝐼)) | |
9 | evlssca.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑆) | |
10 | evlssca.a | . . . . . 6 ⊢ 𝐴 = (algSc‘𝑊) | |
11 | eqid 2798 | . . . . . 6 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) | |
12 | eqid 2798 | . . . . . 6 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ (𝐵 ↑m 𝐼) ↦ (𝑦‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ (𝐵 ↑m 𝐼) ↦ (𝑦‘𝑥))) | |
13 | 4, 5, 6, 7, 8, 9, 10, 11, 12 | evlsval2 20759 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom (𝑆 ↑s (𝐵 ↑m 𝐼))) ∧ ((𝑄 ∘ 𝐴) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ∧ (𝑄 ∘ (𝐼 mVar 𝑈)) = (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ (𝐵 ↑m 𝐼) ↦ (𝑦‘𝑥)))))) |
14 | 1, 2, 3, 13 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝑄 ∈ (𝑊 RingHom (𝑆 ↑s (𝐵 ↑m 𝐼))) ∧ ((𝑄 ∘ 𝐴) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ∧ (𝑄 ∘ (𝐼 mVar 𝑈)) = (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ (𝐵 ↑m 𝐼) ↦ (𝑦‘𝑥)))))) |
15 | 14 | simprld 771 | . . 3 ⊢ (𝜑 → (𝑄 ∘ 𝐴) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))) |
16 | 15 | fveq1d 6647 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝐴)‘𝑋) = ((𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))‘𝑋)) |
17 | eqid 2798 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
18 | eqid 2798 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
19 | 7 | subrgring 19531 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
20 | 3, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ Ring) |
21 | 5, 17, 18, 10, 1, 20 | mplasclf 20736 | . . . 4 ⊢ (𝜑 → 𝐴:(Base‘𝑈)⟶(Base‘𝑊)) |
22 | 9 | subrgss 19529 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵) |
23 | 7, 9 | ressbas2 16547 | . . . . . 6 ⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘𝑈)) |
24 | 3, 22, 23 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑅 = (Base‘𝑈)) |
25 | 24 | feq2d 6473 | . . . 4 ⊢ (𝜑 → (𝐴:𝑅⟶(Base‘𝑊) ↔ 𝐴:(Base‘𝑈)⟶(Base‘𝑊))) |
26 | 21, 25 | mpbird 260 | . . 3 ⊢ (𝜑 → 𝐴:𝑅⟶(Base‘𝑊)) |
27 | evlssca.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑅) | |
28 | fvco3 6737 | . . 3 ⊢ ((𝐴:𝑅⟶(Base‘𝑊) ∧ 𝑋 ∈ 𝑅) → ((𝑄 ∘ 𝐴)‘𝑋) = (𝑄‘(𝐴‘𝑋))) | |
29 | 26, 27, 28 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝐴)‘𝑋) = (𝑄‘(𝐴‘𝑋))) |
30 | sneq 4535 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
31 | 30 | xpeq2d 5549 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐵 ↑m 𝐼) × {𝑥}) = ((𝐵 ↑m 𝐼) × {𝑋})) |
32 | ovex 7168 | . . . . 5 ⊢ (𝐵 ↑m 𝐼) ∈ V | |
33 | snex 5297 | . . . . 5 ⊢ {𝑋} ∈ V | |
34 | 32, 33 | xpex 7456 | . . . 4 ⊢ ((𝐵 ↑m 𝐼) × {𝑋}) ∈ V |
35 | 31, 11, 34 | fvmpt 6745 | . . 3 ⊢ (𝑋 ∈ 𝑅 → ((𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))‘𝑋) = ((𝐵 ↑m 𝐼) × {𝑋})) |
36 | 27, 35 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))‘𝑋) = ((𝐵 ↑m 𝐼) × {𝑋})) |
37 | 16, 29, 36 | 3eqtr3d 2841 | 1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑m 𝐼) × {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 {csn 4525 ↦ cmpt 5110 × cxp 5517 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 Basecbs 16475 ↾s cress 16476 ↑s cpws 16712 Ringcrg 19290 CRingccrg 19291 RingHom crh 19460 SubRingcsubrg 19524 algSccascl 20541 mVar cmvr 20590 mPoly cmpl 20591 evalSub ces 20743 |
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-evls 20745 |
This theorem is referenced by: evlsscasrng 20769 evlsca 20770 mpfconst 20773 mpfind 20779 evls1sca 20947 evl1sca 20958 pf1ind 20979 |
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