![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2728 | . . . . . 6 ⊢ (𝐼 mVar 𝑈) = (𝐼 mVar 𝑈) | |
7 | evlssca.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
8 | eqid 2728 | . . . . . 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 2728 | . . . . . 6 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) | |
12 | eqid 2728 | . . . . . 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 22033 | . . . . 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 1369 | . . . 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 6899 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝐴)‘𝑋) = ((𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))‘𝑋)) |
17 | eqid 2728 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
18 | eqid 2728 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
19 | 7 | subrgring 20513 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
20 | 3, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ Ring) |
21 | 5, 17, 18, 10, 1, 20 | mplasclf 22009 | . . . 4 ⊢ (𝜑 → 𝐴:(Base‘𝑈)⟶(Base‘𝑊)) |
22 | 9 | subrgss 20511 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵) |
23 | 7, 9 | ressbas2 17218 | . . . . . 6 ⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘𝑈)) |
24 | 3, 22, 23 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑅 = (Base‘𝑈)) |
25 | 24 | feq2d 6708 | . . . 4 ⊢ (𝜑 → (𝐴:𝑅⟶(Base‘𝑊) ↔ 𝐴:(Base‘𝑈)⟶(Base‘𝑊))) |
26 | 21, 25 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝐴:𝑅⟶(Base‘𝑊)) |
27 | evlssca.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑅) | |
28 | fvco3 6997 | . . 3 ⊢ ((𝐴:𝑅⟶(Base‘𝑊) ∧ 𝑋 ∈ 𝑅) → ((𝑄 ∘ 𝐴)‘𝑋) = (𝑄‘(𝐴‘𝑋))) | |
29 | 26, 27, 28 | syl2anc 583 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝐴)‘𝑋) = (𝑄‘(𝐴‘𝑋))) |
30 | sneq 4639 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
31 | 30 | xpeq2d 5708 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐵 ↑m 𝐼) × {𝑥}) = ((𝐵 ↑m 𝐼) × {𝑋})) |
32 | ovex 7453 | . . . . 5 ⊢ (𝐵 ↑m 𝐼) ∈ V | |
33 | snex 5433 | . . . . 5 ⊢ {𝑋} ∈ V | |
34 | 32, 33 | xpex 7755 | . . . 4 ⊢ ((𝐵 ↑m 𝐼) × {𝑋}) ∈ V |
35 | 31, 11, 34 | fvmpt 7005 | . . 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 2776 | 1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑m 𝐼) × {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 {csn 4629 ↦ cmpt 5231 × cxp 5676 ∘ ccom 5682 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ↑m cmap 8845 Basecbs 17180 ↾s cress 17209 ↑s cpws 17428 Ringcrg 20173 CRingccrg 20174 RingHom crh 20408 SubRingcsubrg 20506 algSccascl 21786 mVar cmvr 21838 mPoly cmpl 21839 evalSub ces 22016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-ofr 7686 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-srg 20127 df-ring 20175 df-cring 20176 df-rhm 20411 df-subrng 20483 df-subrg 20508 df-lmod 20745 df-lss 20816 df-lsp 20856 df-assa 21787 df-asp 21788 df-ascl 21789 df-psr 21842 df-mvr 21843 df-mpl 21844 df-evls 22018 |
This theorem is referenced by: evlsscasrng 22043 evlsca 22044 mpfconst 22047 mpfind 22053 evls1sca 22242 evl1sca 22253 pf1ind 22274 evlsscaval 41797 |
Copyright terms: Public domain | W3C validator |