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 2739 | . . . . . 6 ⊢ (𝐼 mVar 𝑈) = (𝐼 mVar 𝑈) | |
7 | evlssca.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
8 | eqid 2739 | . . . . . 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 2739 | . . . . . 6 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) | |
12 | eqid 2739 | . . . . . 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 20913 | . . . . 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 1372 | . . . 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 772 | . . 3 ⊢ (𝜑 → (𝑄 ∘ 𝐴) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))) |
16 | 15 | fveq1d 6688 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝐴)‘𝑋) = ((𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))‘𝑋)) |
17 | eqid 2739 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
18 | eqid 2739 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
19 | 7 | subrgring 19669 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
20 | 3, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ Ring) |
21 | 5, 17, 18, 10, 1, 20 | mplasclf 20889 | . . . 4 ⊢ (𝜑 → 𝐴:(Base‘𝑈)⟶(Base‘𝑊)) |
22 | 9 | subrgss 19667 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵) |
23 | 7, 9 | ressbas2 16670 | . . . . . 6 ⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘𝑈)) |
24 | 3, 22, 23 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑅 = (Base‘𝑈)) |
25 | 24 | feq2d 6500 | . . . 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 6779 | . . 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 4536 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
31 | 30 | xpeq2d 5565 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐵 ↑m 𝐼) × {𝑥}) = ((𝐵 ↑m 𝐼) × {𝑋})) |
32 | ovex 7215 | . . . . 5 ⊢ (𝐵 ↑m 𝐼) ∈ V | |
33 | snex 5308 | . . . . 5 ⊢ {𝑋} ∈ V | |
34 | 32, 33 | xpex 7506 | . . . 4 ⊢ ((𝐵 ↑m 𝐼) × {𝑋}) ∈ V |
35 | 31, 11, 34 | fvmpt 6787 | . . 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 2782 | 1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑m 𝐼) × {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ⊆ wss 3853 {csn 4526 ↦ cmpt 5120 × cxp 5533 ∘ ccom 5539 ⟶wf 6345 ‘cfv 6349 (class class class)co 7182 ↑m cmap 8449 Basecbs 16598 ↾s cress 16599 ↑s cpws 16835 Ringcrg 19428 CRingccrg 19429 RingHom crh 19598 SubRingcsubrg 19662 algSccascl 20680 mVar cmvr 20730 mPoly cmpl 20731 evalSub ces 20896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-of 7437 df-ofr 7438 df-om 7612 df-1st 7726 df-2nd 7727 df-supp 7869 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-map 8451 df-pm 8452 df-ixp 8520 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-fsupp 8919 df-sup 8991 df-oi 9059 df-card 9453 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-fz 12994 df-fzo 13137 df-seq 13473 df-hash 13795 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-sca 16696 df-vsca 16697 df-ip 16698 df-tset 16699 df-ple 16700 df-ds 16702 df-hom 16704 df-cco 16705 df-0g 16830 df-gsum 16831 df-prds 16836 df-pws 16838 df-mre 16972 df-mrc 16973 df-acs 16975 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-mhm 18084 df-submnd 18085 df-grp 18234 df-minusg 18235 df-sbg 18236 df-mulg 18355 df-subg 18406 df-ghm 18486 df-cntz 18577 df-cmn 19038 df-abl 19039 df-mgp 19371 df-ur 19383 df-srg 19387 df-ring 19430 df-cring 19431 df-rnghom 19601 df-subrg 19664 df-lmod 19767 df-lss 19835 df-lsp 19875 df-assa 20681 df-asp 20682 df-ascl 20683 df-psr 20734 df-mvr 20735 df-mpl 20736 df-evls 20898 |
This theorem is referenced by: evlsscasrng 20923 evlsca 20924 mpfconst 20927 mpfind 20933 evls1sca 21105 evl1sca 21116 pf1ind 21137 evlsscaval 39892 |
Copyright terms: Public domain | W3C validator |