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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 2737 | . . . . . 6 ⊢ (𝐼 mVar 𝑈) = (𝐼 mVar 𝑈) | |
| 7 | evlssca.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
| 8 | eqid 2737 | . . . . . 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 2737 | . . . . . 6 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) | |
| 12 | eqid 2737 | . . . . . 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 22111 | . . . . 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 1373 | . . . 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 6908 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝐴)‘𝑋) = ((𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥}))‘𝑋)) |
| 17 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 18 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
| 19 | 7 | subrgring 20574 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
| 20 | 3, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ Ring) |
| 21 | 5, 17, 18, 10, 1, 20 | mplasclf 22089 | . . . 4 ⊢ (𝜑 → 𝐴:(Base‘𝑈)⟶(Base‘𝑊)) |
| 22 | 9 | subrgss 20572 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵) |
| 23 | 7, 9 | ressbas2 17283 | . . . . . 6 ⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘𝑈)) |
| 24 | 3, 22, 23 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑅 = (Base‘𝑈)) |
| 25 | 24 | feq2d 6722 | . . . 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 7008 | . . 3 ⊢ ((𝐴:𝑅⟶(Base‘𝑊) ∧ 𝑋 ∈ 𝑅) → ((𝑄 ∘ 𝐴)‘𝑋) = (𝑄‘(𝐴‘𝑋))) | |
| 29 | 26, 27, 28 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝐴)‘𝑋) = (𝑄‘(𝐴‘𝑋))) |
| 30 | sneq 4636 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
| 31 | 30 | xpeq2d 5715 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐵 ↑m 𝐼) × {𝑥}) = ((𝐵 ↑m 𝐼) × {𝑋})) |
| 32 | ovex 7464 | . . . . 5 ⊢ (𝐵 ↑m 𝐼) ∈ V | |
| 33 | snex 5436 | . . . . 5 ⊢ {𝑋} ∈ V | |
| 34 | 32, 33 | xpex 7773 | . . . 4 ⊢ ((𝐵 ↑m 𝐼) × {𝑋}) ∈ V |
| 35 | 31, 11, 34 | fvmpt 7016 | . . 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 2785 | 1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑m 𝐼) × {𝑋})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 {csn 4626 ↦ cmpt 5225 × cxp 5683 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Basecbs 17247 ↾s cress 17274 ↑s cpws 17491 Ringcrg 20230 CRingccrg 20231 RingHom crh 20469 SubRingcsubrg 20569 algSccascl 21872 mVar cmvr 21925 mPoly cmpl 21926 evalSub ces 22096 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-srg 20184 df-ring 20232 df-cring 20233 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-lsp 20970 df-assa 21873 df-asp 21874 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-evls 22098 |
| This theorem is referenced by: evlsscasrng 22121 evlsca 22122 mpfconst 22125 mpfind 22131 evls1sca 22327 evl1sca 22338 pf1ind 22359 evlsscaval 42574 |
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