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Mirrors > Home > MPE Home > Th. List > mpfconst | Structured version Visualization version GIF version |
Description: Constants are multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
mpfconst.b | ⊢ 𝐵 = (Base‘𝑆) |
mpfconst.q | ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) |
mpfconst.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mpfconst.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
mpfconst.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
mpfconst.x | ⊢ (𝜑 → 𝑋 ∈ 𝑅) |
Ref | Expression |
---|---|
mpfconst | ⊢ (𝜑 → ((𝐵 ↑m 𝐼) × {𝑋}) ∈ 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . . 4 ⊢ ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅) | |
2 | eqid 2736 | . . . 4 ⊢ (𝐼 mPoly (𝑆 ↾s 𝑅)) = (𝐼 mPoly (𝑆 ↾s 𝑅)) | |
3 | eqid 2736 | . . . 4 ⊢ (𝑆 ↾s 𝑅) = (𝑆 ↾s 𝑅) | |
4 | mpfconst.b | . . . 4 ⊢ 𝐵 = (Base‘𝑆) | |
5 | eqid 2736 | . . . 4 ⊢ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))) | |
6 | mpfconst.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
7 | mpfconst.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
8 | mpfconst.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
9 | mpfconst.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑅) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | evlssca 21499 | . . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑋)) = ((𝐵 ↑m 𝐼) × {𝑋})) |
11 | eqid 2736 | . . . . . . 7 ⊢ (𝑆 ↑s (𝐵 ↑m 𝐼)) = (𝑆 ↑s (𝐵 ↑m 𝐼)) | |
12 | 1, 2, 3, 11, 4 | evlsrhm 21498 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼)))) |
13 | 6, 7, 8, 12 | syl3anc 1371 | . . . . 5 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼)))) |
14 | eqid 2736 | . . . . . 6 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) | |
15 | eqid 2736 | . . . . . 6 ⊢ (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))) | |
16 | 14, 15 | rhmf 20158 | . . . . 5 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑m 𝐼)))) |
17 | ffn 6668 | . . . . 5 ⊢ (((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) | |
18 | 13, 16, 17 | 3syl 18 | . . . 4 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
19 | 3 | subrgring 20225 | . . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (𝑆 ↾s 𝑅) ∈ Ring) |
20 | 8, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑆 ↾s 𝑅) ∈ Ring) |
21 | eqid 2736 | . . . . . . 7 ⊢ (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))) | |
22 | 2 | mplring 21424 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑆 ↾s 𝑅) ∈ Ring) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring) |
23 | 2 | mpllmod 21423 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑆 ↾s 𝑅) ∈ Ring) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ LMod) |
24 | eqid 2736 | . . . . . . 7 ⊢ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) | |
25 | 5, 21, 22, 23, 24, 14 | asclf 21285 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑆 ↾s 𝑅) ∈ Ring) → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
26 | 6, 20, 25 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
27 | 4 | subrgss 20223 | . . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵) |
28 | 3, 4 | ressbas2 17120 | . . . . . . . 8 ⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘(𝑆 ↾s 𝑅))) |
29 | 8, 27, 28 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → 𝑅 = (Base‘(𝑆 ↾s 𝑅))) |
30 | ovexd 7392 | . . . . . . . . 9 ⊢ (𝜑 → (𝑆 ↾s 𝑅) ∈ V) | |
31 | 2, 6, 30 | mplsca 21417 | . . . . . . . 8 ⊢ (𝜑 → (𝑆 ↾s 𝑅) = (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
32 | 31 | fveq2d 6846 | . . . . . . 7 ⊢ (𝜑 → (Base‘(𝑆 ↾s 𝑅)) = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))) |
33 | 29, 32 | eqtrd 2776 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))) |
34 | 9, 33 | eleqtrd 2840 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))) |
35 | 26, 34 | ffvelcdmd 7036 | . . . 4 ⊢ (𝜑 → ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑋) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
36 | fnfvelrn 7031 | . . . 4 ⊢ ((((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑋) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑋)) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅)) | |
37 | 18, 35, 36 | syl2anc 584 | . . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑋)) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅)) |
38 | 10, 37 | eqeltrrd 2839 | . 2 ⊢ (𝜑 → ((𝐵 ↑m 𝐼) × {𝑋}) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅)) |
39 | mpfconst.q | . 2 ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) | |
40 | 38, 39 | eleqtrrdi 2849 | 1 ⊢ (𝜑 → ((𝐵 ↑m 𝐼) × {𝑋}) ∈ 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ⊆ wss 3910 {csn 4586 × cxp 5631 ran crn 5634 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑m cmap 8765 Basecbs 17083 ↾s cress 17112 Scalarcsca 17136 ↑s cpws 17328 Ringcrg 19964 CRingccrg 19965 RingHom crh 20143 SubRingcsubrg 20218 algSccascl 21258 mPoly cmpl 21308 evalSub ces 21480 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-srg 19918 df-ring 19966 df-cring 19967 df-rnghom 20146 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-assa 21259 df-asp 21260 df-ascl 21261 df-psr 21311 df-mvr 21312 df-mpl 21313 df-evls 21482 |
This theorem is referenced by: mzpmfp 41056 |
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