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Mirrors > Home > MPE Home > Th. List > mplascl | Structured version Visualization version GIF version |
Description: Value of the scalar injection into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
Ref | Expression |
---|---|
mplascl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplascl.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
mplascl.z | ⊢ 0 = (0g‘𝑅) |
mplascl.b | ⊢ 𝐵 = (Base‘𝑅) |
mplascl.a | ⊢ 𝐴 = (algSc‘𝑃) |
mplascl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mplascl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mplascl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
mplascl | ⊢ (𝜑 → (𝐴‘𝑋) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplascl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | mplascl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
3 | mplascl.p | . . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
4 | mplascl.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | mplascl.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
6 | 3, 4, 5 | mplsca 21215 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
7 | 6 | fveq2d 6780 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
8 | 2, 7 | eqtrid 2790 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘𝑃))) |
9 | 1, 8 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝑃))) |
10 | mplascl.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑃) | |
11 | eqid 2738 | . . . 4 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
12 | eqid 2738 | . . . 4 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) | |
13 | eqid 2738 | . . . 4 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
14 | eqid 2738 | . . . 4 ⊢ (1r‘𝑃) = (1r‘𝑃) | |
15 | 10, 11, 12, 13, 14 | asclval 21082 | . . 3 ⊢ (𝑋 ∈ (Base‘(Scalar‘𝑃)) → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑃)(1r‘𝑃))) |
16 | 9, 15 | syl 17 | . 2 ⊢ (𝜑 → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑃)(1r‘𝑃))) |
17 | mplascl.d | . . . 4 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
18 | mplascl.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
19 | eqid 2738 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
20 | 3, 17, 18, 19, 14, 4, 5 | mpl1 21214 | . . 3 ⊢ (𝜑 → (1r‘𝑃) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 ))) |
21 | 20 | oveq2d 7293 | . 2 ⊢ (𝜑 → (𝑋( ·𝑠 ‘𝑃)(1r‘𝑃)) = (𝑋( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 )))) |
22 | 17 | psrbag0 21268 | . . . 4 ⊢ (𝐼 ∈ 𝑊 → (𝐼 × {0}) ∈ 𝐷) |
23 | 4, 22 | syl 17 | . . 3 ⊢ (𝜑 → (𝐼 × {0}) ∈ 𝐷) |
24 | 3, 13, 17, 19, 18, 2, 4, 5, 23, 1 | mplmon2 21267 | . 2 ⊢ (𝜑 → (𝑋( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 ))) |
25 | 16, 21, 24 | 3eqtrd 2782 | 1 ⊢ (𝜑 → (𝐴‘𝑋) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 {crab 3068 ifcif 4461 {csn 4563 ↦ cmpt 5159 × cxp 5589 ◡ccnv 5590 “ cima 5594 ‘cfv 6435 (class class class)co 7277 ↑m cmap 8613 Fincfn 8731 0cc0 10869 ℕcn 11971 ℕ0cn0 12231 Basecbs 16910 Scalarcsca 16963 ·𝑠 cvsca 16964 0gc0g 17148 1rcur 19735 Ringcrg 19781 algSccascl 21057 mPoly cmpl 21107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-isom 6444 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7976 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-map 8615 df-pm 8616 df-ixp 8684 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-fsupp 9127 df-oi 9267 df-card 9695 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-uz 12581 df-fz 13238 df-fzo 13381 df-seq 13720 df-hash 14043 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-tset 16979 df-0g 17150 df-gsum 17151 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-submnd 18429 df-grp 18578 df-minusg 18579 df-mulg 18699 df-subg 18750 df-ghm 18830 df-cntz 18921 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-subrg 20020 df-ascl 21060 df-psr 21110 df-mpl 21112 |
This theorem is referenced by: subrgascl 21272 subrgasclcl 21273 evlslem1 21290 mhpsclcl 21335 mdegle0 25240 |
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