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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 20225 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
7 | 6 | fveq2d 6674 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
8 | 2, 7 | syl5eq 2868 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘𝑃))) |
9 | 1, 8 | eleqtrd 2915 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝑃))) |
10 | mplascl.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑃) | |
11 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
12 | eqid 2821 | . . . 4 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) | |
13 | eqid 2821 | . . . 4 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
14 | eqid 2821 | . . . 4 ⊢ (1r‘𝑃) = (1r‘𝑃) | |
15 | 10, 11, 12, 13, 14 | asclval 20109 | . . 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 2821 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
20 | 3, 17, 18, 19, 14, 4, 5 | mpl1 20224 | . . 3 ⊢ (𝜑 → (1r‘𝑃) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 ))) |
21 | 20 | oveq2d 7172 | . 2 ⊢ (𝜑 → (𝑋( ·𝑠 ‘𝑃)(1r‘𝑃)) = (𝑋( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 )))) |
22 | 17 | psrbag0 20274 | . . . 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 20273 | . 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 2860 | 1 ⊢ (𝜑 → (𝐴‘𝑋) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3142 ifcif 4467 {csn 4567 ↦ cmpt 5146 × cxp 5553 ◡ccnv 5554 “ cima 5558 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 Fincfn 8509 0cc0 10537 ℕcn 11638 ℕ0cn0 11898 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 1rcur 19251 Ringcrg 19297 algSccascl 20084 mPoly cmpl 20133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-tset 16584 df-0g 16715 df-gsum 16716 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-mulg 18225 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-subrg 19533 df-ascl 20087 df-psr 20136 df-mpl 20138 |
This theorem is referenced by: subrgascl 20278 subrgasclcl 20279 evlslem1 20295 mdegle0 24671 |
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