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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 19768 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
7 | 6 | fveq2d 6415 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
8 | 2, 7 | syl5eq 2845 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘𝑃))) |
9 | 1, 8 | eleqtrd 2880 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝑃))) |
10 | mplascl.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑃) | |
11 | eqid 2799 | . . . 4 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
12 | eqid 2799 | . . . 4 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) | |
13 | eqid 2799 | . . . 4 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
14 | eqid 2799 | . . . 4 ⊢ (1r‘𝑃) = (1r‘𝑃) | |
15 | 10, 11, 12, 13, 14 | asclval 19658 | . . 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 2799 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
20 | 3, 17, 18, 19, 14, 4, 5 | mpl1 19767 | . . 3 ⊢ (𝜑 → (1r‘𝑃) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 ))) |
21 | 20 | oveq2d 6894 | . 2 ⊢ (𝜑 → (𝑋( ·𝑠 ‘𝑃)(1r‘𝑃)) = (𝑋( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 )))) |
22 | 17 | psrbag0 19816 | . . . 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 19815 | . 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 2837 | 1 ⊢ (𝜑 → (𝐴‘𝑋) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 {crab 3093 ifcif 4277 {csn 4368 ↦ cmpt 4922 × cxp 5310 ◡ccnv 5311 “ cima 5315 ‘cfv 6101 (class class class)co 6878 ↑𝑚 cmap 8095 Fincfn 8195 0cc0 10224 ℕcn 11312 ℕ0cn0 11580 Basecbs 16184 Scalarcsca 16270 ·𝑠 cvsca 16271 0gc0g 16415 1rcur 18817 Ringcrg 18863 algSccascl 19634 mPoly cmpl 19676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-ofr 7132 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-oi 8657 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-seq 13056 df-hash 13371 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-sca 16283 df-vsca 16284 df-tset 16286 df-0g 16417 df-gsum 16418 df-mre 16561 df-mrc 16562 df-acs 16564 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-mhm 17650 df-submnd 17651 df-grp 17741 df-minusg 17742 df-mulg 17857 df-subg 17904 df-ghm 17971 df-cntz 18062 df-cmn 18510 df-abl 18511 df-mgp 18806 df-ur 18818 df-ring 18865 df-subrg 19096 df-ascl 19637 df-psr 19679 df-mpl 19681 |
This theorem is referenced by: subrgascl 19820 subrgasclcl 19821 evlslem1 19837 mdegle0 24178 |
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