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Mirrors > Home > MPE Home > Th. List > mpl1 | Structured version Visualization version GIF version |
Description: The identity element of the ring of polynomials. (Contributed by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
mpl1.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mpl1.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
mpl1.z | ⊢ 0 = (0g‘𝑅) |
mpl1.o | ⊢ 1 = (1r‘𝑅) |
mpl1.u | ⊢ 𝑈 = (1r‘𝑃) |
mpl1.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mpl1.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
Ref | Expression |
---|---|
mpl1 | ⊢ (𝜑 → 𝑈 = (𝑥 ∈ 𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpl1.u | . 2 ⊢ 𝑈 = (1r‘𝑃) | |
2 | eqid 2825 | . . . . 5 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
3 | mpl1.p | . . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
4 | eqid 2825 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
5 | mpl1.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
6 | mpl1.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
7 | 2, 3, 4, 5, 6 | mplsubrg 19801 | . . . 4 ⊢ (𝜑 → (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅))) |
8 | 3, 2, 4 | mplval2 19792 | . . . . 5 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑃)) |
9 | eqid 2825 | . . . . 5 ⊢ (1r‘(𝐼 mPwSer 𝑅)) = (1r‘(𝐼 mPwSer 𝑅)) | |
10 | 8, 9 | subrg1 19146 | . . . 4 ⊢ ((Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → (1r‘(𝐼 mPwSer 𝑅)) = (1r‘𝑃)) |
11 | 7, 10 | syl 17 | . . 3 ⊢ (𝜑 → (1r‘(𝐼 mPwSer 𝑅)) = (1r‘𝑃)) |
12 | mpl1.d | . . . 4 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
13 | mpl1.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
14 | mpl1.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
15 | 2, 5, 6, 12, 13, 14, 9 | psr1 19773 | . . 3 ⊢ (𝜑 → (1r‘(𝐼 mPwSer 𝑅)) = (𝑥 ∈ 𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 ))) |
16 | 11, 15 | eqtr3d 2863 | . 2 ⊢ (𝜑 → (1r‘𝑃) = (𝑥 ∈ 𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 ))) |
17 | 1, 16 | syl5eq 2873 | 1 ⊢ (𝜑 → 𝑈 = (𝑥 ∈ 𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 {crab 3121 ifcif 4306 {csn 4397 ↦ cmpt 4952 × cxp 5340 ◡ccnv 5341 “ cima 5345 ‘cfv 6123 (class class class)co 6905 ↑𝑚 cmap 8122 Fincfn 8222 0cc0 10252 ℕcn 11350 ℕ0cn0 11618 Basecbs 16222 0gc0g 16453 1rcur 18855 Ringcrg 18901 SubRingcsubrg 19132 mPwSer cmps 19712 mPoly cmpl 19714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-ofr 7158 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-ixp 8176 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-seq 13096 df-hash 13411 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-tset 16324 df-0g 16455 df-gsum 16456 df-mre 16599 df-mrc 16600 df-acs 16602 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-mhm 17688 df-submnd 17689 df-grp 17779 df-minusg 17780 df-mulg 17895 df-subg 17942 df-ghm 18009 df-cntz 18100 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-subrg 19134 df-psr 19717 df-mpl 19719 |
This theorem is referenced by: mplcoe3 19827 mplcoe5 19829 mplascl 19856 ply1nzb 24281 |
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