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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 2727 | . . . . 5 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
3 | mpl1.p | . . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
4 | eqid 2727 | . . . . 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 21940 | . . . 4 ⊢ (𝜑 → (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅))) |
8 | 3, 2, 4 | mplval2 21931 | . . . . 5 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑃)) |
9 | eqid 2727 | . . . . 5 ⊢ (1r‘(𝐼 mPwSer 𝑅)) = (1r‘(𝐼 mPwSer 𝑅)) | |
10 | 8, 9 | subrg1 20514 | . . . 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 21907 | . . 3 ⊢ (𝜑 → (1r‘(𝐼 mPwSer 𝑅)) = (𝑥 ∈ 𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 ))) |
16 | 11, 15 | eqtr3d 2769 | . 2 ⊢ (𝜑 → (1r‘𝑃) = (𝑥 ∈ 𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 ))) |
17 | 1, 16 | eqtrid 2779 | 1 ⊢ (𝜑 → 𝑈 = (𝑥 ∈ 𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {crab 3427 ifcif 4524 {csn 4624 ↦ cmpt 5225 × cxp 5670 ◡ccnv 5671 “ cima 5675 ‘cfv 6542 (class class class)co 7414 ↑m cmap 8838 Fincfn 8957 0cc0 11132 ℕcn 12236 ℕ0cn0 12496 Basecbs 17173 0gc0g 17414 1rcur 20114 Ringcrg 20166 SubRingcsubrg 20499 mPwSer cmps 21830 mPoly cmpl 21832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-hom 17250 df-cco 17251 df-0g 17416 df-gsum 17417 df-prds 17422 df-pws 17424 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-grp 18886 df-minusg 18887 df-mulg 19017 df-subg 19071 df-ghm 19161 df-cntz 19261 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-subrng 20476 df-subrg 20501 df-psr 21835 df-mpl 21837 |
This theorem is referenced by: mplcoe3 21969 mplcoe5 21971 mplascl 22001 ply1nzb 26051 rhmmpl 41758 |
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