![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ressply1bas2 | Structured version Visualization version GIF version |
Description: The base set of a restricted polynomial algebra consists of power series in the subring which are also polynomials (in the parent ring). (Contributed by Mario Carneiro, 3-Jul-2015.) |
Ref | Expression |
---|---|
ressply1.s | ⊢ 𝑆 = (Poly1‘𝑅) |
ressply1.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
ressply1.u | ⊢ 𝑈 = (Poly1‘𝐻) |
ressply1.b | ⊢ 𝐵 = (Base‘𝑈) |
ressply1.2 | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) |
ressply1bas2.w | ⊢ 𝑊 = (PwSer1‘𝐻) |
ressply1bas2.c | ⊢ 𝐶 = (Base‘𝑊) |
ressply1bas2.k | ⊢ 𝐾 = (Base‘𝑆) |
Ref | Expression |
---|---|
ressply1bas2 | ⊢ (𝜑 → 𝐵 = (𝐶 ∩ 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . 2 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
2 | ressply1.h | . 2 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
3 | eqid 2727 | . 2 ⊢ (1o mPoly 𝐻) = (1o mPoly 𝐻) | |
4 | ressply1.u | . . 3 ⊢ 𝑈 = (Poly1‘𝐻) | |
5 | ressply1bas2.w | . . 3 ⊢ 𝑊 = (PwSer1‘𝐻) | |
6 | ressply1.b | . . 3 ⊢ 𝐵 = (Base‘𝑈) | |
7 | 4, 5, 6 | ply1bas 22101 | . 2 ⊢ 𝐵 = (Base‘(1o mPoly 𝐻)) |
8 | 1on 8492 | . . 3 ⊢ 1o ∈ On | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → 1o ∈ On) |
10 | ressply1.2 | . 2 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
11 | eqid 2727 | . 2 ⊢ (1o mPwSer 𝐻) = (1o mPwSer 𝐻) | |
12 | ressply1bas2.c | . . 3 ⊢ 𝐶 = (Base‘𝑊) | |
13 | 5, 12, 11 | psr1bas2 22096 | . 2 ⊢ 𝐶 = (Base‘(1o mPwSer 𝐻)) |
14 | ressply1.s | . . 3 ⊢ 𝑆 = (Poly1‘𝑅) | |
15 | eqid 2727 | . . 3 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
16 | ressply1bas2.k | . . 3 ⊢ 𝐾 = (Base‘𝑆) | |
17 | 14, 15, 16 | ply1bas 22101 | . 2 ⊢ 𝐾 = (Base‘(1o mPoly 𝑅)) |
18 | 1, 2, 3, 7, 9, 10, 11, 13, 17 | ressmplbas2 21952 | 1 ⊢ (𝜑 → 𝐵 = (𝐶 ∩ 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∩ cin 3943 Oncon0 6363 ‘cfv 6542 (class class class)co 7414 1oc1o 8473 Basecbs 17171 ↾s cress 17200 SubRingcsubrg 20495 mPwSer cmps 21824 mPoly cmpl 21826 PwSer1cps1 22081 Poly1cpl1 22083 |
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 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
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 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-fzo 13652 df-seq 13991 df-hash 14314 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-hom 17248 df-cco 17249 df-0g 17414 df-gsum 17415 df-prds 17420 df-pws 17422 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-mulg 19015 df-subg 19069 df-ghm 19159 df-cntz 19259 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-subrng 20472 df-subrg 20497 df-psr 21829 df-mpl 21831 df-opsr 21833 df-psr1 22086 df-ply1 22088 |
This theorem is referenced by: ressply1bas 22134 ressdeg1 33177 ressply1invg 33180 ressply1evl 33183 evls1addd 33184 evls1subd 33185 evls1muld 33186 evls1vsca 33187 irngss 33297 evls1maprhm 33305 algextdeglem7 33327 algextdeglem8 33328 |
Copyright terms: Public domain | W3C validator |