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Mirrors > Home > MPE Home > Th. List > ressmplbas2 | 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 |
---|---|
ressmpl.s | ⊢ 𝑆 = (𝐼 mPoly 𝑅) |
ressmpl.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
ressmpl.u | ⊢ 𝑈 = (𝐼 mPoly 𝐻) |
ressmpl.b | ⊢ 𝐵 = (Base‘𝑈) |
ressmpl.1 | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
ressmpl.2 | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) |
ressmplbas2.w | ⊢ 𝑊 = (𝐼 mPwSer 𝐻) |
ressmplbas2.c | ⊢ 𝐶 = (Base‘𝑊) |
ressmplbas2.k | ⊢ 𝐾 = (Base‘𝑆) |
Ref | Expression |
---|---|
ressmplbas2 | ⊢ (𝜑 → 𝐵 = (𝐶 ∩ 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressmpl.1 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
2 | ressmpl.2 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
3 | eqid 2798 | . . . . . . . 8 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
4 | ressmpl.h | . . . . . . . 8 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
5 | ressmplbas2.w | . . . . . . . 8 ⊢ 𝑊 = (𝐼 mPwSer 𝐻) | |
6 | ressmplbas2.c | . . . . . . . 8 ⊢ 𝐶 = (Base‘𝑊) | |
7 | 3, 4, 5, 6 | subrgpsr 20657 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐶 ∈ (SubRing‘(𝐼 mPwSer 𝑅))) |
8 | 1, 2, 7 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (SubRing‘(𝐼 mPwSer 𝑅))) |
9 | eqid 2798 | . . . . . . 7 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
10 | 9 | subrgss 19529 | . . . . . 6 ⊢ (𝐶 ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → 𝐶 ⊆ (Base‘(𝐼 mPwSer 𝑅))) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (Base‘(𝐼 mPwSer 𝑅))) |
12 | df-ss 3898 | . . . . 5 ⊢ (𝐶 ⊆ (Base‘(𝐼 mPwSer 𝑅)) ↔ (𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) = 𝐶) | |
13 | 11, 12 | sylib 221 | . . . 4 ⊢ (𝜑 → (𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) = 𝐶) |
14 | eqid 2798 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | 4, 14 | subrg0 19535 | . . . . . . 7 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0g‘𝑅) = (0g‘𝐻)) |
16 | 2, 15 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐻)) |
17 | 16 | breq2d 5042 | . . . . 5 ⊢ (𝜑 → (𝑓 finSupp (0g‘𝑅) ↔ 𝑓 finSupp (0g‘𝐻))) |
18 | 17 | abbidv 2862 | . . . 4 ⊢ (𝜑 → {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)} = {𝑓 ∣ 𝑓 finSupp (0g‘𝐻)}) |
19 | 13, 18 | ineq12d 4140 | . . 3 ⊢ (𝜑 → ((𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)}) = (𝐶 ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝐻)})) |
20 | 19 | eqcomd 2804 | . 2 ⊢ (𝜑 → (𝐶 ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝐻)}) = ((𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)})) |
21 | ressmpl.u | . . . 4 ⊢ 𝑈 = (𝐼 mPoly 𝐻) | |
22 | eqid 2798 | . . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
23 | ressmpl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑈) | |
24 | 21, 5, 6, 22, 23 | mplbas 20667 | . . 3 ⊢ 𝐵 = {𝑓 ∈ 𝐶 ∣ 𝑓 finSupp (0g‘𝐻)} |
25 | dfrab3 4230 | . . 3 ⊢ {𝑓 ∈ 𝐶 ∣ 𝑓 finSupp (0g‘𝐻)} = (𝐶 ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝐻)}) | |
26 | 24, 25 | eqtri 2821 | . 2 ⊢ 𝐵 = (𝐶 ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝐻)}) |
27 | ressmpl.s | . . . . . 6 ⊢ 𝑆 = (𝐼 mPoly 𝑅) | |
28 | ressmplbas2.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
29 | 27, 3, 9, 14, 28 | mplbas 20667 | . . . . 5 ⊢ 𝐾 = {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)} |
30 | dfrab3 4230 | . . . . 5 ⊢ {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)} = ((Base‘(𝐼 mPwSer 𝑅)) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)}) | |
31 | 29, 30 | eqtri 2821 | . . . 4 ⊢ 𝐾 = ((Base‘(𝐼 mPwSer 𝑅)) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)}) |
32 | 31 | ineq2i 4136 | . . 3 ⊢ (𝐶 ∩ 𝐾) = (𝐶 ∩ ((Base‘(𝐼 mPwSer 𝑅)) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)})) |
33 | inass 4146 | . . 3 ⊢ ((𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)}) = (𝐶 ∩ ((Base‘(𝐼 mPwSer 𝑅)) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)})) | |
34 | 32, 33 | eqtr4i 2824 | . 2 ⊢ (𝐶 ∩ 𝐾) = ((𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)}) |
35 | 20, 26, 34 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → 𝐵 = (𝐶 ∩ 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {cab 2776 {crab 3110 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 finSupp cfsupp 8817 Basecbs 16475 ↾s cress 16476 0gc0g 16705 SubRingcsubrg 19524 mPwSer cmps 20589 mPoly cmpl 20591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-tset 16576 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-subrg 19526 df-psr 20594 df-mpl 20596 |
This theorem is referenced by: ressmplbas 20696 subrgmpl 20700 ressply1bas2 20857 |
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