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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 2821 | . . . . . . . 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 20199 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐶 ∈ (SubRing‘(𝐼 mPwSer 𝑅))) |
8 | 1, 2, 7 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (SubRing‘(𝐼 mPwSer 𝑅))) |
9 | eqid 2821 | . . . . . . 7 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
10 | 9 | subrgss 19536 | . . . . . 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 3952 | . . . . 5 ⊢ (𝐶 ⊆ (Base‘(𝐼 mPwSer 𝑅)) ↔ (𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) = 𝐶) | |
13 | 11, 12 | sylib 220 | . . . 4 ⊢ (𝜑 → (𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) = 𝐶) |
14 | eqid 2821 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | 4, 14 | subrg0 19542 | . . . . . . 7 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0g‘𝑅) = (0g‘𝐻)) |
16 | 2, 15 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐻)) |
17 | 16 | breq2d 5078 | . . . . 5 ⊢ (𝜑 → (𝑓 finSupp (0g‘𝑅) ↔ 𝑓 finSupp (0g‘𝐻))) |
18 | 17 | abbidv 2885 | . . . 4 ⊢ (𝜑 → {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)} = {𝑓 ∣ 𝑓 finSupp (0g‘𝐻)}) |
19 | 13, 18 | ineq12d 4190 | . . 3 ⊢ (𝜑 → ((𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)}) = (𝐶 ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝐻)})) |
20 | 19 | eqcomd 2827 | . 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 2821 | . . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
23 | ressmpl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑈) | |
24 | 21, 5, 6, 22, 23 | mplbas 20209 | . . 3 ⊢ 𝐵 = {𝑓 ∈ 𝐶 ∣ 𝑓 finSupp (0g‘𝐻)} |
25 | dfrab3 4278 | . . 3 ⊢ {𝑓 ∈ 𝐶 ∣ 𝑓 finSupp (0g‘𝐻)} = (𝐶 ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝐻)}) | |
26 | 24, 25 | eqtri 2844 | . 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 20209 | . . . . 5 ⊢ 𝐾 = {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)} |
30 | dfrab3 4278 | . . . . 5 ⊢ {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)} = ((Base‘(𝐼 mPwSer 𝑅)) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)}) | |
31 | 29, 30 | eqtri 2844 | . . . 4 ⊢ 𝐾 = ((Base‘(𝐼 mPwSer 𝑅)) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)}) |
32 | 31 | ineq2i 4186 | . . 3 ⊢ (𝐶 ∩ 𝐾) = (𝐶 ∩ ((Base‘(𝐼 mPwSer 𝑅)) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)})) |
33 | inass 4196 | . . 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 2847 | . 2 ⊢ (𝐶 ∩ 𝐾) = ((𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)}) |
35 | 20, 26, 34 | 3eqtr4g 2881 | 1 ⊢ (𝜑 → 𝐵 = (𝐶 ∩ 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {cab 2799 {crab 3142 ∩ cin 3935 ⊆ wss 3936 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 finSupp cfsupp 8833 Basecbs 16483 ↾s cress 16484 0gc0g 16713 SubRingcsubrg 19531 mPwSer cmps 20131 mPoly cmpl 20133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-tset 16584 df-0g 16715 df-gsum 16716 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-mulg 18225 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-subrg 19533 df-psr 20136 df-mpl 20138 |
This theorem is referenced by: ressmplbas 20237 subrgmpl 20241 ressply1bas2 20396 |
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