Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2610 | . . . . . . . 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 19240 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐶 ∈ (SubRing‘(𝐼 mPwSer 𝑅))) |
8 | 1, 2, 7 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (SubRing‘(𝐼 mPwSer 𝑅))) |
9 | eqid 2610 | . . . . . . 7 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
10 | 9 | subrgss 18604 | . . . . . 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 3554 | . . . . 5 ⊢ (𝐶 ⊆ (Base‘(𝐼 mPwSer 𝑅)) ↔ (𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) = 𝐶) | |
13 | 11, 12 | sylib 207 | . . . 4 ⊢ (𝜑 → (𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) = 𝐶) |
14 | eqid 2610 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | 4, 14 | subrg0 18610 | . . . . . . 7 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0g‘𝑅) = (0g‘𝐻)) |
16 | 2, 15 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐻)) |
17 | 16 | breq2d 4595 | . . . . 5 ⊢ (𝜑 → (𝑓 finSupp (0g‘𝑅) ↔ 𝑓 finSupp (0g‘𝐻))) |
18 | 17 | abbidv 2728 | . . . 4 ⊢ (𝜑 → {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)} = {𝑓 ∣ 𝑓 finSupp (0g‘𝐻)}) |
19 | 13, 18 | ineq12d 3777 | . . 3 ⊢ (𝜑 → ((𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)}) = (𝐶 ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝐻)})) |
20 | 19 | eqcomd 2616 | . 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 2610 | . . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
23 | ressmpl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑈) | |
24 | 21, 5, 6, 22, 23 | mplbas 19250 | . . 3 ⊢ 𝐵 = {𝑓 ∈ 𝐶 ∣ 𝑓 finSupp (0g‘𝐻)} |
25 | dfrab3 3861 | . . 3 ⊢ {𝑓 ∈ 𝐶 ∣ 𝑓 finSupp (0g‘𝐻)} = (𝐶 ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝐻)}) | |
26 | 24, 25 | eqtri 2632 | . 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 19250 | . . . . 5 ⊢ 𝐾 = {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)} |
30 | dfrab3 3861 | . . . . 5 ⊢ {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)} = ((Base‘(𝐼 mPwSer 𝑅)) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)}) | |
31 | 29, 30 | eqtri 2632 | . . . 4 ⊢ 𝐾 = ((Base‘(𝐼 mPwSer 𝑅)) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)}) |
32 | 31 | ineq2i 3773 | . . 3 ⊢ (𝐶 ∩ 𝐾) = (𝐶 ∩ ((Base‘(𝐼 mPwSer 𝑅)) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)})) |
33 | inass 3785 | . . 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 2635 | . 2 ⊢ (𝐶 ∩ 𝐾) = ((𝐶 ∩ (Base‘(𝐼 mPwSer 𝑅))) ∩ {𝑓 ∣ 𝑓 finSupp (0g‘𝑅)}) |
35 | 20, 26, 34 | 3eqtr4g 2669 | 1 ⊢ (𝜑 → 𝐵 = (𝐶 ∩ 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {cab 2596 {crab 2900 ∩ cin 3539 ⊆ wss 3540 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 finSupp cfsupp 8158 Basecbs 15695 ↾s cress 15696 0gc0g 15923 SubRingcsubrg 18599 mPwSer cmps 19172 mPoly cmpl 19174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-subrg 18601 df-psr 19177 df-mpl 19179 |
This theorem is referenced by: ressmplbas 19277 subrgmpl 19281 ressply1bas2 19419 |
Copyright terms: Public domain | W3C validator |