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Mirrors > Home > MPE Home > Th. List > resspsrvsca | Structured version Visualization version GIF version |
Description: A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.) |
Ref | Expression |
---|---|
resspsr.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
resspsr.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
resspsr.u | ⊢ 𝑈 = (𝐼 mPwSer 𝐻) |
resspsr.b | ⊢ 𝐵 = (Base‘𝑈) |
resspsr.p | ⊢ 𝑃 = (𝑆 ↾s 𝐵) |
resspsr.2 | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) |
Ref | Expression |
---|---|
resspsrvsca | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑈)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resspsr.u | . . 3 ⊢ 𝑈 = (𝐼 mPwSer 𝐻) | |
2 | eqid 2622 | . . 3 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
3 | eqid 2622 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
4 | resspsr.b | . . 3 ⊢ 𝐵 = (Base‘𝑈) | |
5 | eqid 2622 | . . 3 ⊢ (.r‘𝐻) = (.r‘𝐻) | |
6 | eqid 2622 | . . 3 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
7 | simprl 794 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝑇) | |
8 | resspsr.2 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
9 | 8 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑇 ∈ (SubRing‘𝑅)) |
10 | resspsr.h | . . . . . 6 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
11 | 10 | subrgbas 18789 | . . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻)) |
12 | 9, 11 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑇 = (Base‘𝐻)) |
13 | 7, 12 | eleqtrd 2703 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ (Base‘𝐻)) |
14 | simprr 796 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
15 | 1, 2, 3, 4, 5, 6, 13, 14 | psrvsca 19391 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑈)𝑌) = (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝐻)𝑌)) |
16 | resspsr.s | . . . 4 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
17 | eqid 2622 | . . . 4 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆) | |
18 | eqid 2622 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
19 | eqid 2622 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
20 | eqid 2622 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
21 | 18 | subrgss 18781 | . . . . . 6 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅)) |
22 | 9, 21 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑇 ⊆ (Base‘𝑅)) |
23 | 22, 7 | sseldd 3604 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ (Base‘𝑅)) |
24 | resspsr.p | . . . . . . . 8 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
25 | 16, 10, 1, 4, 24, 8 | resspsrbas 19415 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑃)) |
26 | 24, 19 | ressbasss 15932 | . . . . . . 7 ⊢ (Base‘𝑃) ⊆ (Base‘𝑆) |
27 | 25, 26 | syl6eqss 3655 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑆)) |
28 | 27 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝐵 ⊆ (Base‘𝑆)) |
29 | 28, 14 | sseldd 3604 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ (Base‘𝑆)) |
30 | 16, 17, 18, 19, 20, 6, 23, 29 | psrvsca 19391 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑆)𝑌) = (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝑌)) |
31 | 10, 20 | ressmulr 16006 | . . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝐻)) |
32 | ofeq 6899 | . . . . 5 ⊢ ((.r‘𝑅) = (.r‘𝐻) → ∘𝑓 (.r‘𝑅) = ∘𝑓 (.r‘𝐻)) | |
33 | 9, 31, 32 | 3syl 18 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → ∘𝑓 (.r‘𝑅) = ∘𝑓 (.r‘𝐻)) |
34 | 33 | oveqd 6667 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝑌) = (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝐻)𝑌)) |
35 | 30, 34 | eqtrd 2656 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑆)𝑌) = (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝐻)𝑌)) |
36 | fvex 6201 | . . . . 5 ⊢ (Base‘𝑈) ∈ V | |
37 | 4, 36 | eqeltri 2697 | . . . 4 ⊢ 𝐵 ∈ V |
38 | 24, 17 | ressvsca 16032 | . . . 4 ⊢ (𝐵 ∈ V → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑃)) |
39 | 37, 38 | mp1i 13 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑃)) |
40 | 39 | oveqd 6667 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑆)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌)) |
41 | 15, 35, 40 | 3eqtr2d 2662 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑈)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 ⊆ wss 3574 {csn 4177 × cxp 5112 ◡ccnv 5113 “ cima 5117 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 ↑𝑚 cmap 7857 Fincfn 7955 ℕcn 11020 ℕ0cn0 11292 Basecbs 15857 ↾s cress 15858 .rcmulr 15942 ·𝑠 cvsca 15945 SubRingcsubrg 18776 mPwSer cmps 19351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-tset 15960 df-subg 17591 df-ring 18549 df-subrg 18778 df-psr 19356 |
This theorem is referenced by: ressmplvsca 19459 |
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