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Mirrors > Home > MPE Home > Th. List > psrvscacl | Structured version Visualization version GIF version |
Description: Closure of the power series scalar multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
psrvscacl.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
psrvscacl.n | ⊢ · = ( ·𝑠 ‘𝑆) |
psrvscacl.k | ⊢ 𝐾 = (Base‘𝑅) |
psrvscacl.b | ⊢ 𝐵 = (Base‘𝑆) |
psrvscacl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
psrvscacl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
psrvscacl.y | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
Ref | Expression |
---|---|
psrvscacl | ⊢ (𝜑 → (𝑋 · 𝐹) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrvscacl.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | psrvscacl.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝑅) | |
3 | eqid 2622 | . . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | 2, 3 | ringcl 18561 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) |
5 | 4 | 3expb 1266 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) |
6 | 1, 5 | sylan 488 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) |
7 | psrvscacl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
8 | fconst6g 6094 | . . . . 5 ⊢ (𝑋 ∈ 𝐾 → ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
10 | psrvscacl.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
11 | eqid 2622 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
12 | psrvscacl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
13 | psrvscacl.y | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
14 | 10, 2, 11, 12, 13 | psrelbas 19379 | . . . 4 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
15 | ovex 6678 | . . . . . 6 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V | |
16 | 15 | rabex 4813 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) |
18 | inidm 3822 | . . . 4 ⊢ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∩ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
19 | 6, 9, 14, 17, 17, 18 | off 6912 | . . 3 ⊢ (𝜑 → (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝐹):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
20 | fvex 6201 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
21 | 2, 20 | eqeltri 2697 | . . . 4 ⊢ 𝐾 ∈ V |
22 | 21, 16 | elmap 7886 | . . 3 ⊢ ((({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝐹) ∈ (𝐾 ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ↔ (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝐹):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
23 | 19, 22 | sylibr 224 | . 2 ⊢ (𝜑 → (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝐹) ∈ (𝐾 ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
24 | psrvscacl.n | . . 3 ⊢ · = ( ·𝑠 ‘𝑆) | |
25 | 10, 24, 2, 12, 3, 11, 7, 13 | psrvsca 19391 | . 2 ⊢ (𝜑 → (𝑋 · 𝐹) = (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝐹)) |
26 | reldmpsr 19361 | . . . . . 6 ⊢ Rel dom mPwSer | |
27 | 26, 10, 12 | elbasov 15921 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
28 | 13, 27 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
29 | 28 | simpld 475 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
30 | 10, 2, 11, 12, 29 | psrbas 19378 | . 2 ⊢ (𝜑 → 𝐵 = (𝐾 ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
31 | 23, 25, 30 | 3eltr4d 2716 | 1 ⊢ (𝜑 → (𝑋 · 𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 {csn 4177 × cxp 5112 ◡ccnv 5113 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 ↑𝑚 cmap 7857 Fincfn 7955 ℕcn 11020 ℕ0cn0 11292 Basecbs 15857 .rcmulr 15942 ·𝑠 cvsca 15945 Ringcrg 18547 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-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-tset 15960 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mgp 18490 df-ring 18549 df-psr 19356 |
This theorem is referenced by: psrlmod 19401 psrass23l 19408 psrass23 19410 mpllsslem 19435 |
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