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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 2760 | . . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | 2, 3 | ringcl 18761 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) |
5 | 4 | 3expb 1114 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) |
6 | 1, 5 | sylan 489 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) |
7 | psrvscacl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
8 | fconst6g 6255 | . . . . 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 2760 | . . . . 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 19581 | . . . 4 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
15 | ovex 6841 | . . . . . 6 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V | |
16 | 15 | rabex 4964 | . . . . 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 3965 | . . . 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 7077 | . . 3 ⊢ (𝜑 → (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝐹):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
20 | fvex 6362 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
21 | 2, 20 | eqeltri 2835 | . . . 4 ⊢ 𝐾 ∈ V |
22 | 21, 16 | elmap 8052 | . . 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 19593 | . 2 ⊢ (𝜑 → (𝑋 · 𝐹) = (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝐹)) |
26 | reldmpsr 19563 | . . . . . 6 ⊢ Rel dom mPwSer | |
27 | 26, 10, 12 | elbasov 16123 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
28 | 13, 27 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
29 | 28 | simpld 477 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
30 | 10, 2, 11, 12, 29 | psrbas 19580 | . 2 ⊢ (𝜑 → 𝐵 = (𝐾 ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
31 | 23, 25, 30 | 3eltr4d 2854 | 1 ⊢ (𝜑 → (𝑋 · 𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 {crab 3054 Vcvv 3340 {csn 4321 × cxp 5264 ◡ccnv 5265 “ cima 5269 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 ∘𝑓 cof 7060 ↑𝑚 cmap 8023 Fincfn 8121 ℕcn 11212 ℕ0cn0 11484 Basecbs 16059 .rcmulr 16144 ·𝑠 cvsca 16147 Ringcrg 18747 mPwSer cmps 19553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-tset 16162 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-mgp 18690 df-ring 18749 df-psr 19558 |
This theorem is referenced by: psrlmod 19603 psrass23l 19610 psrass23 19612 mpllsslem 19637 |
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