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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 2798 | . . 3 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
3 | eqid 2798 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
4 | resspsr.b | . . 3 ⊢ 𝐵 = (Base‘𝑈) | |
5 | eqid 2798 | . . 3 ⊢ (.r‘𝐻) = (.r‘𝐻) | |
6 | eqid 2798 | . . 3 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
7 | simprl 770 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝑇) | |
8 | resspsr.2 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
9 | 8 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑇 ∈ (SubRing‘𝑅)) |
10 | resspsr.h | . . . . . 6 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
11 | 10 | subrgbas 19537 | . . . . 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 2892 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ (Base‘𝐻)) |
14 | simprr 772 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
15 | 1, 2, 3, 4, 5, 6, 13, 14 | psrvsca 20629 | . 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 2798 | . . . 4 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆) | |
18 | eqid 2798 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
19 | eqid 2798 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
20 | eqid 2798 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
21 | 18 | subrgss 19529 | . . . . . 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 3916 | . . . 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 20653 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑃)) |
26 | 24, 19 | ressbasss 16548 | . . . . . . 7 ⊢ (Base‘𝑃) ⊆ (Base‘𝑆) |
27 | 25, 26 | eqsstrdi 3969 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑆)) |
28 | 27 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝐵 ⊆ (Base‘𝑆)) |
29 | 28, 14 | sseldd 3916 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ (Base‘𝑆)) |
30 | 16, 17, 18, 19, 20, 6, 23, 29 | psrvsca 20629 | . . 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 16617 | . . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝐻)) |
32 | ofeq 7391 | . . . . 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 7152 | . . 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 2833 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑆)𝑌) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘f (.r‘𝐻)𝑌)) |
36 | 4 | fvexi 6659 | . . . 4 ⊢ 𝐵 ∈ V |
37 | 24, 17 | ressvsca 16643 | . . . 4 ⊢ (𝐵 ∈ V → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑃)) |
38 | 36, 37 | mp1i 13 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑃)) |
39 | 38 | oveqd 7152 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑆)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌)) |
40 | 15, 35, 39 | 3eqtr2d 2839 | 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 399 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ⊆ wss 3881 {csn 4525 × cxp 5517 ◡ccnv 5518 “ cima 5522 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 ↑m cmap 8389 Fincfn 8492 ℕcn 11625 ℕ0cn0 11885 Basecbs 16475 ↾s cress 16476 .rcmulr 16558 ·𝑠 cvsca 16561 SubRingcsubrg 19524 mPwSer cmps 20589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-tset 16576 df-subg 18268 df-ring 19292 df-subrg 19526 df-psr 20594 |
This theorem is referenced by: ressmplvsca 20699 |
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