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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 2821 | . . 3 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
3 | eqid 2821 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
4 | resspsr.b | . . 3 ⊢ 𝐵 = (Base‘𝑈) | |
5 | eqid 2821 | . . 3 ⊢ (.r‘𝐻) = (.r‘𝐻) | |
6 | eqid 2821 | . . 3 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
7 | simprl 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝑇) | |
8 | resspsr.2 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
9 | 8 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑇 ∈ (SubRing‘𝑅)) |
10 | resspsr.h | . . . . . 6 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
11 | 10 | subrgbas 19544 | . . . . 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 2915 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ (Base‘𝐻)) |
14 | simprr 771 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
15 | 1, 2, 3, 4, 5, 6, 13, 14 | psrvsca 20171 | . 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 2821 | . . . 4 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆) | |
18 | eqid 2821 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
19 | eqid 2821 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
20 | eqid 2821 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
21 | 18 | subrgss 19536 | . . . . . 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 3968 | . . . 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 20195 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑃)) |
26 | 24, 19 | ressbasss 16556 | . . . . . . 7 ⊢ (Base‘𝑃) ⊆ (Base‘𝑆) |
27 | 25, 26 | eqsstrdi 4021 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑆)) |
28 | 27 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝐵 ⊆ (Base‘𝑆)) |
29 | 28, 14 | sseldd 3968 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ (Base‘𝑆)) |
30 | 16, 17, 18, 19, 20, 6, 23, 29 | psrvsca 20171 | . . 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 16625 | . . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝐻)) |
32 | ofeq 7411 | . . . . 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 7173 | . . 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 2856 | . 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 6684 | . . . 4 ⊢ 𝐵 ∈ V |
37 | 24, 17 | ressvsca 16651 | . . . 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 7173 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑆)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌)) |
40 | 15, 35, 39 | 3eqtr2d 2862 | 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 398 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3494 ⊆ wss 3936 {csn 4567 × cxp 5553 ◡ccnv 5554 “ cima 5558 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 ↑m cmap 8406 Fincfn 8509 ℕcn 11638 ℕ0cn0 11898 Basecbs 16483 ↾s cress 16484 .rcmulr 16566 ·𝑠 cvsca 16569 SubRingcsubrg 19531 mPwSer cmps 20131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-tset 16584 df-subg 18276 df-ring 19299 df-subrg 19533 df-psr 20136 |
This theorem is referenced by: ressmplvsca 20240 |
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