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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 2826 | . . 3 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
3 | eqid 2826 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
4 | resspsr.b | . . 3 ⊢ 𝐵 = (Base‘𝑈) | |
5 | eqid 2826 | . . 3 ⊢ (.r‘𝐻) = (.r‘𝐻) | |
6 | eqid 2826 | . . 3 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
7 | simprl 789 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝑇) | |
8 | resspsr.2 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
9 | 8 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑇 ∈ (SubRing‘𝑅)) |
10 | resspsr.h | . . . . . 6 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
11 | 10 | subrgbas 19146 | . . . . 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 2909 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ (Base‘𝐻)) |
14 | simprr 791 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
15 | 1, 2, 3, 4, 5, 6, 13, 14 | psrvsca 19753 | . 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 2826 | . . . 4 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆) | |
18 | eqid 2826 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
19 | eqid 2826 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
20 | eqid 2826 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
21 | 18 | subrgss 19138 | . . . . . 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 3829 | . . . 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 19777 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑃)) |
26 | 24, 19 | ressbasss 16296 | . . . . . . 7 ⊢ (Base‘𝑃) ⊆ (Base‘𝑆) |
27 | 25, 26 | syl6eqss 3881 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑆)) |
28 | 27 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝐵 ⊆ (Base‘𝑆)) |
29 | 28, 14 | sseldd 3829 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ (Base‘𝑆)) |
30 | 16, 17, 18, 19, 20, 6, 23, 29 | psrvsca 19753 | . . 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 16366 | . . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝐻)) |
32 | ofeq 7160 | . . . . 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 6923 | . . 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 2862 | . 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 6448 | . . . 4 ⊢ 𝐵 ∈ V |
37 | 24, 17 | ressvsca 16392 | . . . 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 6923 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑆)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌)) |
40 | 15, 35, 39 | 3eqtr2d 2868 | 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 386 = wceq 1658 ∈ wcel 2166 {crab 3122 Vcvv 3415 ⊆ wss 3799 {csn 4398 × cxp 5341 ◡ccnv 5342 “ cima 5346 ‘cfv 6124 (class class class)co 6906 ∘𝑓 cof 7156 ↑𝑚 cmap 8123 Fincfn 8223 ℕcn 11351 ℕ0cn0 11619 Basecbs 16223 ↾s cress 16224 .rcmulr 16307 ·𝑠 cvsca 16310 SubRingcsubrg 19133 mPwSer cmps 19713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-of 7158 df-om 7328 df-1st 7429 df-2nd 7430 df-supp 7561 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-fsupp 8546 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-uz 11970 df-fz 12621 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-sca 16322 df-vsca 16323 df-tset 16325 df-subg 17943 df-ring 18904 df-subrg 19135 df-psr 19718 |
This theorem is referenced by: ressmplvsca 19821 |
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