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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 2823 | . . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | 2, 3 | ringcl 19313 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) |
5 | 4 | 3expb 1116 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) |
6 | 1, 5 | sylan 582 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) |
7 | psrvscacl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
8 | fconst6g 6570 | . . . . 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 2823 | . . . . 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 20161 | . . . 4 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
15 | ovex 7191 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
16 | 15 | rabex 5237 | . . . . 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 4197 | . . . 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 7426 | . . 3 ⊢ (𝜑 → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘f (.r‘𝑅)𝐹):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
20 | 2 | fvexi 6686 | . . . 4 ⊢ 𝐾 ∈ V |
21 | 20, 16 | elmap 8437 | . . 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) |
22 | 19, 21 | sylibr 236 | . 2 ⊢ (𝜑 → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘f (.r‘𝑅)𝐹) ∈ (𝐾 ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
23 | psrvscacl.n | . . 3 ⊢ · = ( ·𝑠 ‘𝑆) | |
24 | 10, 23, 2, 12, 3, 11, 7, 13 | psrvsca 20173 | . 2 ⊢ (𝜑 → (𝑋 · 𝐹) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘f (.r‘𝑅)𝐹)) |
25 | reldmpsr 20143 | . . . . . 6 ⊢ Rel dom mPwSer | |
26 | 25, 10, 12 | elbasov 16547 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
27 | 13, 26 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
28 | 27 | simpld 497 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
29 | 10, 2, 11, 12, 28 | psrbas 20160 | . 2 ⊢ (𝜑 → 𝐵 = (𝐾 ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
30 | 22, 24, 29 | 3eltr4d 2930 | 1 ⊢ (𝜑 → (𝑋 · 𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 {csn 4569 × cxp 5555 ◡ccnv 5556 “ cima 5560 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 ↑m cmap 8408 Fincfn 8511 ℕcn 11640 ℕ0cn0 11900 Basecbs 16485 .rcmulr 16568 ·𝑠 cvsca 16571 Ringcrg 19299 mPwSer cmps 20133 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-tset 16586 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mgp 19242 df-ring 19301 df-psr 20138 |
This theorem is referenced by: psrlmod 20183 psrass23l 20190 psrass23 20192 mpllsslem 20217 |
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