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Mirrors > Home > MPE Home > Th. List > psrelbas | Structured version Visualization version GIF version |
Description: An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
psrbas.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
psrbas.k | ⊢ 𝐾 = (Base‘𝑅) |
psrbas.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
psrbas.b | ⊢ 𝐵 = (Base‘𝑆) |
psrelbas.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
psrelbas | ⊢ (𝜑 → 𝑋:𝐷⟶𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrelbas.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | psrbas.s | . . . 4 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
3 | psrbas.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
4 | psrbas.d | . . . 4 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
5 | psrbas.b | . . . 4 ⊢ 𝐵 = (Base‘𝑆) | |
6 | reldmpsr 21846 | . . . . . . 7 ⊢ Rel dom mPwSer | |
7 | 6, 2, 5 | elbasov 17186 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
9 | 8 | simpld 494 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ V) |
10 | 2, 3, 4, 5, 9 | psrbas 21877 | . . 3 ⊢ (𝜑 → 𝐵 = (𝐾 ↑m 𝐷)) |
11 | 1, 10 | eleqtrd 2831 | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝐾 ↑m 𝐷)) |
12 | 3 | fvexi 6911 | . . 3 ⊢ 𝐾 ∈ V |
13 | ovex 7453 | . . . 4 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
14 | 4, 13 | rabex2 5336 | . . 3 ⊢ 𝐷 ∈ V |
15 | 12, 14 | elmap 8889 | . 2 ⊢ (𝑋 ∈ (𝐾 ↑m 𝐷) ↔ 𝑋:𝐷⟶𝐾) |
16 | 11, 15 | sylib 217 | 1 ⊢ (𝜑 → 𝑋:𝐷⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3429 Vcvv 3471 ◡ccnv 5677 “ cima 5681 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ↑m cmap 8844 Fincfn 8963 ℕcn 12242 ℕ0cn0 12502 Basecbs 17179 mPwSer cmps 21836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-tset 17251 df-psr 21841 |
This theorem is referenced by: psrelbasfun 21879 psraddcl 21882 psraddclOLD 21883 psrmulcllem 21887 psrvscaval 21892 psrvscacl 21893 psr0lid 21895 psrnegcl 21896 psrlinv 21897 psrgrpOLD 21899 psrlmod 21902 psrlidm 21904 psrridm 21905 psrass1 21906 psrdi 21907 psrdir 21908 psrass23l 21909 psrcom 21910 psrass23 21911 resspsrmul 21918 mvrcl 21933 mplelf 21939 mplsubglem 21940 mpllsslem 21941 mplsubrglem 21945 subrgasclcl 22010 psdcl 22084 psdadd 22086 psdvsca 22087 psdmul 22089 psrplusgpropd 22153 psropprmul 22155 |
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