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Mirrors > Home > MPE Home > Th. List > coe1fval3 | Structured version Visualization version GIF version |
Description: Univariate power series coefficient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
Ref | Expression |
---|---|
coe1fval.a | ⊢ 𝐴 = (coe1‘𝐹) |
coe1f2.b | ⊢ 𝐵 = (Base‘𝑃) |
coe1f2.p | ⊢ 𝑃 = (PwSer1‘𝑅) |
coe1fval3.g | ⊢ 𝐺 = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) |
Ref | Expression |
---|---|
coe1fval3 | ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coe1fval.a | . . 3 ⊢ 𝐴 = (coe1‘𝐹) | |
2 | 1 | coe1fval 19971 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝑦 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑦})))) |
3 | coe1f2.p | . . . . 5 ⊢ 𝑃 = (PwSer1‘𝑅) | |
4 | coe1f2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑃) | |
5 | eqid 2777 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 3, 4, 5 | psr1basf 19967 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑𝑚 1o)⟶(Base‘𝑅)) |
7 | ssv 3843 | . . . 4 ⊢ (Base‘𝑅) ⊆ V | |
8 | fss 6304 | . . . 4 ⊢ ((𝐹:(ℕ0 ↑𝑚 1o)⟶(Base‘𝑅) ∧ (Base‘𝑅) ⊆ V) → 𝐹:(ℕ0 ↑𝑚 1o)⟶V) | |
9 | 6, 7, 8 | sylancl 580 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑𝑚 1o)⟶V) |
10 | fconst6g 6344 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (1o × {𝑦}):1o⟶ℕ0) | |
11 | 10 | adantl 475 | . . . . 5 ⊢ ((𝐹:(ℕ0 ↑𝑚 1o)⟶V ∧ 𝑦 ∈ ℕ0) → (1o × {𝑦}):1o⟶ℕ0) |
12 | nn0ex 11649 | . . . . . 6 ⊢ ℕ0 ∈ V | |
13 | 1oex 7851 | . . . . . 6 ⊢ 1o ∈ V | |
14 | 12, 13 | elmap 8169 | . . . . 5 ⊢ ((1o × {𝑦}) ∈ (ℕ0 ↑𝑚 1o) ↔ (1o × {𝑦}):1o⟶ℕ0) |
15 | 11, 14 | sylibr 226 | . . . 4 ⊢ ((𝐹:(ℕ0 ↑𝑚 1o)⟶V ∧ 𝑦 ∈ ℕ0) → (1o × {𝑦}) ∈ (ℕ0 ↑𝑚 1o)) |
16 | coe1fval3.g | . . . . 5 ⊢ 𝐺 = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) | |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝐹:(ℕ0 ↑𝑚 1o)⟶V → 𝐺 = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦}))) |
18 | id 22 | . . . . 5 ⊢ (𝐹:(ℕ0 ↑𝑚 1o)⟶V → 𝐹:(ℕ0 ↑𝑚 1o)⟶V) | |
19 | 18 | feqmptd 6509 | . . . 4 ⊢ (𝐹:(ℕ0 ↑𝑚 1o)⟶V → 𝐹 = (𝑥 ∈ (ℕ0 ↑𝑚 1o) ↦ (𝐹‘𝑥))) |
20 | fveq2 6446 | . . . 4 ⊢ (𝑥 = (1o × {𝑦}) → (𝐹‘𝑥) = (𝐹‘(1o × {𝑦}))) | |
21 | 15, 17, 19, 20 | fmptco 6661 | . . 3 ⊢ (𝐹:(ℕ0 ↑𝑚 1o)⟶V → (𝐹 ∘ 𝐺) = (𝑦 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑦})))) |
22 | 9, 21 | syl 17 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐹 ∘ 𝐺) = (𝑦 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑦})))) |
23 | 2, 22 | eqtr4d 2816 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ⊆ wss 3791 {csn 4397 ↦ cmpt 4965 × cxp 5353 ∘ ccom 5359 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 1oc1o 7836 ↑𝑚 cmap 8140 ℕ0cn0 11642 Basecbs 16255 PwSer1cps1 19941 coe1cco1 19944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-tset 16357 df-ple 16358 df-psr 19753 df-opsr 19757 df-psr1 19946 df-coe1 19949 |
This theorem is referenced by: coe1f2 19975 coe1fval2 19976 coe1mul2 20035 |
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