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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 19623 | . 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 2651 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 3, 4, 5 | psr1basf 19619 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑𝑚 1𝑜)⟶(Base‘𝑅)) |
7 | ssv 3658 | . . . 4 ⊢ (Base‘𝑅) ⊆ V | |
8 | fss 6094 | . . . 4 ⊢ ((𝐹:(ℕ0 ↑𝑚 1𝑜)⟶(Base‘𝑅) ∧ (Base‘𝑅) ⊆ V) → 𝐹:(ℕ0 ↑𝑚 1𝑜)⟶V) | |
9 | 6, 7, 8 | sylancl 695 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑𝑚 1𝑜)⟶V) |
10 | fconst6g 6132 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (1𝑜 × {𝑦}):1𝑜⟶ℕ0) | |
11 | 10 | adantl 481 | . . . . 5 ⊢ ((𝐹:(ℕ0 ↑𝑚 1𝑜)⟶V ∧ 𝑦 ∈ ℕ0) → (1𝑜 × {𝑦}):1𝑜⟶ℕ0) |
12 | nn0ex 11336 | . . . . . 6 ⊢ ℕ0 ∈ V | |
13 | 1on 7612 | . . . . . . 7 ⊢ 1𝑜 ∈ On | |
14 | 13 | elexi 3244 | . . . . . 6 ⊢ 1𝑜 ∈ V |
15 | 12, 14 | elmap 7928 | . . . . 5 ⊢ ((1𝑜 × {𝑦}) ∈ (ℕ0 ↑𝑚 1𝑜) ↔ (1𝑜 × {𝑦}):1𝑜⟶ℕ0) |
16 | 11, 15 | sylibr 224 | . . . 4 ⊢ ((𝐹:(ℕ0 ↑𝑚 1𝑜)⟶V ∧ 𝑦 ∈ ℕ0) → (1𝑜 × {𝑦}) ∈ (ℕ0 ↑𝑚 1𝑜)) |
17 | coe1fval3.g | . . . . 5 ⊢ 𝐺 = (𝑦 ∈ ℕ0 ↦ (1𝑜 × {𝑦})) | |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝐹:(ℕ0 ↑𝑚 1𝑜)⟶V → 𝐺 = (𝑦 ∈ ℕ0 ↦ (1𝑜 × {𝑦}))) |
19 | id 22 | . . . . 5 ⊢ (𝐹:(ℕ0 ↑𝑚 1𝑜)⟶V → 𝐹:(ℕ0 ↑𝑚 1𝑜)⟶V) | |
20 | 19 | feqmptd 6288 | . . . 4 ⊢ (𝐹:(ℕ0 ↑𝑚 1𝑜)⟶V → 𝐹 = (𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝐹‘𝑥))) |
21 | fveq2 6229 | . . . 4 ⊢ (𝑥 = (1𝑜 × {𝑦}) → (𝐹‘𝑥) = (𝐹‘(1𝑜 × {𝑦}))) | |
22 | 16, 18, 20, 21 | fmptco 6436 | . . 3 ⊢ (𝐹:(ℕ0 ↑𝑚 1𝑜)⟶V → (𝐹 ∘ 𝐺) = (𝑦 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑦})))) |
23 | 9, 22 | syl 17 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐹 ∘ 𝐺) = (𝑦 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑦})))) |
24 | 2, 23 | eqtr4d 2688 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ⊆ wss 3607 {csn 4210 ↦ cmpt 4762 × cxp 5141 ∘ ccom 5147 Oncon0 5761 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 1𝑜c1o 7598 ↑𝑚 cmap 7899 ℕ0cn0 11330 Basecbs 15904 PwSer1cps1 19593 coe1cco1 19596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-tset 16007 df-ple 16008 df-psr 19404 df-opsr 19408 df-psr1 19598 df-coe1 19601 |
This theorem is referenced by: coe1f2 19627 coe1fval2 19628 coe1mul2 19687 |
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