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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 19575 | . 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 2622 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 3, 4, 5 | psr1basf 19571 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑𝑚 1𝑜)⟶(Base‘𝑅)) |
7 | ssv 3625 | . . . 4 ⊢ (Base‘𝑅) ⊆ V | |
8 | fss 6056 | . . . 4 ⊢ ((𝐹:(ℕ0 ↑𝑚 1𝑜)⟶(Base‘𝑅) ∧ (Base‘𝑅) ⊆ V) → 𝐹:(ℕ0 ↑𝑚 1𝑜)⟶V) | |
9 | 6, 7, 8 | sylancl 694 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑𝑚 1𝑜)⟶V) |
10 | fconst6g 6094 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (1𝑜 × {𝑦}):1𝑜⟶ℕ0) | |
11 | 10 | adantl 482 | . . . . 5 ⊢ ((𝐹:(ℕ0 ↑𝑚 1𝑜)⟶V ∧ 𝑦 ∈ ℕ0) → (1𝑜 × {𝑦}):1𝑜⟶ℕ0) |
12 | nn0ex 11298 | . . . . . 6 ⊢ ℕ0 ∈ V | |
13 | 1on 7567 | . . . . . . 7 ⊢ 1𝑜 ∈ On | |
14 | 13 | elexi 3213 | . . . . . 6 ⊢ 1𝑜 ∈ V |
15 | 12, 14 | elmap 7886 | . . . . 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 6249 | . . . 4 ⊢ (𝐹:(ℕ0 ↑𝑚 1𝑜)⟶V → 𝐹 = (𝑥 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝐹‘𝑥))) |
21 | fveq2 6191 | . . . 4 ⊢ (𝑥 = (1𝑜 × {𝑦}) → (𝐹‘𝑥) = (𝐹‘(1𝑜 × {𝑦}))) | |
22 | 16, 18, 20, 21 | fmptco 6396 | . . 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 2659 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 {csn 4177 ↦ cmpt 4729 × cxp 5112 ∘ ccom 5118 Oncon0 5723 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1𝑜c1o 7553 ↑𝑚 cmap 7857 ℕ0cn0 11292 Basecbs 15857 PwSer1cps1 19545 coe1cco1 19548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-tset 15960 df-ple 15961 df-psr 19356 df-opsr 19360 df-psr1 19550 df-coe1 19553 |
This theorem is referenced by: coe1f2 19579 coe1fval2 19580 coe1mul2 19639 |
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