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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 20973 | . 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 2738 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 3, 4, 5 | psr1basf 20969 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑m 1o)⟶(Base‘𝑅)) |
7 | ssv 3899 | . . . 4 ⊢ (Base‘𝑅) ⊆ V | |
8 | fss 6515 | . . . 4 ⊢ ((𝐹:(ℕ0 ↑m 1o)⟶(Base‘𝑅) ∧ (Base‘𝑅) ⊆ V) → 𝐹:(ℕ0 ↑m 1o)⟶V) | |
9 | 6, 7, 8 | sylancl 589 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑m 1o)⟶V) |
10 | fconst6g 6561 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (1o × {𝑦}):1o⟶ℕ0) | |
11 | 10 | adantl 485 | . . . . 5 ⊢ ((𝐹:(ℕ0 ↑m 1o)⟶V ∧ 𝑦 ∈ ℕ0) → (1o × {𝑦}):1o⟶ℕ0) |
12 | nn0ex 11975 | . . . . . 6 ⊢ ℕ0 ∈ V | |
13 | 1oex 8137 | . . . . . 6 ⊢ 1o ∈ V | |
14 | 12, 13 | elmap 8474 | . . . . 5 ⊢ ((1o × {𝑦}) ∈ (ℕ0 ↑m 1o) ↔ (1o × {𝑦}):1o⟶ℕ0) |
15 | 11, 14 | sylibr 237 | . . . 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 6731 | . . . 4 ⊢ (𝐹:(ℕ0 ↑m 1o)⟶V → 𝐹 = (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝐹‘𝑥))) |
20 | fveq2 6668 | . . . 4 ⊢ (𝑥 = (1o × {𝑦}) → (𝐹‘𝑥) = (𝐹‘(1o × {𝑦}))) | |
21 | 15, 17, 19, 20 | fmptco 6895 | . . 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 2776 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 Vcvv 3397 ⊆ wss 3841 {csn 4513 ↦ cmpt 5107 × cxp 5517 ∘ ccom 5523 ⟶wf 6329 ‘cfv 6333 (class class class)co 7164 1oc1o 8117 ↑m cmap 8430 ℕ0cn0 11969 Basecbs 16579 PwSer1cps1 20943 coe1cco1 20946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-supp 7850 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fsupp 8900 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-tset 16680 df-ple 16681 df-psr 20715 df-opsr 20719 df-psr1 20948 df-coe1 20951 |
This theorem is referenced by: coe1f2 20977 coe1fval2 20978 coe1mul2 21037 |
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