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Mirrors > Home > MPE Home > Th. List > coe1f2 | Structured version Visualization version GIF version |
Description: Functionality of univariate power series coefficient vectors. (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) |
coe1f2.k | ⊢ 𝐾 = (Base‘𝑅) |
Ref | Expression |
---|---|
coe1f2 | ⊢ (𝐹 ∈ 𝐵 → 𝐴:ℕ0⟶𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coe1f2.p | . . . 4 ⊢ 𝑃 = (PwSer1‘𝑅) | |
2 | coe1f2.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
3 | coe1f2.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
4 | 1, 2, 3 | psr1basf 22064 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑m 1o)⟶𝐾) |
5 | df1o2 8469 | . . . . 5 ⊢ 1o = {∅} | |
6 | nn0ex 12477 | . . . . 5 ⊢ ℕ0 ∈ V | |
7 | 0ex 5298 | . . . . 5 ⊢ ∅ ∈ V | |
8 | eqid 2724 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})) | |
9 | 5, 6, 7, 8 | mapsnf1o3 8886 | . . . 4 ⊢ (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})):ℕ0–1-1-onto→(ℕ0 ↑m 1o) |
10 | f1of 6824 | . . . 4 ⊢ ((𝑥 ∈ ℕ0 ↦ (1o × {𝑥})):ℕ0–1-1-onto→(ℕ0 ↑m 1o) → (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})):ℕ0⟶(ℕ0 ↑m 1o)) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})):ℕ0⟶(ℕ0 ↑m 1o) |
12 | fco 6732 | . . 3 ⊢ ((𝐹:(ℕ0 ↑m 1o)⟶𝐾 ∧ (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})):ℕ0⟶(ℕ0 ↑m 1o)) → (𝐹 ∘ (𝑥 ∈ ℕ0 ↦ (1o × {𝑥}))):ℕ0⟶𝐾) | |
13 | 4, 11, 12 | sylancl 585 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐹 ∘ (𝑥 ∈ ℕ0 ↦ (1o × {𝑥}))):ℕ0⟶𝐾) |
14 | coe1fval.a | . . . 4 ⊢ 𝐴 = (coe1‘𝐹) | |
15 | 14, 2, 1, 8 | coe1fval3 22071 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})))) |
16 | 15 | feq1d 6693 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐴:ℕ0⟶𝐾 ↔ (𝐹 ∘ (𝑥 ∈ ℕ0 ↦ (1o × {𝑥}))):ℕ0⟶𝐾)) |
17 | 13, 16 | mpbird 257 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴:ℕ0⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∅c0 4315 {csn 4621 ↦ cmpt 5222 × cxp 5665 ∘ ccom 5671 ⟶wf 6530 –1-1-onto→wf1o 6533 ‘cfv 6534 (class class class)co 7402 1oc1o 8455 ↑m cmap 8817 ℕ0cn0 12471 Basecbs 17149 PwSer1cps1 22038 coe1cco1 22041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-tset 17221 df-ple 17222 df-psr 21792 df-opsr 21796 df-psr1 22043 df-coe1 22046 |
This theorem is referenced by: coe1f 22074 coe1mul2 22132 |
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