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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 22117 | . 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 2728 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 3, 4, 5 | psr1basf 22113 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑m 1o)⟶(Base‘𝑅)) |
7 | ssv 4002 | . . . 4 ⊢ (Base‘𝑅) ⊆ V | |
8 | fss 6733 | . . . 4 ⊢ ((𝐹:(ℕ0 ↑m 1o)⟶(Base‘𝑅) ∧ (Base‘𝑅) ⊆ V) → 𝐹:(ℕ0 ↑m 1o)⟶V) | |
9 | 6, 7, 8 | sylancl 585 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑m 1o)⟶V) |
10 | fconst6g 6780 | . . . . . 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 12502 | . . . . . 6 ⊢ ℕ0 ∈ V | |
13 | 1oex 8490 | . . . . . 6 ⊢ 1o ∈ V | |
14 | 12, 13 | elmap 8883 | . . . . 5 ⊢ ((1o × {𝑦}) ∈ (ℕ0 ↑m 1o) ↔ (1o × {𝑦}):1o⟶ℕ0) |
15 | 11, 14 | sylibr 233 | . . . 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 6961 | . . . 4 ⊢ (𝐹:(ℕ0 ↑m 1o)⟶V → 𝐹 = (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝐹‘𝑥))) |
20 | fveq2 6891 | . . . 4 ⊢ (𝑥 = (1o × {𝑦}) → (𝐹‘𝑥) = (𝐹‘(1o × {𝑦}))) | |
21 | 15, 17, 19, 20 | fmptco 7132 | . . 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 2771 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ⊆ wss 3945 {csn 4624 ↦ cmpt 5225 × cxp 5670 ∘ ccom 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 1oc1o 8473 ↑m cmap 8838 ℕ0cn0 12496 Basecbs 17173 PwSer1cps1 22087 coe1cco1 22090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-tset 17245 df-ple 17246 df-psr 21835 df-opsr 21839 df-psr1 22092 df-coe1 22095 |
This theorem is referenced by: coe1f2 22121 coe1fval2 22122 coe1mul2 22181 |
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