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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 | β’ π΄ = (coe1βπΉ) |
coe1f2.b | β’ π΅ = (Baseβπ) |
coe1f2.p | β’ π = (PwSer1βπ ) |
coe1fval3.g | β’ πΊ = (π¦ β β0 β¦ (1o Γ {π¦})) |
Ref | Expression |
---|---|
coe1fval3 | β’ (πΉ β π΅ β π΄ = (πΉ β πΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coe1fval.a | . . 3 β’ π΄ = (coe1βπΉ) | |
2 | 1 | coe1fval 21592 | . 2 β’ (πΉ β π΅ β π΄ = (π¦ β β0 β¦ (πΉβ(1o Γ {π¦})))) |
3 | coe1f2.p | . . . . 5 β’ π = (PwSer1βπ ) | |
4 | coe1f2.b | . . . . 5 β’ π΅ = (Baseβπ) | |
5 | eqid 2733 | . . . . 5 β’ (Baseβπ ) = (Baseβπ ) | |
6 | 3, 4, 5 | psr1basf 21588 | . . . 4 β’ (πΉ β π΅ β πΉ:(β0 βm 1o)βΆ(Baseβπ )) |
7 | ssv 3969 | . . . 4 β’ (Baseβπ ) β V | |
8 | fss 6686 | . . . 4 β’ ((πΉ:(β0 βm 1o)βΆ(Baseβπ ) β§ (Baseβπ ) β V) β πΉ:(β0 βm 1o)βΆV) | |
9 | 6, 7, 8 | sylancl 587 | . . 3 β’ (πΉ β π΅ β πΉ:(β0 βm 1o)βΆV) |
10 | fconst6g 6732 | . . . . . 6 β’ (π¦ β β0 β (1o Γ {π¦}):1oβΆβ0) | |
11 | 10 | adantl 483 | . . . . 5 β’ ((πΉ:(β0 βm 1o)βΆV β§ π¦ β β0) β (1o Γ {π¦}):1oβΆβ0) |
12 | nn0ex 12424 | . . . . . 6 β’ β0 β V | |
13 | 1oex 8423 | . . . . . 6 β’ 1o β V | |
14 | 12, 13 | elmap 8812 | . . . . 5 β’ ((1o Γ {π¦}) β (β0 βm 1o) β (1o Γ {π¦}):1oβΆβ0) |
15 | 11, 14 | sylibr 233 | . . . 4 β’ ((πΉ:(β0 βm 1o)βΆV β§ π¦ β β0) β (1o Γ {π¦}) β (β0 βm 1o)) |
16 | coe1fval3.g | . . . . 5 β’ πΊ = (π¦ β β0 β¦ (1o Γ {π¦})) | |
17 | 16 | a1i 11 | . . . 4 β’ (πΉ:(β0 βm 1o)βΆV β πΊ = (π¦ β β0 β¦ (1o Γ {π¦}))) |
18 | id 22 | . . . . 5 β’ (πΉ:(β0 βm 1o)βΆV β πΉ:(β0 βm 1o)βΆV) | |
19 | 18 | feqmptd 6911 | . . . 4 β’ (πΉ:(β0 βm 1o)βΆV β πΉ = (π₯ β (β0 βm 1o) β¦ (πΉβπ₯))) |
20 | fveq2 6843 | . . . 4 β’ (π₯ = (1o Γ {π¦}) β (πΉβπ₯) = (πΉβ(1o Γ {π¦}))) | |
21 | 15, 17, 19, 20 | fmptco 7076 | . . 3 β’ (πΉ:(β0 βm 1o)βΆV β (πΉ β πΊ) = (π¦ β β0 β¦ (πΉβ(1o Γ {π¦})))) |
22 | 9, 21 | syl 17 | . 2 β’ (πΉ β π΅ β (πΉ β πΊ) = (π¦ β β0 β¦ (πΉβ(1o Γ {π¦})))) |
23 | 2, 22 | eqtr4d 2776 | 1 β’ (πΉ β π΅ β π΄ = (πΉ β πΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3444 β wss 3911 {csn 4587 β¦ cmpt 5189 Γ cxp 5632 β ccom 5638 βΆwf 6493 βcfv 6497 (class class class)co 7358 1oc1o 8406 βm cmap 8768 β0cn0 12418 Basecbs 17088 PwSer1cps1 21562 coe1cco1 21565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-tset 17157 df-ple 17158 df-psr 21327 df-opsr 21331 df-psr1 21567 df-coe1 21570 |
This theorem is referenced by: coe1f2 21596 coe1fval2 21597 coe1mul2 21656 |
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