![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > taylplem1 | Structured version Visualization version GIF version |
Description: Lemma for taylpfval 26317 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.) |
Ref | Expression |
---|---|
taylpfval.s | β’ (π β π β {β, β}) |
taylpfval.f | β’ (π β πΉ:π΄βΆβ) |
taylpfval.a | β’ (π β π΄ β π) |
taylpfval.n | β’ (π β π β β0) |
taylpfval.b | β’ (π β π΅ β dom ((π Dπ πΉ)βπ)) |
Ref | Expression |
---|---|
taylplem1 | β’ ((π β§ π β ((0[,]π) β© β€)) β π΅ β dom ((π Dπ πΉ)βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 12599 | . . . . 5 β’ 0 β β€ | |
2 | taylpfval.n | . . . . . 6 β’ (π β π β β0) | |
3 | 2 | nn0zd 12614 | . . . . 5 β’ (π β π β β€) |
4 | fzval2 13519 | . . . . 5 β’ ((0 β β€ β§ π β β€) β (0...π) = ((0[,]π) β© β€)) | |
5 | 1, 3, 4 | sylancr 585 | . . . 4 β’ (π β (0...π) = ((0[,]π) β© β€)) |
6 | 5 | eleq2d 2811 | . . 3 β’ (π β (π β (0...π) β π β ((0[,]π) β© β€))) |
7 | 6 | biimpar 476 | . 2 β’ ((π β§ π β ((0[,]π) β© β€)) β π β (0...π)) |
8 | taylpfval.s | . . . . 5 β’ (π β π β {β, β}) | |
9 | cnex 11219 | . . . . . . 7 β’ β β V | |
10 | 9 | a1i 11 | . . . . . 6 β’ (π β β β V) |
11 | taylpfval.f | . . . . . 6 β’ (π β πΉ:π΄βΆβ) | |
12 | taylpfval.a | . . . . . 6 β’ (π β π΄ β π) | |
13 | elpm2r 8862 | . . . . . 6 β’ (((β β V β§ π β {β, β}) β§ (πΉ:π΄βΆβ β§ π΄ β π)) β πΉ β (β βpm π)) | |
14 | 10, 8, 11, 12, 13 | syl22anc 837 | . . . . 5 β’ (π β πΉ β (β βpm π)) |
15 | 8, 14 | jca 510 | . . . 4 β’ (π β (π β {β, β} β§ πΉ β (β βpm π))) |
16 | dvn2bss 25878 | . . . . 5 β’ ((π β {β, β} β§ πΉ β (β βpm π) β§ π β (0...π)) β dom ((π Dπ πΉ)βπ) β dom ((π Dπ πΉ)βπ)) | |
17 | 16 | 3expa 1115 | . . . 4 β’ (((π β {β, β} β§ πΉ β (β βpm π)) β§ π β (0...π)) β dom ((π Dπ πΉ)βπ) β dom ((π Dπ πΉ)βπ)) |
18 | 15, 17 | sylan 578 | . . 3 β’ ((π β§ π β (0...π)) β dom ((π Dπ πΉ)βπ) β dom ((π Dπ πΉ)βπ)) |
19 | taylpfval.b | . . . 4 β’ (π β π΅ β dom ((π Dπ πΉ)βπ)) | |
20 | 19 | adantr 479 | . . 3 β’ ((π β§ π β (0...π)) β π΅ β dom ((π Dπ πΉ)βπ)) |
21 | 18, 20 | sseldd 3973 | . 2 β’ ((π β§ π β (0...π)) β π΅ β dom ((π Dπ πΉ)βπ)) |
22 | 7, 21 | syldan 589 | 1 β’ ((π β§ π β ((0[,]π) β© β€)) β π΅ β dom ((π Dπ πΉ)βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3463 β© cin 3938 β wss 3939 {cpr 4626 dom cdm 5672 βΆwf 6539 βcfv 6543 (class class class)co 7416 βpm cpm 8844 βcc 11136 βcr 11137 0cc0 11138 β0cn0 12502 β€cz 12588 [,]cicc 13359 ...cfz 13516 Dπ cdvn 25811 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 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-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fi 9434 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-icc 13363 df-fz 13517 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-mulr 17246 df-starv 17247 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-rest 17403 df-topn 17404 df-topgen 17424 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cnp 23150 df-haus 23237 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-limc 25813 df-dv 25814 df-dvn 25815 |
This theorem is referenced by: taylplem2 26316 taylpfval 26317 dvtaylp 26323 dvntaylp0 26325 |
Copyright terms: Public domain | W3C validator |