![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > eldvap | GIF version |
Description: The differentiable predicate. A function πΉ is differentiable at π΅ with derivative πΆ iff πΉ is defined in a neighborhood of π΅ and the difference quotient has limit πΆ at π΅. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.) |
Ref | Expression |
---|---|
dvval.t | β’ π = (πΎ βΎt π) |
dvval.k | β’ πΎ = (MetOpenβ(abs β β )) |
eldvap.g | β’ πΊ = (π§ β {π€ β π΄ β£ π€ # π΅} β¦ (((πΉβπ§) β (πΉβπ΅)) / (π§ β π΅))) |
eldv.s | β’ (π β π β β) |
eldv.f | β’ (π β πΉ:π΄βΆβ) |
eldv.a | β’ (π β π΄ β π) |
Ref | Expression |
---|---|
eldvap | β’ (π β (π΅(π D πΉ)πΆ β (π΅ β ((intβπ)βπ΄) β§ πΆ β (πΊ limβ π΅)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldv.s | . . . . 5 β’ (π β π β β) | |
2 | eldv.f | . . . . 5 β’ (π β πΉ:π΄βΆβ) | |
3 | eldv.a | . . . . 5 β’ (π β π΄ β π) | |
4 | dvval.t | . . . . . 6 β’ π = (πΎ βΎt π) | |
5 | dvval.k | . . . . . 6 β’ πΎ = (MetOpenβ(abs β β )) | |
6 | 4, 5 | dvfvalap 14235 | . . . . 5 β’ ((π β β β§ πΉ:π΄βΆβ β§ π΄ β π) β ((π D πΉ) = βͺ π₯ β ((intβπ)βπ΄)({π₯} Γ ((π§ β {π€ β π΄ β£ π€ # π₯} β¦ (((πΉβπ§) β (πΉβπ₯)) / (π§ β π₯))) limβ π₯)) β§ (π D πΉ) β (((intβπ)βπ΄) Γ β))) |
7 | 1, 2, 3, 6 | syl3anc 1238 | . . . 4 β’ (π β ((π D πΉ) = βͺ π₯ β ((intβπ)βπ΄)({π₯} Γ ((π§ β {π€ β π΄ β£ π€ # π₯} β¦ (((πΉβπ§) β (πΉβπ₯)) / (π§ β π₯))) limβ π₯)) β§ (π D πΉ) β (((intβπ)βπ΄) Γ β))) |
8 | 7 | simpld 112 | . . 3 β’ (π β (π D πΉ) = βͺ π₯ β ((intβπ)βπ΄)({π₯} Γ ((π§ β {π€ β π΄ β£ π€ # π₯} β¦ (((πΉβπ§) β (πΉβπ₯)) / (π§ β π₯))) limβ π₯))) |
9 | 8 | eleq2d 2247 | . 2 β’ (π β (β¨π΅, πΆβ© β (π D πΉ) β β¨π΅, πΆβ© β βͺ π₯ β ((intβπ)βπ΄)({π₯} Γ ((π§ β {π€ β π΄ β£ π€ # π₯} β¦ (((πΉβπ§) β (πΉβπ₯)) / (π§ β π₯))) limβ π₯)))) |
10 | df-br 4006 | . . 3 β’ (π΅(π D πΉ)πΆ β β¨π΅, πΆβ© β (π D πΉ)) | |
11 | 10 | bicomi 132 | . 2 β’ (β¨π΅, πΆβ© β (π D πΉ) β π΅(π D πΉ)πΆ) |
12 | breq2 4009 | . . . . . . 7 β’ (π₯ = π΅ β (π€ # π₯ β π€ # π΅)) | |
13 | 12 | rabbidv 2728 | . . . . . 6 β’ (π₯ = π΅ β {π€ β π΄ β£ π€ # π₯} = {π€ β π΄ β£ π€ # π΅}) |
14 | fveq2 5517 | . . . . . . . 8 β’ (π₯ = π΅ β (πΉβπ₯) = (πΉβπ΅)) | |
15 | 14 | oveq2d 5893 | . . . . . . 7 β’ (π₯ = π΅ β ((πΉβπ§) β (πΉβπ₯)) = ((πΉβπ§) β (πΉβπ΅))) |
16 | oveq2 5885 | . . . . . . 7 β’ (π₯ = π΅ β (π§ β π₯) = (π§ β π΅)) | |
17 | 15, 16 | oveq12d 5895 | . . . . . 6 β’ (π₯ = π΅ β (((πΉβπ§) β (πΉβπ₯)) / (π§ β π₯)) = (((πΉβπ§) β (πΉβπ΅)) / (π§ β π΅))) |
18 | 13, 17 | mpteq12dv 4087 | . . . . 5 β’ (π₯ = π΅ β (π§ β {π€ β π΄ β£ π€ # π₯} β¦ (((πΉβπ§) β (πΉβπ₯)) / (π§ β π₯))) = (π§ β {π€ β π΄ β£ π€ # π΅} β¦ (((πΉβπ§) β (πΉβπ΅)) / (π§ β π΅)))) |
19 | eldvap.g | . . . . 5 β’ πΊ = (π§ β {π€ β π΄ β£ π€ # π΅} β¦ (((πΉβπ§) β (πΉβπ΅)) / (π§ β π΅))) | |
20 | 18, 19 | eqtr4di 2228 | . . . 4 β’ (π₯ = π΅ β (π§ β {π€ β π΄ β£ π€ # π₯} β¦ (((πΉβπ§) β (πΉβπ₯)) / (π§ β π₯))) = πΊ) |
21 | id 19 | . . . 4 β’ (π₯ = π΅ β π₯ = π΅) | |
22 | 20, 21 | oveq12d 5895 | . . 3 β’ (π₯ = π΅ β ((π§ β {π€ β π΄ β£ π€ # π₯} β¦ (((πΉβπ§) β (πΉβπ₯)) / (π§ β π₯))) limβ π₯) = (πΊ limβ π΅)) |
23 | 22 | opeliunxp2 4769 | . 2 β’ (β¨π΅, πΆβ© β βͺ π₯ β ((intβπ)βπ΄)({π₯} Γ ((π§ β {π€ β π΄ β£ π€ # π₯} β¦ (((πΉβπ§) β (πΉβπ₯)) / (π§ β π₯))) limβ π₯)) β (π΅ β ((intβπ)βπ΄) β§ πΆ β (πΊ limβ π΅))) |
24 | 9, 11, 23 | 3bitr3g 222 | 1 β’ (π β (π΅(π D πΉ)πΆ β (π΅ β ((intβπ)βπ΄) β§ πΆ β (πΊ limβ π΅)))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wb 105 = wceq 1353 β wcel 2148 {crab 2459 β wss 3131 {csn 3594 β¨cop 3597 βͺ ciun 3888 class class class wbr 4005 β¦ cmpt 4066 Γ cxp 4626 β ccom 4632 βΆwf 5214 βcfv 5218 (class class class)co 5877 βcc 7811 β cmin 8130 # cap 8540 / cdiv 8631 abscabs 11008 βΎt crest 12693 MetOpencmopn 13530 intcnt 13678 limβ climc 14208 D cdv 14209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-map 6652 df-pm 6653 df-sup 6985 df-inf 6986 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-xneg 9774 df-xadd 9775 df-seqfrec 10448 df-exp 10522 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-rest 12695 df-topgen 12714 df-psmet 13532 df-xmet 13533 df-met 13534 df-bl 13535 df-mopn 13536 df-top 13583 df-topon 13596 df-bases 13628 df-ntr 13681 df-limced 14210 df-dvap 14211 |
This theorem is referenced by: dvcl 14237 dvfgg 14242 dvidlemap 14245 dvcnp2cntop 14248 dvaddxxbr 14250 dvmulxxbr 14251 dvcoapbr 14256 dvcjbr 14257 dvrecap 14262 dveflem 14272 |
Copyright terms: Public domain | W3C validator |