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Mirrors > Home > ILE Home > Th. List > dvcjbr | Unicode version |
Description: The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 12842. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
Ref | Expression |
---|---|
dvcj.f | |
dvcj.x | |
dvcj.c |
Ref | Expression |
---|---|
dvcjbr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-resscn 7712 | . . . . 5 | |
2 | 1 | a1i 9 | . . . 4 |
3 | dvcj.f | . . . 4 | |
4 | dvcj.x | . . . 4 | |
5 | eqid 2139 | . . . . 5 | |
6 | 5 | tgioo2cntop 12718 | . . . 4 ↾t |
7 | 2, 3, 4, 6, 5 | dvbssntrcntop 12822 | . . 3 |
8 | dvcj.c | . . 3 | |
9 | 7, 8 | sseldd 3098 | . 2 |
10 | 4, 1 | sstrdi 3109 | . . . . . 6 |
11 | 1 | a1i 9 | . . . . . . . . 9 |
12 | simpl 108 | . . . . . . . . 9 | |
13 | simpr 109 | . . . . . . . . 9 | |
14 | 11, 12, 13 | dvbss 12823 | . . . . . . . 8 |
15 | 3, 4, 14 | syl2anc 408 | . . . . . . 7 |
16 | 15, 8 | sseldd 3098 | . . . . . 6 |
17 | 3, 10, 16 | dvlemap 12818 | . . . . 5 # |
18 | 17 | fmpttd 5575 | . . . 4 # # |
19 | ssidd 3118 | . . . 4 | |
20 | 5 | cntoptopon 12701 | . . . . 5 TopOn |
21 | 20 | toponrestid 12188 | . . . 4 ↾t |
22 | 3 | fdmd 5279 | . . . . . . . . . . . . 13 |
23 | 22 | feq2d 5260 | . . . . . . . . . . . 12 |
24 | 3, 23 | mpbird 166 | . . . . . . . . . . 11 |
25 | 22, 4 | eqsstrd 3133 | . . . . . . . . . . 11 |
26 | cnex 7744 | . . . . . . . . . . . 12 | |
27 | reex 7754 | . . . . . . . . . . . 12 | |
28 | 26, 27 | elpm2 6574 | . . . . . . . . . . 11 |
29 | 24, 25, 28 | sylanbrc 413 | . . . . . . . . . 10 |
30 | dvfpm 12827 | . . . . . . . . . 10 | |
31 | 29, 30 | syl 14 | . . . . . . . . 9 |
32 | 31 | ffund 5276 | . . . . . . . 8 |
33 | funfvbrb 5533 | . . . . . . . 8 | |
34 | 32, 33 | syl 14 | . . . . . . 7 |
35 | 8, 34 | mpbid 146 | . . . . . 6 |
36 | eqid 2139 | . . . . . . 7 # # | |
37 | 6, 5, 36, 2, 3, 4 | eldvap 12820 | . . . . . 6 # lim |
38 | 35, 37 | mpbid 146 | . . . . 5 # lim |
39 | 38 | simprd 113 | . . . 4 # lim |
40 | cjcncf 12744 | . . . . . 6 | |
41 | 5 | cncfcn1cntop 12750 | . . . . . 6 |
42 | 40, 41 | eleqtri 2214 | . . . . 5 |
43 | 31, 8 | ffvelrnd 5556 | . . . . 5 |
44 | unicntopcntop 12705 | . . . . . 6 | |
45 | 44 | cncnpi 12397 | . . . . 5 |
46 | 42, 43, 45 | sylancr 410 | . . . 4 |
47 | 18, 19, 5, 21, 39, 46 | limccnpcntop 12813 | . . 3 # lim |
48 | cjf 10619 | . . . . . . 7 | |
49 | 48 | a1i 9 | . . . . . 6 |
50 | 49, 17 | cofmpt 5589 | . . . . 5 # # |
51 | 3 | adantr 274 | . . . . . . . . . 10 # |
52 | elrabi 2837 | . . . . . . . . . . 11 # | |
53 | 52 | adantl 275 | . . . . . . . . . 10 # |
54 | 51, 53 | ffvelrnd 5556 | . . . . . . . . 9 # |
55 | 3, 16 | ffvelrnd 5556 | . . . . . . . . . 10 |
56 | 55 | adantr 274 | . . . . . . . . 9 # |
57 | 54, 56 | subcld 8073 | . . . . . . . 8 # |
58 | 4 | sselda 3097 | . . . . . . . . . . 11 |
59 | 52, 58 | sylan2 284 | . . . . . . . . . 10 # |
60 | 4, 16 | sseldd 3098 | . . . . . . . . . . 11 |
61 | 60 | adantr 274 | . . . . . . . . . 10 # |
62 | 59, 61 | resubcld 8143 | . . . . . . . . 9 # |
63 | 62 | recnd 7794 | . . . . . . . 8 # |
64 | 59 | recnd 7794 | . . . . . . . . 9 # |
65 | 61 | recnd 7794 | . . . . . . . . 9 # |
66 | breq1 3932 | . . . . . . . . . . . 12 # # | |
67 | 66 | elrab 2840 | . . . . . . . . . . 11 # # |
68 | 67 | simprbi 273 | . . . . . . . . . 10 # # |
69 | 68 | adantl 275 | . . . . . . . . 9 # # |
70 | 64, 65, 69 | subap0d 8406 | . . . . . . . 8 # # |
71 | 57, 63, 70 | cjdivapd 10740 | . . . . . . 7 # |
72 | cjsub 10664 | . . . . . . . . . 10 | |
73 | 54, 56, 72 | syl2anc 408 | . . . . . . . . 9 # |
74 | fvco3 5492 | . . . . . . . . . . 11 | |
75 | 3, 52, 74 | syl2an 287 | . . . . . . . . . 10 # |
76 | fvco3 5492 | . . . . . . . . . . . 12 | |
77 | 3, 16, 76 | syl2anc 408 | . . . . . . . . . . 11 |
78 | 77 | adantr 274 | . . . . . . . . . 10 # |
79 | 75, 78 | oveq12d 5792 | . . . . . . . . 9 # |
80 | 73, 79 | eqtr4d 2175 | . . . . . . . 8 # |
81 | 62 | cjred 10743 | . . . . . . . 8 # |
82 | 80, 81 | oveq12d 5792 | . . . . . . 7 # |
83 | 71, 82 | eqtrd 2172 | . . . . . 6 # |
84 | 83 | mpteq2dva 4018 | . . . . 5 # # |
85 | 50, 84 | eqtrd 2172 | . . . 4 # # |
86 | 85 | oveq1d 5789 | . . 3 # lim # lim |
87 | 47, 86 | eleqtrd 2218 | . 2 # lim |
88 | eqid 2139 | . . 3 # # | |
89 | fco 5288 | . . . 4 | |
90 | 48, 3, 89 | sylancr 410 | . . 3 |
91 | 6, 5, 88, 2, 90, 4 | eldvap 12820 | . 2 # lim |
92 | 9, 87, 91 | mpbir2and 928 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 crab 2420 wss 3071 class class class wbr 3929 cmpt 3989 cdm 4539 crn 4540 ccom 4543 wfun 5117 wf 5119 cfv 5123 (class class class)co 5774 cpm 6543 cc 7618 cr 7619 cmin 7933 # cap 8343 cdiv 8432 cioo 9671 ccj 10611 cabs 10769 ctg 12135 cmopn 12154 cnt 12262 ccn 12354 ccnp 12355 ccncf 12726 lim climc 12792 cdv 12793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-map 6544 df-pm 6545 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-xneg 9559 df-xadd 9560 df-ioo 9675 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-rest 12122 df-topgen 12141 df-psmet 12156 df-xmet 12157 df-met 12158 df-bl 12159 df-mopn 12160 df-top 12165 df-topon 12178 df-bases 12210 df-ntr 12265 df-cn 12357 df-cnp 12358 df-cncf 12727 df-limced 12794 df-dvap 12795 |
This theorem is referenced by: dvcj 12842 |
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