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Mirrors > Home > ILE Home > Th. List > limcimo | Unicode version |
Description: Conditions which ensure there is at most one limit value of at . (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 8-Jul-2023.) |
Ref | Expression |
---|---|
limcflf.f | |
limcflf.a | |
limcimo.b | |
limcimo.bc | |
limcimo.bs | |
limcimo.c | ↾t |
limcimo.s | |
limcimo.ca | # |
limcflfcntop.k |
Ref | Expression |
---|---|
limcimo | lim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 3971 | . . . . . . . . . 10 | |
2 | 1 | imbi2d 229 | . . . . . . . . 9 # # |
3 | 2 | rexralbidv 2483 | . . . . . . . 8 # # |
4 | limcflf.f | . . . . . . . . . . . . 13 | |
5 | limcflf.a | . . . . . . . . . . . . 13 | |
6 | limcimo.b | . . . . . . . . . . . . 13 | |
7 | 4, 5, 6 | ellimc3ap 13100 | . . . . . . . . . . . 12 lim # |
8 | 7 | biimpa 294 | . . . . . . . . . . 11 lim # |
9 | 8 | adantrr 471 | . . . . . . . . . 10 lim lim # |
10 | 9 | simprd 113 | . . . . . . . . 9 lim lim # |
11 | 10 | adantr 274 | . . . . . . . 8 lim lim # # |
12 | 9 | simpld 111 | . . . . . . . . . . . 12 lim lim |
13 | 12 | adantr 274 | . . . . . . . . . . 11 lim lim # |
14 | 4, 5, 6 | ellimc3ap 13100 | . . . . . . . . . . . . . . 15 lim # |
15 | 14 | biimpa 294 | . . . . . . . . . . . . . 14 lim # |
16 | 15 | adantrl 470 | . . . . . . . . . . . . 13 lim lim # |
17 | 16 | simpld 111 | . . . . . . . . . . . 12 lim lim |
18 | 17 | adantr 274 | . . . . . . . . . . 11 lim lim # |
19 | 13, 18 | subcld 8190 | . . . . . . . . . 10 lim lim # |
20 | simpr 109 | . . . . . . . . . . 11 lim lim # # | |
21 | 13, 18, 20 | subap0d 8523 | . . . . . . . . . 10 lim lim # # |
22 | 19, 21 | absrpclapd 11099 | . . . . . . . . 9 lim lim # |
23 | 22 | rphalfcld 9622 | . . . . . . . 8 lim lim # |
24 | 3, 11, 23 | rspcdva 2821 | . . . . . . 7 lim lim # # |
25 | breq2 3971 | . . . . . . . . . . . 12 | |
26 | 25 | imbi2d 229 | . . . . . . . . . . 11 # # |
27 | 26 | rexralbidv 2483 | . . . . . . . . . 10 # # |
28 | 16 | simprd 113 | . . . . . . . . . . 11 lim lim # |
29 | 28 | adantr 274 | . . . . . . . . . 10 lim lim # # |
30 | 27, 29, 23 | rspcdva 2821 | . . . . . . . . 9 lim lim # # |
31 | 30 | adantr 274 | . . . . . . . 8 lim lim # # # |
32 | 4 | ad4antr 486 | . . . . . . . . 9 lim lim # # # |
33 | 5 | ad4antr 486 | . . . . . . . . 9 lim lim # # # |
34 | 6 | ad4antr 486 | . . . . . . . . 9 lim lim # # # |
35 | limcimo.bc | . . . . . . . . . 10 | |
36 | 35 | ad4antr 486 | . . . . . . . . 9 lim lim # # # |
37 | limcimo.bs | . . . . . . . . . 10 | |
38 | 37 | ad4antr 486 | . . . . . . . . 9 lim lim # # # |
39 | limcimo.c | . . . . . . . . . 10 ↾t | |
40 | 39 | ad4antr 486 | . . . . . . . . 9 lim lim # # # ↾t |
41 | limcimo.s | . . . . . . . . . 10 | |
42 | 41 | ad4antr 486 | . . . . . . . . 9 lim lim # # # |
43 | limcimo.ca | . . . . . . . . . 10 # | |
44 | 43 | ad4antr 486 | . . . . . . . . 9 lim lim # # # # |
45 | limcflfcntop.k | . . . . . . . . 9 | |
46 | simplrl 525 | . . . . . . . . 9 lim lim # # # | |
47 | simprl 521 | . . . . . . . . . 10 lim lim lim | |
48 | 47 | ad3antrrr 484 | . . . . . . . . 9 lim lim # # # lim |
49 | simprr 522 | . . . . . . . . . 10 lim lim lim | |
50 | 49 | ad3antrrr 484 | . . . . . . . . 9 lim lim # # # lim |
51 | simplrr 526 | . . . . . . . . 9 lim lim # # # # | |
52 | simprl 521 | . . . . . . . . 9 lim lim # # # | |
53 | simprr 522 | . . . . . . . . 9 lim lim # # # # | |
54 | 32, 33, 34, 36, 38, 40, 42, 44, 45, 46, 48, 50, 51, 52, 53 | limcimolemlt 13103 | . . . . . . . 8 lim lim # # # |
55 | 31, 54 | rexlimddv 2579 | . . . . . . 7 lim lim # # |
56 | 24, 55 | rexlimddv 2579 | . . . . . 6 lim lim # |
57 | 22 | rpred 9609 | . . . . . . 7 lim lim # |
58 | 57 | ltnrd 7991 | . . . . . 6 lim lim # |
59 | 56, 58 | pm2.65da 651 | . . . . 5 lim lim # |
60 | apti 8501 | . . . . . 6 # | |
61 | 12, 17, 60 | syl2anc 409 | . . . . 5 lim lim # |
62 | 59, 61 | mpbird 166 | . . . 4 lim lim |
63 | 62 | ex 114 | . . 3 lim lim |
64 | 63 | alrimivv 1855 | . 2 lim lim |
65 | eleq1w 2218 | . . 3 lim lim | |
66 | 65 | mo4 2067 | . 2 lim lim lim |
67 | 64, 66 | sylibr 133 | 1 lim |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wal 1333 wceq 1335 wmo 2007 wcel 2128 wral 2435 wrex 2436 crab 2439 wss 3102 cpr 3562 class class class wbr 3967 ccom 4592 wf 5168 cfv 5172 (class class class)co 5826 cc 7732 cr 7733 clt 7914 cmin 8050 # cap 8460 cdiv 8549 c2 8889 crp 9566 cabs 10908 ↾t crest 12421 cmopn 12455 lim climc 13093 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-precex 7844 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-apti 7849 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 ax-pre-mulext 7852 ax-arch 7853 ax-caucvg 7854 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-id 4255 df-po 4258 df-iso 4259 df-iord 4328 df-on 4330 df-ilim 4331 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-isom 5181 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-recs 6254 df-frec 6340 df-map 6597 df-pm 6598 df-sup 6930 df-inf 6931 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-reap 8454 df-ap 8461 df-div 8550 df-inn 8839 df-2 8897 df-3 8898 df-4 8899 df-n0 9096 df-z 9173 df-uz 9445 df-q 9535 df-rp 9567 df-xneg 9685 df-xadd 9686 df-seqfrec 10354 df-exp 10428 df-cj 10753 df-re 10754 df-im 10755 df-rsqrt 10909 df-abs 10910 df-rest 12423 df-topgen 12442 df-psmet 12457 df-xmet 12458 df-met 12459 df-bl 12460 df-mopn 12461 df-top 12466 df-topon 12479 df-bases 12511 df-limced 13095 |
This theorem is referenced by: dvfgg 13127 |
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