Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > limcdifap | Unicode version |
Description: It suffices to consider functions which are not defined at to define the limit of a function. In particular, the value of the original function at does not affect the limit of . (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.) |
Ref | Expression |
---|---|
limccl.f | |
limcdifap.a |
Ref | Expression |
---|---|
limcdifap | lim # lim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limcrcl 12796 | . . . . 5 lim | |
2 | 1 | simp3d 995 | . . . 4 lim |
3 | 2 | a1i 9 | . . 3 lim |
4 | limcrcl 12796 | . . . . 5 # lim # # # | |
5 | 4 | simp3d 995 | . . . 4 # lim |
6 | 5 | a1i 9 | . . 3 # lim |
7 | breq1 3932 | . . . . . . . . . . . . . . . . 17 # # | |
8 | simplr 519 | . . . . . . . . . . . . . . . . 17 # | |
9 | simpr 109 | . . . . . . . . . . . . . . . . 17 # # | |
10 | 7, 8, 9 | elrabd 2842 | . . . . . . . . . . . . . . . 16 # # |
11 | fvres 5445 | . . . . . . . . . . . . . . . . 17 # # | |
12 | 11 | eqcomd 2145 | . . . . . . . . . . . . . . . 16 # # |
13 | 10, 12 | syl 14 | . . . . . . . . . . . . . . 15 # # |
14 | 13 | fvoveq1d 5796 | . . . . . . . . . . . . . 14 # # |
15 | 14 | breq1d 3939 | . . . . . . . . . . . . 13 # # |
16 | 15 | imbi2d 229 | . . . . . . . . . . . 12 # # |
17 | 16 | pm5.74da 439 | . . . . . . . . . . 11 # # # |
18 | impexp 261 | . . . . . . . . . . 11 # # | |
19 | impexp 261 | . . . . . . . . . . . . 13 # # # # | |
20 | 19 | imbi2i 225 | . . . . . . . . . . . 12 # # # # # # |
21 | pm5.4 248 | . . . . . . . . . . . 12 # # # # # | |
22 | 20, 21 | bitri 183 | . . . . . . . . . . 11 # # # # # |
23 | 17, 18, 22 | 3bitr4g 222 | . . . . . . . . . 10 # # # # |
24 | 23 | ralbidva 2433 | . . . . . . . . 9 # # # # |
25 | 7 | ralrab 2845 | . . . . . . . . 9 # # # # # # |
26 | 24, 25 | syl6bbr 197 | . . . . . . . 8 # # # # |
27 | 26 | rexbidv 2438 | . . . . . . 7 # # # # |
28 | 27 | ralbidv 2437 | . . . . . 6 # # # # |
29 | 28 | anbi2d 459 | . . . . 5 # # # # |
30 | limccl.f | . . . . . . 7 | |
31 | 30 | adantr 274 | . . . . . 6 |
32 | limcdifap.a | . . . . . . 7 | |
33 | 32 | adantr 274 | . . . . . 6 |
34 | simpr 109 | . . . . . 6 | |
35 | 31, 33, 34 | ellimc3ap 12799 | . . . . 5 lim # |
36 | ssrab2 3182 | . . . . . . 7 # | |
37 | fssres 5298 | . . . . . . 7 # # # | |
38 | 31, 36, 37 | sylancl 409 | . . . . . 6 # # |
39 | 36, 33 | sstrid 3108 | . . . . . 6 # |
40 | 38, 39, 34 | ellimc3ap 12799 | . . . . 5 # lim # # # |
41 | 29, 35, 40 | 3bitr4d 219 | . . . 4 lim # lim |
42 | 41 | ex 114 | . . 3 lim # lim |
43 | 3, 6, 42 | pm5.21ndd 694 | . 2 lim # lim |
44 | 43 | eqrdv 2137 | 1 lim # lim |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wrex 2417 crab 2420 wss 3071 class class class wbr 3929 cdm 4539 cres 4541 wf 5119 cfv 5123 (class class class)co 5774 cc 7618 clt 7800 cmin 7933 # cap 8343 crp 9441 cabs 10769 lim climc 12792 |
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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 |
This theorem depends on definitions: df-bi 116 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-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pm 6545 df-limced 12794 |
This theorem is referenced by: dvcnp2cntop 12832 dvmulxxbr 12835 dvrecap 12846 |
Copyright terms: Public domain | W3C validator |