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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 12785 | . . . . 5 lim | |
2 | 1 | simp3d 995 | . . . 4 lim |
3 | 2 | a1i 9 | . . 3 lim |
4 | limcrcl 12785 | . . . . 5 # lim # # # | |
5 | 4 | simp3d 995 | . . . 4 # lim |
6 | 5 | a1i 9 | . . 3 # lim |
7 | breq1 3927 | . . . . . . . . . . . . . . . . 17 # # | |
8 | simplr 519 | . . . . . . . . . . . . . . . . 17 # | |
9 | simpr 109 | . . . . . . . . . . . . . . . . 17 # # | |
10 | 7, 8, 9 | elrabd 2837 | . . . . . . . . . . . . . . . 16 # # |
11 | fvres 5438 | . . . . . . . . . . . . . . . . 17 # # | |
12 | 11 | eqcomd 2143 | . . . . . . . . . . . . . . . 16 # # |
13 | 10, 12 | syl 14 | . . . . . . . . . . . . . . 15 # # |
14 | 13 | fvoveq1d 5789 | . . . . . . . . . . . . . 14 # # |
15 | 14 | breq1d 3934 | . . . . . . . . . . . . 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 2431 | . . . . . . . . 9 # # # # |
25 | 7 | ralrab 2840 | . . . . . . . . 9 # # # # # # |
26 | 24, 25 | syl6bbr 197 | . . . . . . . 8 # # # # |
27 | 26 | rexbidv 2436 | . . . . . . 7 # # # # |
28 | 27 | ralbidv 2435 | . . . . . 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 12788 | . . . . 5 lim # |
36 | ssrab2 3177 | . . . . . . 7 # | |
37 | fssres 5293 | . . . . . . 7 # # # | |
38 | 31, 36, 37 | sylancl 409 | . . . . . 6 # # |
39 | 36, 33 | sstrid 3103 | . . . . . 6 # |
40 | 38, 39, 34 | ellimc3ap 12788 | . . . . 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 2135 | 1 lim # lim |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2414 wrex 2415 crab 2418 wss 3066 class class class wbr 3924 cdm 4534 cres 4536 wf 5114 cfv 5118 (class class class)co 5767 cc 7611 clt 7793 cmin 7926 # cap 8336 crp 9434 cabs 10762 lim climc 12781 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pm 6538 df-limced 12783 |
This theorem is referenced by: dvcnp2cntop 12821 dvmulxxbr 12824 dvrecap 12835 |
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