Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climf2 | Structured version Visualization version Unicode version |
Description: Express the predicate: The limit of complex number sequence is , or converges to . Similar to clim 14225, but without the disjoint var constraint and . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
climf2.1 | |
climf2.nf | |
climf2.f | |
climf2.fv |
Ref | Expression |
---|---|
climf2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climrel 14223 | . . . . 5 | |
2 | 1 | brrelex2i 5159 | . . . 4 |
3 | 2 | a1i 11 | . . 3 |
4 | elex 3212 | . . . . 5 | |
5 | 4 | adantr 481 | . . . 4 |
6 | 5 | a1i 11 | . . 3 |
7 | climf2.f | . . . 4 | |
8 | simpr 477 | . . . . . . . 8 | |
9 | 8 | eleq1d 2686 | . . . . . . 7 |
10 | nfv 1843 | . . . . . . . 8 | |
11 | climf2.nf | . . . . . . . . . . . 12 | |
12 | 11 | nfeq2 2780 | . . . . . . . . . . 11 |
13 | nfv 1843 | . . . . . . . . . . 11 | |
14 | 12, 13 | nfan 1828 | . . . . . . . . . 10 |
15 | fveq1 6190 | . . . . . . . . . . . . 13 | |
16 | 15 | adantr 481 | . . . . . . . . . . . 12 |
17 | 16 | eleq1d 2686 | . . . . . . . . . . 11 |
18 | oveq12 6659 | . . . . . . . . . . . . . 14 | |
19 | 15, 18 | sylan 488 | . . . . . . . . . . . . 13 |
20 | 19 | fveq2d 6195 | . . . . . . . . . . . 12 |
21 | 20 | breq1d 4663 | . . . . . . . . . . 11 |
22 | 17, 21 | anbi12d 747 | . . . . . . . . . 10 |
23 | 14, 22 | ralbid 2983 | . . . . . . . . 9 |
24 | 23 | rexbidv 3052 | . . . . . . . 8 |
25 | 10, 24 | ralbid 2983 | . . . . . . 7 |
26 | 9, 25 | anbi12d 747 | . . . . . 6 |
27 | df-clim 14219 | . . . . . 6 | |
28 | 26, 27 | brabga 4989 | . . . . 5 |
29 | 28 | ex 450 | . . . 4 |
30 | 7, 29 | syl 17 | . . 3 |
31 | 3, 6, 30 | pm5.21ndd 369 | . 2 |
32 | climf2.1 | . . . . . 6 | |
33 | eluzelz 11697 | . . . . . . 7 | |
34 | climf2.fv | . . . . . . . . 9 | |
35 | 34 | eleq1d 2686 | . . . . . . . 8 |
36 | 34 | oveq1d 6665 | . . . . . . . . . 10 |
37 | 36 | fveq2d 6195 | . . . . . . . . 9 |
38 | 37 | breq1d 4663 | . . . . . . . 8 |
39 | 35, 38 | anbi12d 747 | . . . . . . 7 |
40 | 33, 39 | sylan2 491 | . . . . . 6 |
41 | 32, 40 | ralbida 2982 | . . . . 5 |
42 | 41 | rexbidv 3052 | . . . 4 |
43 | 42 | ralbidv 2986 | . . 3 |
44 | 43 | anbi2d 740 | . 2 |
45 | 31, 44 | bitrd 268 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wnf 1708 wcel 1990 wnfc 2751 wral 2912 wrex 2913 cvv 3200 class class class wbr 4653 cfv 5888 (class class class)co 6650 cc 9934 clt 10074 cmin 10266 cz 11377 cuz 11687 crp 11832 cabs 13974 cli 14215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-neg 10269 df-z 11378 df-uz 11688 df-clim 14219 |
This theorem is referenced by: clim2f2 39902 |
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