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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-minftyccb | Structured version Visualization version GIF version |
Description: The class -∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.) |
Ref | Expression |
---|---|
bj-minftyccb | ⊢ -∞ ∈ ℂ̅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-ccinftyssccbar 35487 | . 2 ⊢ ℂ∞ ⊆ ℂ̅ | |
2 | df-bj-inftyexpi 35476 | . . . . 5 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩) | |
3 | 2 | funmpt2 6517 | . . . 4 ⊢ Fun +∞ei |
4 | pire 25713 | . . . . . . . 8 ⊢ π ∈ ℝ | |
5 | 4 | renegcli 11375 | . . . . . . 7 ⊢ -π ∈ ℝ |
6 | 5 | rexri 11126 | . . . . . 6 ⊢ -π ∈ ℝ* |
7 | 4 | rexri 11126 | . . . . . 6 ⊢ π ∈ ℝ* |
8 | pipos 25715 | . . . . . . . . 9 ⊢ 0 < π | |
9 | 0re 11070 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
10 | 9, 4 | ltnegi 11612 | . . . . . . . . 9 ⊢ (0 < π ↔ -π < -0) |
11 | 8, 10 | mpbi 229 | . . . . . . . 8 ⊢ -π < -0 |
12 | neg0 11360 | . . . . . . . 8 ⊢ -0 = 0 | |
13 | 11, 12 | breqtri 5114 | . . . . . . 7 ⊢ -π < 0 |
14 | 5, 9, 4 | lttri 11194 | . . . . . . 7 ⊢ ((-π < 0 ∧ 0 < π) → -π < π) |
15 | 13, 8, 14 | mp2an 689 | . . . . . 6 ⊢ -π < π |
16 | ubioc1 13225 | . . . . . 6 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ* ∧ -π < π) → π ∈ (-π(,]π)) | |
17 | 6, 7, 15, 16 | mp3an 1460 | . . . . 5 ⊢ π ∈ (-π(,]π) |
18 | opex 5403 | . . . . . 6 ⊢ ⟨𝑥, ℂ⟩ ∈ V | |
19 | 18, 2 | dmmpti 6622 | . . . . 5 ⊢ dom +∞ei = (-π(,]π) |
20 | 17, 19 | eleqtrri 2836 | . . . 4 ⊢ π ∈ dom +∞ei |
21 | fvelrn 7004 | . . . 4 ⊢ ((Fun +∞ei ∧ π ∈ dom +∞ei) → (+∞ei‘π) ∈ ran +∞ei) | |
22 | 3, 20, 21 | mp2an 689 | . . 3 ⊢ (+∞ei‘π) ∈ ran +∞ei |
23 | df-bj-minfty 35493 | . . 3 ⊢ -∞ = (+∞ei‘π) | |
24 | df-bj-ccinfty 35481 | . . 3 ⊢ ℂ∞ = ran +∞ei | |
25 | 22, 23, 24 | 3eltr4i 2850 | . 2 ⊢ -∞ ∈ ℂ∞ |
26 | 1, 25 | sselii 3928 | 1 ⊢ -∞ ∈ ℂ̅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ⟨cop 4578 class class class wbr 5089 dom cdm 5614 ran crn 5615 Fun wfun 6467 ‘cfv 6473 (class class class)co 7329 ℂcc 10962 0cc0 10964 ℝ*cxr 11101 < clt 11102 -cneg 11299 (,]cioc 13173 πcpi 15867 +∞eicinftyexpi 35475 ℂ∞cccinfty 35480 ℂ̅cccbar 35484 -∞cminfty 35492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-inf2 9490 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-pre-sup 11042 ax-addf 11043 ax-mulf 11044 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-om 7773 df-1st 7891 df-2nd 7892 df-supp 8040 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-2o 8360 df-er 8561 df-map 8680 df-pm 8681 df-ixp 8749 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-fsupp 9219 df-fi 9260 df-sup 9291 df-inf 9292 df-oi 9359 df-card 9788 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-div 11726 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-q 12782 df-rp 12824 df-xneg 12941 df-xadd 12942 df-xmul 12943 df-ioo 13176 df-ioc 13177 df-ico 13178 df-icc 13179 df-fz 13333 df-fzo 13476 df-fl 13605 df-seq 13815 df-exp 13876 df-fac 14081 df-bc 14110 df-hash 14138 df-shft 14869 df-cj 14901 df-re 14902 df-im 14903 df-sqrt 15037 df-abs 15038 df-limsup 15271 df-clim 15288 df-rlim 15289 df-sum 15489 df-ef 15868 df-sin 15870 df-cos 15871 df-pi 15873 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-starv 17066 df-sca 17067 df-vsca 17068 df-ip 17069 df-tset 17070 df-ple 17071 df-ds 17073 df-unif 17074 df-hom 17075 df-cco 17076 df-rest 17222 df-topn 17223 df-0g 17241 df-gsum 17242 df-topgen 17243 df-pt 17244 df-prds 17247 df-xrs 17302 df-qtop 17307 df-imas 17308 df-xps 17310 df-mre 17384 df-mrc 17385 df-acs 17387 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-submnd 18520 df-mulg 18789 df-cntz 19011 df-cmn 19475 df-psmet 20687 df-xmet 20688 df-met 20689 df-bl 20690 df-mopn 20691 df-fbas 20692 df-fg 20693 df-cnfld 20696 df-top 22141 df-topon 22158 df-topsp 22180 df-bases 22194 df-cld 22268 df-ntr 22269 df-cls 22270 df-nei 22347 df-lp 22385 df-perf 22386 df-cn 22476 df-cnp 22477 df-haus 22564 df-tx 22811 df-hmeo 23004 df-fil 23095 df-fm 23187 df-flim 23188 df-flf 23189 df-xms 23571 df-ms 23572 df-tms 23573 df-cncf 24139 df-limc 25128 df-dv 25129 df-bj-inftyexpi 35476 df-bj-ccinfty 35481 df-bj-ccbar 35485 df-bj-minfty 35493 |
This theorem is referenced by: (None) |
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