![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iihalf2cn | Structured version Visualization version GIF version |
Description: The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) |
Ref | Expression |
---|---|
iihalf2cn.1 | ⊢ 𝐽 = ((topGen‘ran (,)) ↾t ((1 / 2)[,]1)) |
Ref | Expression |
---|---|
iihalf2cn | ⊢ (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | iihalf2cn.1 | . . 3 ⊢ 𝐽 = ((topGen‘ran (,)) ↾t ((1 / 2)[,]1)) | |
3 | dfii2 24173 | . . 3 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1)) | |
4 | halfre 12301 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
5 | 1re 11089 | . . . . 5 ⊢ 1 ∈ ℝ | |
6 | iccssre 13276 | . . . . 5 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ) | |
7 | 4, 5, 6 | mp2an 691 | . . . 4 ⊢ ((1 / 2)[,]1) ⊆ ℝ |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → ((1 / 2)[,]1) ⊆ ℝ) |
9 | unitssre 13346 | . . . 4 ⊢ (0[,]1) ⊆ ℝ | |
10 | 9 | a1i 11 | . . 3 ⊢ (⊤ → (0[,]1) ⊆ ℝ) |
11 | iihalf2 24224 | . . . 4 ⊢ (𝑥 ∈ ((1 / 2)[,]1) → ((2 · 𝑥) − 1) ∈ (0[,]1)) | |
12 | 11 | adantl 483 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ((1 / 2)[,]1)) → ((2 · 𝑥) − 1) ∈ (0[,]1)) |
13 | 1 | cnfldtopon 24074 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
14 | 13 | a1i 11 | . . . 4 ⊢ (⊤ → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
15 | 2cnd 12165 | . . . . . 6 ⊢ (⊤ → 2 ∈ ℂ) | |
16 | 14, 14, 15 | cnmptc 22941 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 2) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
17 | 14 | cnmptid 22940 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 𝑥) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
18 | 1 | mulcn 24158 | . . . . . 6 ⊢ · ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) |
19 | 18 | a1i 11 | . . . . 5 ⊢ (⊤ → · ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))) |
20 | 14, 16, 17, 19 | cnmpt12f 22945 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (2 · 𝑥)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
21 | 1cnd 11084 | . . . . 5 ⊢ (⊤ → 1 ∈ ℂ) | |
22 | 14, 14, 21 | cnmptc 22941 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
23 | 1 | subcn 24157 | . . . . 5 ⊢ − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) |
24 | 23 | a1i 11 | . . . 4 ⊢ (⊤ → − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))) |
25 | 14, 20, 22, 24 | cnmpt12f 22945 | . . 3 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((2 · 𝑥) − 1)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
26 | 1, 2, 3, 8, 10, 12, 25 | cnmptre 24218 | . 2 ⊢ (⊤ → (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II)) |
27 | 26 | mptru 1549 | 1 ⊢ (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ⊆ wss 3909 ↦ cmpt 5187 ran crn 5632 ‘cfv 6492 (class class class)co 7350 ℂcc 10983 ℝcr 10984 0cc0 10985 1c1 10986 · cmul 10990 − cmin 11319 / cdiv 11746 2c2 12142 (,)cioo 13194 [,]cicc 13197 ↾t crest 17238 TopOpenctopn 17239 topGenctg 17255 ℂfldccnfld 20725 TopOnctopon 22187 Cn ccn 22503 ×t ctx 22839 IIcii 24166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 ax-mulf 11065 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-of 7608 df-om 7794 df-1st 7912 df-2nd 7913 df-supp 8061 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8582 df-map 8701 df-ixp 8770 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-fsupp 9240 df-fi 9281 df-sup 9312 df-inf 9313 df-oi 9380 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12553 df-uz 12698 df-q 12804 df-rp 12846 df-xneg 12963 df-xadd 12964 df-xmul 12965 df-ioo 13198 df-icc 13201 df-fz 13355 df-fzo 13498 df-seq 13837 df-exp 13898 df-hash 14160 df-cj 14919 df-re 14920 df-im 14921 df-sqrt 15055 df-abs 15056 df-struct 16955 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-ress 17049 df-plusg 17082 df-mulr 17083 df-starv 17084 df-sca 17085 df-vsca 17086 df-ip 17087 df-tset 17088 df-ple 17089 df-ds 17091 df-unif 17092 df-hom 17093 df-cco 17094 df-rest 17240 df-topn 17241 df-0g 17259 df-gsum 17260 df-topgen 17261 df-pt 17262 df-prds 17265 df-xrs 17320 df-qtop 17325 df-imas 17326 df-xps 17328 df-mre 17402 df-mrc 17403 df-acs 17405 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-submnd 18538 df-mulg 18808 df-cntz 19032 df-cmn 19499 df-psmet 20717 df-xmet 20718 df-met 20719 df-bl 20720 df-mopn 20721 df-cnfld 20726 df-top 22171 df-topon 22188 df-topsp 22210 df-bases 22224 df-cn 22506 df-cnp 22507 df-tx 22841 df-hmeo 23034 df-xms 23601 df-ms 23602 df-tms 23603 df-ii 24168 |
This theorem is referenced by: htpycc 24271 pcocn 24308 pcohtpylem 24310 pcopt 24313 pcorevlem 24317 |
Copyright terms: Public domain | W3C validator |