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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.) Avoid ax-mulf 11218. (Revised by GG, 16-Mar-2025.) |
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 2725 | . . 3 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | iihalf2cn.1 | . . 3 ⊢ 𝐽 = ((topGen‘ran (,)) ↾t ((1 / 2)[,]1)) | |
3 | dfii2 24820 | . . 3 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1)) | |
4 | halfre 12456 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
5 | 1red 11245 | . . . 4 ⊢ (⊤ → 1 ∈ ℝ) | |
6 | iccssre 13438 | . . . 4 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ) | |
7 | 4, 5, 6 | sylancr 585 | . . 3 ⊢ (⊤ → ((1 / 2)[,]1) ⊆ ℝ) |
8 | unitssre 13508 | . . . 4 ⊢ (0[,]1) ⊆ ℝ | |
9 | 8 | a1i 11 | . . 3 ⊢ (⊤ → (0[,]1) ⊆ ℝ) |
10 | iihalf2 24873 | . . . 4 ⊢ (𝑥 ∈ ((1 / 2)[,]1) → ((2 · 𝑥) − 1) ∈ (0[,]1)) | |
11 | 10 | adantl 480 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ((1 / 2)[,]1)) → ((2 · 𝑥) − 1) ∈ (0[,]1)) |
12 | 1 | cnfldtopon 24717 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
13 | 12 | a1i 11 | . . . 4 ⊢ (⊤ → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
14 | 2cnd 12320 | . . . . . 6 ⊢ (⊤ → 2 ∈ ℂ) | |
15 | 13, 13, 14 | cnmptc 23584 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 2) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
16 | 13 | cnmptid 23583 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 𝑥) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
17 | 1 | mpomulcn 24803 | . . . . . 6 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) |
18 | 17 | a1i 11 | . . . . 5 ⊢ (⊤ → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))) |
19 | oveq12 7425 | . . . . 5 ⊢ ((𝑢 = 2 ∧ 𝑣 = 𝑥) → (𝑢 · 𝑣) = (2 · 𝑥)) | |
20 | 13, 15, 16, 13, 13, 18, 19 | cnmpt12 23589 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (2 · 𝑥)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
21 | 1cnd 11239 | . . . . 5 ⊢ (⊤ → 1 ∈ ℂ) | |
22 | 13, 13, 21 | cnmptc 23584 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
23 | 1 | subcn 24800 | . . . . 5 ⊢ − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) |
24 | 23 | a1i 11 | . . . 4 ⊢ (⊤ → − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))) |
25 | 13, 20, 22, 24 | cnmpt12f 23588 | . . 3 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((2 · 𝑥) − 1)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
26 | 1, 2, 3, 7, 9, 11, 25 | cnmptre 24866 | . 2 ⊢ (⊤ → (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II)) |
27 | 26 | mptru 1540 | 1 ⊢ (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ⊆ wss 3939 ↦ cmpt 5226 ran crn 5673 ‘cfv 6543 (class class class)co 7416 ∈ cmpo 7418 ℂcc 11136 ℝcr 11137 0cc0 11138 1c1 11139 · cmul 11143 − cmin 11474 / cdiv 11901 2c2 12297 (,)cioo 13356 [,]cicc 13359 ↾t crest 17401 TopOpenctopn 17402 topGenctg 17418 ℂfldccnfld 21283 TopOnctopon 22830 Cn ccn 23146 ×t ctx 23482 IIcii 24813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-icc 13363 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cn 23149 df-cnp 23150 df-tx 23484 df-hmeo 23677 df-xms 24244 df-ms 24245 df-tms 24246 df-ii 24815 |
This theorem is referenced by: htpycc 24924 pcocn 24962 pcohtpylem 24964 pcopt 24967 pcorevlem 24971 |
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