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Mirrors > Home > MPE Home > Th. List > iihalf1cn | Structured version Visualization version GIF version |
Description: The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.) |
Ref | Expression |
---|---|
iihalf1cn.1 | ⊢ 𝐽 = ((topGen‘ran (,)) ↾t (0[,](1 / 2))) |
Ref | Expression |
---|---|
iihalf1cn | ⊢ (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (𝐽 Cn II) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | iihalf1cn.1 | . . 3 ⊢ 𝐽 = ((topGen‘ran (,)) ↾t (0[,](1 / 2))) | |
3 | dfii2 24757 | . . 3 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1)) | |
4 | 0red 11221 | . . . 4 ⊢ (⊤ → 0 ∈ ℝ) | |
5 | halfre 12430 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
6 | iccssre 13412 | . . . 4 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ) | |
7 | 4, 5, 6 | sylancl 585 | . . 3 ⊢ (⊤ → (0[,](1 / 2)) ⊆ ℝ) |
8 | unitssre 13482 | . . . 4 ⊢ (0[,]1) ⊆ ℝ | |
9 | 8 | a1i 11 | . . 3 ⊢ (⊤ → (0[,]1) ⊆ ℝ) |
10 | iihalf1 24807 | . . . 4 ⊢ (𝑥 ∈ (0[,](1 / 2)) → (2 · 𝑥) ∈ (0[,]1)) | |
11 | 10 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,](1 / 2))) → (2 · 𝑥) ∈ (0[,]1)) |
12 | 1 | cnfldtopon 24654 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
13 | 12 | a1i 11 | . . . 4 ⊢ (⊤ → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
14 | 2cnd 12294 | . . . . 5 ⊢ (⊤ → 2 ∈ ℂ) | |
15 | 13, 13, 14 | cnmptc 23521 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 2) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
16 | 13 | cnmptid 23520 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 𝑥) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
17 | 1 | mpomulcn 24740 | . . . . 5 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) |
18 | 17 | a1i 11 | . . . 4 ⊢ (⊤ → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))) |
19 | oveq12 7414 | . . . 4 ⊢ ((𝑢 = 2 ∧ 𝑣 = 𝑥) → (𝑢 · 𝑣) = (2 · 𝑥)) | |
20 | 13, 15, 16, 13, 13, 18, 19 | cnmpt12 23526 | . . 3 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (2 · 𝑥)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
21 | 1, 2, 3, 7, 9, 11, 20 | cnmptre 24803 | . 2 ⊢ (⊤ → (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (𝐽 Cn II)) |
22 | 21 | mptru 1540 | 1 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (𝐽 Cn II) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ⊆ wss 3943 ↦ cmpt 5224 ran crn 5670 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 · cmul 11117 / cdiv 11875 2c2 12271 (,)cioo 13330 [,]cicc 13333 ↾t crest 17375 TopOpenctopn 17376 topGenctg 17392 ℂfldccnfld 21240 TopOnctopon 22767 Cn ccn 23083 ×t ctx 23419 IIcii 24750 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-icc 13337 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cn 23086 df-cnp 23087 df-tx 23421 df-hmeo 23614 df-xms 24181 df-ms 24182 df-tms 24183 df-ii 24752 |
This theorem is referenced by: htpycc 24861 pcocn 24899 pcohtpylem 24901 pcopt2 24905 pcoass 24906 pcorevlem 24908 |
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