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Mirrors > Home > MPE Home > Th. List > cncfcnvcn | Structured version Visualization version GIF version |
Description: Rewrite cmphaushmeo 21805 for functions on the complex numbers. (Contributed by Mario Carneiro, 19-Feb-2015.) |
Ref | Expression |
---|---|
cncfcnvcn.j | ⊢ 𝐽 = (TopOpen‘ℂfld) |
cncfcnvcn.k | ⊢ 𝐾 = (𝐽 ↾t 𝑋) |
Ref | Expression |
---|---|
cncfcnvcn | ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑌–cn→𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 479 | . . . 4 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → 𝐹 ∈ (𝑋–cn→𝑌)) | |
2 | cncfrss 22895 | . . . . . 6 ⊢ (𝐹 ∈ (𝑋–cn→𝑌) → 𝑋 ⊆ ℂ) | |
3 | 2 | adantl 473 | . . . . 5 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → 𝑋 ⊆ ℂ) |
4 | cncfrss2 22896 | . . . . . 6 ⊢ (𝐹 ∈ (𝑋–cn→𝑌) → 𝑌 ⊆ ℂ) | |
5 | 4 | adantl 473 | . . . . 5 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → 𝑌 ⊆ ℂ) |
6 | cncfcnvcn.j | . . . . . 6 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
7 | cncfcnvcn.k | . . . . . 6 ⊢ 𝐾 = (𝐽 ↾t 𝑋) | |
8 | eqid 2760 | . . . . . 6 ⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t 𝑌) | |
9 | 6, 7, 8 | cncfcn 22913 | . . . . 5 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋–cn→𝑌) = (𝐾 Cn (𝐽 ↾t 𝑌))) |
10 | 3, 5, 9 | syl2anc 696 | . . . 4 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝑋–cn→𝑌) = (𝐾 Cn (𝐽 ↾t 𝑌))) |
11 | 1, 10 | eleqtrd 2841 | . . 3 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → 𝐹 ∈ (𝐾 Cn (𝐽 ↾t 𝑌))) |
12 | ishmeo 21764 | . . . 4 ⊢ (𝐹 ∈ (𝐾Homeo(𝐽 ↾t 𝑌)) ↔ (𝐹 ∈ (𝐾 Cn (𝐽 ↾t 𝑌)) ∧ ◡𝐹 ∈ ((𝐽 ↾t 𝑌) Cn 𝐾))) | |
13 | 12 | baib 982 | . . 3 ⊢ (𝐹 ∈ (𝐾 Cn (𝐽 ↾t 𝑌)) → (𝐹 ∈ (𝐾Homeo(𝐽 ↾t 𝑌)) ↔ ◡𝐹 ∈ ((𝐽 ↾t 𝑌) Cn 𝐾))) |
14 | 11, 13 | syl 17 | . 2 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝐹 ∈ (𝐾Homeo(𝐽 ↾t 𝑌)) ↔ ◡𝐹 ∈ ((𝐽 ↾t 𝑌) Cn 𝐾))) |
15 | 6 | cnfldtop 22788 | . . . . . 6 ⊢ 𝐽 ∈ Top |
16 | 6 | cnfldtopon 22787 | . . . . . . . 8 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
17 | 16 | toponunii 20923 | . . . . . . 7 ⊢ ℂ = ∪ 𝐽 |
18 | 17 | restuni 21168 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ℂ) → 𝑋 = ∪ (𝐽 ↾t 𝑋)) |
19 | 15, 3, 18 | sylancr 698 | . . . . 5 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → 𝑋 = ∪ (𝐽 ↾t 𝑋)) |
20 | 7 | unieqi 4597 | . . . . 5 ⊢ ∪ 𝐾 = ∪ (𝐽 ↾t 𝑋) |
21 | 19, 20 | syl6eqr 2812 | . . . 4 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → 𝑋 = ∪ 𝐾) |
22 | f1oeq2 6289 | . . . 4 ⊢ (𝑋 = ∪ 𝐾 → (𝐹:𝑋–1-1-onto→∪ (𝐽 ↾t 𝑌) ↔ 𝐹:∪ 𝐾–1-1-onto→∪ (𝐽 ↾t 𝑌))) | |
23 | 21, 22 | syl 17 | . . 3 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝐹:𝑋–1-1-onto→∪ (𝐽 ↾t 𝑌) ↔ 𝐹:∪ 𝐾–1-1-onto→∪ (𝐽 ↾t 𝑌))) |
24 | 17 | restuni 21168 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ ℂ) → 𝑌 = ∪ (𝐽 ↾t 𝑌)) |
25 | 15, 5, 24 | sylancr 698 | . . . 4 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → 𝑌 = ∪ (𝐽 ↾t 𝑌)) |
26 | f1oeq3 6290 | . . . 4 ⊢ (𝑌 = ∪ (𝐽 ↾t 𝑌) → (𝐹:𝑋–1-1-onto→𝑌 ↔ 𝐹:𝑋–1-1-onto→∪ (𝐽 ↾t 𝑌))) | |
27 | 25, 26 | syl 17 | . . 3 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝐹:𝑋–1-1-onto→𝑌 ↔ 𝐹:𝑋–1-1-onto→∪ (𝐽 ↾t 𝑌))) |
28 | simpl 474 | . . . 4 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → 𝐾 ∈ Comp) | |
29 | 6 | cnfldhaus 22789 | . . . . 5 ⊢ 𝐽 ∈ Haus |
30 | cnex 10209 | . . . . . . 7 ⊢ ℂ ∈ V | |
31 | 30 | ssex 4954 | . . . . . 6 ⊢ (𝑌 ⊆ ℂ → 𝑌 ∈ V) |
32 | 5, 31 | syl 17 | . . . . 5 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → 𝑌 ∈ V) |
33 | resthaus 21374 | . . . . 5 ⊢ ((𝐽 ∈ Haus ∧ 𝑌 ∈ V) → (𝐽 ↾t 𝑌) ∈ Haus) | |
34 | 29, 32, 33 | sylancr 698 | . . . 4 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝐽 ↾t 𝑌) ∈ Haus) |
35 | eqid 2760 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
36 | eqid 2760 | . . . . 5 ⊢ ∪ (𝐽 ↾t 𝑌) = ∪ (𝐽 ↾t 𝑌) | |
37 | 35, 36 | cmphaushmeo 21805 | . . . 4 ⊢ ((𝐾 ∈ Comp ∧ (𝐽 ↾t 𝑌) ∈ Haus ∧ 𝐹 ∈ (𝐾 Cn (𝐽 ↾t 𝑌))) → (𝐹 ∈ (𝐾Homeo(𝐽 ↾t 𝑌)) ↔ 𝐹:∪ 𝐾–1-1-onto→∪ (𝐽 ↾t 𝑌))) |
38 | 28, 34, 11, 37 | syl3anc 1477 | . . 3 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝐹 ∈ (𝐾Homeo(𝐽 ↾t 𝑌)) ↔ 𝐹:∪ 𝐾–1-1-onto→∪ (𝐽 ↾t 𝑌))) |
39 | 23, 27, 38 | 3bitr4d 300 | . 2 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝐹:𝑋–1-1-onto→𝑌 ↔ 𝐹 ∈ (𝐾Homeo(𝐽 ↾t 𝑌)))) |
40 | 6, 8, 7 | cncfcn 22913 | . . . 4 ⊢ ((𝑌 ⊆ ℂ ∧ 𝑋 ⊆ ℂ) → (𝑌–cn→𝑋) = ((𝐽 ↾t 𝑌) Cn 𝐾)) |
41 | 5, 3, 40 | syl2anc 696 | . . 3 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝑌–cn→𝑋) = ((𝐽 ↾t 𝑌) Cn 𝐾)) |
42 | 41 | eleq2d 2825 | . 2 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (◡𝐹 ∈ (𝑌–cn→𝑋) ↔ ◡𝐹 ∈ ((𝐽 ↾t 𝑌) Cn 𝐾))) |
43 | 14, 39, 42 | 3bitr4d 300 | 1 ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑌–cn→𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ⊆ wss 3715 ∪ cuni 4588 ◡ccnv 5265 –1-1-onto→wf1o 6048 ‘cfv 6049 (class class class)co 6813 ℂcc 10126 ↾t crest 16283 TopOpenctopn 16284 ℂfldccnfld 19948 Topctop 20900 Cn ccn 21230 Hauscha 21314 Compccmp 21391 Homeochmeo 21758 –cn→ccncf 22880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fi 8482 df-sup 8513 df-inf 8514 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-q 11982 df-rp 12026 df-xneg 12139 df-xadd 12140 df-xmul 12141 df-icc 12375 df-fz 12520 df-seq 12996 df-exp 13055 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-plusg 16156 df-mulr 16157 df-starv 16158 df-tset 16162 df-ple 16163 df-ds 16166 df-unif 16167 df-rest 16285 df-topn 16286 df-topgen 16306 df-psmet 19940 df-xmet 19941 df-met 19942 df-bl 19943 df-mopn 19944 df-cnfld 19949 df-top 20901 df-topon 20918 df-topsp 20939 df-bases 20952 df-cld 21025 df-cls 21027 df-cn 21233 df-cnp 21234 df-haus 21321 df-cmp 21392 df-hmeo 21760 df-xms 22326 df-ms 22327 df-cncf 22882 |
This theorem is referenced by: dvcnvrelem2 23980 |
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