Metamath Proof Explorer


Theorem cnrehmeo

Description: The canonical bijection from ( RR X. RR ) to CC described in cnref1o is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if ( RR X. RR ) is metrized with the l2 norm.) (Contributed by Mario Carneiro, 25-Aug-2014) Avoid ax-mulf . (Revised by GG, 16-Mar-2025)

Ref Expression
Hypotheses cnrehmeo.1 F = x , y x + i y
cnrehmeo.2 J = topGen ran .
cnrehmeo.3 K = TopOpen fld
Assertion cnrehmeo F J × t J Homeo K

Proof

Step Hyp Ref Expression
1 cnrehmeo.1 F = x , y x + i y
2 cnrehmeo.2 J = topGen ran .
3 cnrehmeo.3 K = TopOpen fld
4 retopon topGen ran . TopOn
5 2 4 eqeltri J TopOn
6 5 a1i J TopOn
7 3 cnfldtop K Top
8 cnrest2r K Top J × t J Cn K 𝑡 J × t J Cn K
9 7 8 mp1i J × t J Cn K 𝑡 J × t J Cn K
10 6 6 cnmpt1st x , y x J × t J Cn J
11 3 tgioo2 topGen ran . = K 𝑡
12 2 11 eqtri J = K 𝑡
13 12 oveq2i J × t J Cn J = J × t J Cn K 𝑡
14 10 13 eleqtrdi x , y x J × t J Cn K 𝑡
15 9 14 sseldd x , y x J × t J Cn K
16 3 cnfldtopon K TopOn
17 16 a1i K TopOn
18 ax-icn i
19 18 a1i i
20 6 6 17 19 cnmpt2c x , y i J × t J Cn K
21 6 6 cnmpt2nd x , y y J × t J Cn J
22 21 13 eleqtrdi x , y y J × t J Cn K 𝑡
23 9 22 sseldd x , y y J × t J Cn K
24 3 mpomulcn u , v u v K × t K Cn K
25 24 a1i u , v u v K × t K Cn K
26 oveq12 u = i v = y u v = i y
27 6 6 20 23 17 17 25 26 cnmpt22 x , y i y J × t J Cn K
28 3 addcn + K × t K Cn K
29 28 a1i + K × t K Cn K
30 6 6 15 27 29 cnmpt22f x , y x + i y J × t J Cn K
31 1 30 eqeltrid F J × t J Cn K
32 1 cnrecnv F -1 = z z z
33 ref :
34 33 a1i :
35 34 feqmptd = z z
36 recncf : cn
37 ssid
38 ax-resscn
39 16 toponrestid K = K 𝑡
40 3 39 12 cncfcn cn = K Cn J
41 37 38 40 mp2an cn = K Cn J
42 36 41 eleqtri K Cn J
43 35 42 eqeltrrdi z z K Cn J
44 imf :
45 44 a1i :
46 45 feqmptd = z z
47 imcncf : cn
48 47 41 eleqtri K Cn J
49 46 48 eqeltrrdi z z K Cn J
50 17 43 49 cnmpt1t z z z K Cn J × t J
51 32 50 eqeltrid F -1 K Cn J × t J
52 ishmeo F J × t J Homeo K F J × t J Cn K F -1 K Cn J × t J
53 31 51 52 sylanbrc F J × t J Homeo K
54 53 mptru F J × t J Homeo K