cnrehmeo - Metamath Proof Explorer

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Theorem cnrehmeo 24909

Description: The canonical bijection from (ℝ × ℝ) to ℂ described in cnref1o 12900 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if (ℝ × ℝ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.) Avoid ax-mulf 11108. (Revised by GG, 16-Mar-2025.)

**Hypotheses**
Ref	Expression
cnrehmeo.1	⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
cnrehmeo.2	⊢ 𝐽 = (topGen‘ran (,))
cnrehmeo.3	⊢ 𝐾 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
cnrehmeo	⊢ 𝐹 ∈ ((𝐽 ×_t 𝐽)Homeo𝐾)

Distinct variable group: 𝑥,𝑦,𝐾 Allowed substitution hints: 𝐹(𝑥,𝑦) 𝐽(𝑥,𝑦)

Proof of Theorem cnrehmeo Dummy variables 𝑧 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnrehmeo.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
2		cnrehmeo.2	. . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,))
3		retopon 24709	. . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
4	2, 3	eqeltri 2831	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℝ)
5	4	a1i 11	. . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℝ))
6		cnrehmeo.3	. . . . . . . 8 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
7	6	cnfldtop 24729	. . . . . . 7 ⊢ 𝐾 ∈ Top
8		cnrest2r 23233	. . . . . . 7 ⊢ (𝐾 ∈ Top → ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ)) ⊆ ((𝐽 ×_t 𝐽) Cn 𝐾))
9	7, 8	mp1i 13	. . . . . 6 ⊢ (⊤ → ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ)) ⊆ ((𝐽 ×_t 𝐽) Cn 𝐾))
10	5, 5	cnmpt1st 23614	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
11	6	tgioo2 24749	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐾 ↾_t ℝ)
12	2, 11	eqtri 2758	. . . . . . . 8 ⊢ 𝐽 = (𝐾 ↾_t ℝ)
13	12	oveq2i 7369	. . . . . . 7 ⊢ ((𝐽 ×_t 𝐽) Cn 𝐽) = ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ))
14	10, 13	eleqtrdi 2845	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ)))
15	9, 14	sseldd 3933	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
16	6	cnfldtopon 24728	. . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ)
17	16	a1i 11	. . . . . . 7 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ))
18		ax-icn 11087	. . . . . . . 8 ⊢ i ∈ ℂ
19	18	a1i 11	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
20	5, 5, 17, 19	cnmpt2c 23616	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ i) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
21	5, 5	cnmpt2nd 23615	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
22	21, 13	eleqtrdi 2845	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ)))
23	9, 22	sseldd 3933	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
24	6	mpomulcn 24816	. . . . . . 7 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
25	24	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
26		oveq12 7367	. . . . . 6 ⊢ ((𝑢 = i ∧ 𝑣 = 𝑦) → (𝑢 · 𝑣) = (i · 𝑦))
27	5, 5, 20, 23, 17, 17, 25, 26	cnmpt22 23620	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (i · 𝑦)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
28	6	addcn 24812	. . . . . 6 ⊢ + ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
29	28	a1i 11	. . . . 5 ⊢ (⊤ → + ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
30	5, 5, 15, 27, 29	cnmpt22f 23621	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
31	1, 30	eqeltrid 2839	. . 3 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
32	1	cnrecnv 15090	. . . 4 ⊢ ^◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
33		ref 15037	. . . . . . . 8 ⊢ ℜ:ℂ⟶ℝ
34	33	a1i 11	. . . . . . 7 ⊢ (⊤ → ℜ:ℂ⟶ℝ)
35	34	feqmptd 6901	. . . . . 6 ⊢ (⊤ → ℜ = (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)))
36		recncf 24853	. . . . . . 7 ⊢ ℜ ∈ (ℂ–cn→ℝ)
37		ssid 3955	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
38		ax-resscn 11085	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
39	16	toponrestid 22867	. . . . . . . . 9 ⊢ 𝐾 = (𝐾 ↾_t ℂ)
40	6, 39, 12	cncfcn 24861	. . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = (𝐾 Cn 𝐽))
41	37, 38, 40	mp2an 693	. . . . . . 7 ⊢ (ℂ–cn→ℝ) = (𝐾 Cn 𝐽)
42	36, 41	eleqtri 2833	. . . . . 6 ⊢ ℜ ∈ (𝐾 Cn 𝐽)
43	35, 42	eqeltrrdi 2844	. . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)) ∈ (𝐾 Cn 𝐽))
44		imf 15038	. . . . . . . 8 ⊢ ℑ:ℂ⟶ℝ
45	44	a1i 11	. . . . . . 7 ⊢ (⊤ → ℑ:ℂ⟶ℝ)
46	45	feqmptd 6901	. . . . . 6 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)))
47		imcncf 24854	. . . . . . 7 ⊢ ℑ ∈ (ℂ–cn→ℝ)
48	47, 41	eleqtri 2833	. . . . . 6 ⊢ ℑ ∈ (𝐾 Cn 𝐽)
49	46, 48	eqeltrrdi 2844	. . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)) ∈ (𝐾 Cn 𝐽))
50	17, 43, 49	cnmpt1t 23611	. . . 4 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) ∈ (𝐾 Cn (𝐽 ×_t 𝐽)))
51	32, 50	eqeltrid 2839	. . 3 ⊢ (⊤ → ^◡𝐹 ∈ (𝐾 Cn (𝐽 ×_t 𝐽)))
52		ishmeo 23705	. . 3 ⊢ (𝐹 ∈ ((𝐽 ×_t 𝐽)Homeo𝐾) ↔ (𝐹 ∈ ((𝐽 ×_t 𝐽) Cn 𝐾) ∧ ^◡𝐹 ∈ (𝐾 Cn (𝐽 ×_t 𝐽))))
53	31, 51, 52	sylanbrc 584	. 2 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×_t 𝐽)Homeo𝐾))
54	53	mptru 1549	1 ⊢ 𝐹 ∈ ((𝐽 ×_t 𝐽)Homeo𝐾)

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ⊆ wss 3900 ⟨cop 4585 ↦ cmpt 5178 ^◡ccnv 5622 ran crn 5624 ⟶wf 6487 ‘cfv 6491 (class class class)co 7358 ∈ cmpo 7360 ℂcc 11026 ℝcr 11027 ici 11030 + caddc 11031 · cmul 11033 (,)cioo 13263 ℜcre 15022 ℑcim 15023 ↾_t crest 17342 TopOpenctopn 17343 topGenctg 17359 ℂ_fldccnfld 21311 Topctop 22839 TopOnctopon 22856 Cn ccn 23170 ×_t ctx 23506 Homeochmeo 23699 –cn→ccncf 24827

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-iin 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8767 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ioo 13267 df-icc 13270 df-fz 13426 df-fzo 13573 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19248 df-cmn 19713 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-cnfld 21312 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cn 23173 df-cnp 23174 df-tx 23508 df-hmeo 23701 df-xms 24266 df-ms 24267 df-tms 24268 df-cncf 24829

This theorem is referenced by: cnheiborlem 24911 mbfimaopnlem 25614 tpr2rico 34048

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