cnrehmeo - Metamath Proof Explorer

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Theorem cnrehmeo 24878

Description: The canonical bijection from (ℝ × ℝ) to ℂ described in cnref1o 12883 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 11086. (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 24678	. . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
4	2, 3	eqeltri 2827	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℝ)
5	4	a1i 11	. . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℝ))
6		cnrehmeo.3	. . . . . . . 8 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
7	6	cnfldtop 24698	. . . . . . 7 ⊢ 𝐾 ∈ Top
8		cnrest2r 23202	. . . . . . 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 23583	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
11	6	tgioo2 24718	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐾 ↾_t ℝ)
12	2, 11	eqtri 2754	. . . . . . . 8 ⊢ 𝐽 = (𝐾 ↾_t ℝ)
13	12	oveq2i 7357	. . . . . . 7 ⊢ ((𝐽 ×_t 𝐽) Cn 𝐽) = ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ))
14	10, 13	eleqtrdi 2841	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ)))
15	9, 14	sseldd 3930	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
16	6	cnfldtopon 24697	. . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ)
17	16	a1i 11	. . . . . . 7 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ))
18		ax-icn 11065	. . . . . . . 8 ⊢ i ∈ ℂ
19	18	a1i 11	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
20	5, 5, 17, 19	cnmpt2c 23585	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ i) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
21	5, 5	cnmpt2nd 23584	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
22	21, 13	eleqtrdi 2841	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ)))
23	9, 22	sseldd 3930	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
24	6	mpomulcn 24785	. . . . . . 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 7355	. . . . . 6 ⊢ ((𝑢 = i ∧ 𝑣 = 𝑦) → (𝑢 · 𝑣) = (i · 𝑦))
27	5, 5, 20, 23, 17, 17, 25, 26	cnmpt22 23589	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (i · 𝑦)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
28	6	addcn 24781	. . . . . 6 ⊢ + ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
29	28	a1i 11	. . . . 5 ⊢ (⊤ → + ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
30	5, 5, 15, 27, 29	cnmpt22f 23590	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
31	1, 30	eqeltrid 2835	. . 3 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
32	1	cnrecnv 15072	. . . 4 ⊢ ^◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
33		ref 15019	. . . . . . . 8 ⊢ ℜ:ℂ⟶ℝ
34	33	a1i 11	. . . . . . 7 ⊢ (⊤ → ℜ:ℂ⟶ℝ)
35	34	feqmptd 6890	. . . . . 6 ⊢ (⊤ → ℜ = (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)))
36		recncf 24822	. . . . . . 7 ⊢ ℜ ∈ (ℂ–cn→ℝ)
37		ssid 3952	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
38		ax-resscn 11063	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
39	16	toponrestid 22836	. . . . . . . . 9 ⊢ 𝐾 = (𝐾 ↾_t ℂ)
40	6, 39, 12	cncfcn 24830	. . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = (𝐾 Cn 𝐽))
41	37, 38, 40	mp2an 692	. . . . . . 7 ⊢ (ℂ–cn→ℝ) = (𝐾 Cn 𝐽)
42	36, 41	eleqtri 2829	. . . . . 6 ⊢ ℜ ∈ (𝐾 Cn 𝐽)
43	35, 42	eqeltrrdi 2840	. . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)) ∈ (𝐾 Cn 𝐽))
44		imf 15020	. . . . . . . 8 ⊢ ℑ:ℂ⟶ℝ
45	44	a1i 11	. . . . . . 7 ⊢ (⊤ → ℑ:ℂ⟶ℝ)
46	45	feqmptd 6890	. . . . . 6 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)))
47		imcncf 24823	. . . . . . 7 ⊢ ℑ ∈ (ℂ–cn→ℝ)
48	47, 41	eleqtri 2829	. . . . . 6 ⊢ ℑ ∈ (𝐾 Cn 𝐽)
49	46, 48	eqeltrrdi 2840	. . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)) ∈ (𝐾 Cn 𝐽))
50	17, 43, 49	cnmpt1t 23580	. . . 4 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) ∈ (𝐾 Cn (𝐽 ×_t 𝐽)))
51	32, 50	eqeltrid 2835	. . 3 ⊢ (⊤ → ^◡𝐹 ∈ (𝐾 Cn (𝐽 ×_t 𝐽)))
52		ishmeo 23674	. . 3 ⊢ (𝐹 ∈ ((𝐽 ×_t 𝐽)Homeo𝐾) ↔ (𝐹 ∈ ((𝐽 ×_t 𝐽) Cn 𝐾) ∧ ^◡𝐹 ∈ (𝐾 Cn (𝐽 ×_t 𝐽))))
53	31, 51, 52	sylanbrc 583	. 2 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×_t 𝐽)Homeo𝐾))
54	53	mptru 1548	1 ⊢ 𝐹 ∈ ((𝐽 ×_t 𝐽)Homeo𝐾)

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ⊤wtru 1542 ∈ wcel 2111 ⊆ wss 3897 ⟨cop 4579 ↦ cmpt 5170 ^◡ccnv 5613 ran crn 5615 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 ℂcc 11004 ℝcr 11005 ici 11008 + caddc 11009 · cmul 11011 (,)cioo 13245 ℜcre 15004 ℑcim 15005 ↾_t crest 17324 TopOpenctopn 17325 topGenctg 17341 ℂ_fldccnfld 21291 Topctop 22808 TopOnctopon 22825 Cn ccn 23139 ×_t ctx 23475 Homeochmeo 23668 –cn→ccncf 24796

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-icc 13252 df-fz 13408 df-fzo 13555 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cn 23142 df-cnp 23143 df-tx 23477 df-hmeo 23670 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798

This theorem is referenced by: cnheiborlem 24880 mbfimaopnlem 25583 tpr2rico 33925

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