| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > cnrehmeoOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of cnrehmeo 24984 as of 9-Apr-2025. (Contributed by Mario Carneiro, 25-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| cnrehmeoOLD.1 | ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) |
| cnrehmeoOLD.2 | ⊢ 𝐽 = (topGen‘ran (,)) |
| cnrehmeoOLD.3 | ⊢ 𝐾 = (TopOpen‘ℂfld) |
| Ref | Expression |
|---|---|
| cnrehmeoOLD | ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnrehmeoOLD.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) | |
| 2 | cnrehmeoOLD.2 | . . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,)) | |
| 3 | retopon 24784 | . . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
| 4 | 2, 3 | eqeltri 2837 | . . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℝ) |
| 5 | 4 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℝ)) |
| 6 | cnrehmeoOLD.3 | . . . . . . . 8 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
| 7 | 6 | cnfldtop 24804 | . . . . . . 7 ⊢ 𝐾 ∈ Top |
| 8 | cnrest2r 23295 | . . . . . . 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 23676 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
| 11 | 6 | tgioo2 24824 | . . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐾 ↾t ℝ) |
| 12 | 2, 11 | eqtri 2765 | . . . . . . . 8 ⊢ 𝐽 = (𝐾 ↾t ℝ) |
| 13 | 12 | oveq2i 7442 | . . . . . . 7 ⊢ ((𝐽 ×t 𝐽) Cn 𝐽) = ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ)) |
| 14 | 10, 13 | eleqtrdi 2851 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ))) |
| 15 | 9, 14 | sseldd 3984 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
| 16 | 6 | cnfldtopon 24803 | . . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
| 17 | 16 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ)) |
| 18 | ax-icn 11214 | . . . . . . . 8 ⊢ i ∈ ℂ | |
| 19 | 18 | a1i 11 | . . . . . . 7 ⊢ (⊤ → i ∈ ℂ) |
| 20 | 5, 5, 17, 19 | cnmpt2c 23678 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ i) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
| 21 | 5, 5 | cnmpt2nd 23677 | . . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
| 22 | 21, 13 | eleqtrdi 2851 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ))) |
| 23 | 9, 22 | sseldd 3984 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
| 24 | 6 | mulcn 24889 | . . . . . . 7 ⊢ · ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
| 25 | 24 | a1i 11 | . . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
| 26 | 5, 5, 20, 23, 25 | cnmpt22f 23683 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (i · 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
| 27 | 6 | addcn 24887 | . . . . . 6 ⊢ + ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
| 28 | 27 | a1i 11 | . . . . 5 ⊢ (⊤ → + ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
| 29 | 5, 5, 15, 26, 28 | cnmpt22f 23683 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
| 30 | 1, 29 | eqeltrid 2845 | . . 3 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
| 31 | 1 | cnrecnv 15204 | . . . 4 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
| 32 | ref 15151 | . . . . . . . 8 ⊢ ℜ:ℂ⟶ℝ | |
| 33 | 32 | a1i 11 | . . . . . . 7 ⊢ (⊤ → ℜ:ℂ⟶ℝ) |
| 34 | 33 | feqmptd 6977 | . . . . . 6 ⊢ (⊤ → ℜ = (𝑧 ∈ ℂ ↦ (ℜ‘𝑧))) |
| 35 | recncf 24928 | . . . . . . 7 ⊢ ℜ ∈ (ℂ–cn→ℝ) | |
| 36 | ssid 4006 | . . . . . . . 8 ⊢ ℂ ⊆ ℂ | |
| 37 | ax-resscn 11212 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
| 38 | 16 | toponrestid 22927 | . . . . . . . . 9 ⊢ 𝐾 = (𝐾 ↾t ℂ) |
| 39 | 6, 38, 12 | cncfcn 24936 | . . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = (𝐾 Cn 𝐽)) |
| 40 | 36, 37, 39 | mp2an 692 | . . . . . . 7 ⊢ (ℂ–cn→ℝ) = (𝐾 Cn 𝐽) |
| 41 | 35, 40 | eleqtri 2839 | . . . . . 6 ⊢ ℜ ∈ (𝐾 Cn 𝐽) |
| 42 | 34, 41 | eqeltrrdi 2850 | . . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)) ∈ (𝐾 Cn 𝐽)) |
| 43 | imf 15152 | . . . . . . . 8 ⊢ ℑ:ℂ⟶ℝ | |
| 44 | 43 | a1i 11 | . . . . . . 7 ⊢ (⊤ → ℑ:ℂ⟶ℝ) |
| 45 | 44 | feqmptd 6977 | . . . . . 6 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℑ‘𝑧))) |
| 46 | imcncf 24929 | . . . . . . 7 ⊢ ℑ ∈ (ℂ–cn→ℝ) | |
| 47 | 46, 40 | eleqtri 2839 | . . . . . 6 ⊢ ℑ ∈ (𝐾 Cn 𝐽) |
| 48 | 45, 47 | eqeltrrdi 2850 | . . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)) ∈ (𝐾 Cn 𝐽)) |
| 49 | 17, 42, 48 | cnmpt1t 23673 | . . . 4 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) ∈ (𝐾 Cn (𝐽 ×t 𝐽))) |
| 50 | 31, 49 | eqeltrid 2845 | . . 3 ⊢ (⊤ → ◡𝐹 ∈ (𝐾 Cn (𝐽 ×t 𝐽))) |
| 51 | ishmeo 23767 | . . 3 ⊢ (𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) ↔ (𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn (𝐽 ×t 𝐽)))) | |
| 52 | 30, 50, 51 | sylanbrc 583 | . 2 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾)) |
| 53 | 52 | mptru 1547 | 1 ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ⊆ wss 3951 〈cop 4632 ↦ cmpt 5225 ◡ccnv 5684 ran crn 5686 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ℂcc 11153 ℝcr 11154 ici 11157 + caddc 11158 · cmul 11160 (,)cioo 13387 ℜcre 15136 ℑcim 15137 ↾t crest 17465 TopOpenctopn 17466 topGenctg 17482 ℂfldccnfld 21364 Topctop 22899 TopOnctopon 22916 Cn ccn 23232 ×t ctx 23568 Homeochmeo 23761 –cn→ccncf 24902 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cn 23235 df-cnp 23236 df-tx 23570 df-hmeo 23763 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |