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Mirrors > Home > MPE Home > Th. List > resscdrg | Structured version Visualization version GIF version |
Description: The real numbers are a subset of any complete subfield in the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
resscdrg.1 | ⊢ 𝐹 = (ℂfld ↾s 𝐾) |
Ref | Expression |
---|---|
resscdrg | ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → ℝ ⊆ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2752 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | 1 | cnfldtop 22780 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ Top |
3 | ax-resscn 10177 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
4 | qssre 11983 | . . . . 5 ⊢ ℚ ⊆ ℝ | |
5 | 1 | cnfldtopon 22779 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
6 | 5 | toponunii 20915 | . . . . . 6 ⊢ ℂ = ∪ (TopOpen‘ℂfld) |
7 | 1 | tgioo2 22799 | . . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
8 | 6, 7 | restcls 21179 | . . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ ℝ ⊆ ℂ ∧ ℚ ⊆ ℝ) → ((cls‘(topGen‘ran (,)))‘ℚ) = (((cls‘(TopOpen‘ℂfld))‘ℚ) ∩ ℝ)) |
9 | 2, 3, 4, 8 | mp3an 1565 | . . . 4 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = (((cls‘(TopOpen‘ℂfld))‘ℚ) ∩ ℝ) |
10 | qdensere 22766 | . . . 4 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ | |
11 | 9, 10 | eqtr3i 2776 | . . 3 ⊢ (((cls‘(TopOpen‘ℂfld))‘ℚ) ∩ ℝ) = ℝ |
12 | sseqin2 3952 | . . 3 ⊢ (ℝ ⊆ ((cls‘(TopOpen‘ℂfld))‘ℚ) ↔ (((cls‘(TopOpen‘ℂfld))‘ℚ) ∩ ℝ) = ℝ) | |
13 | 11, 12 | mpbir 221 | . 2 ⊢ ℝ ⊆ ((cls‘(TopOpen‘ℂfld))‘ℚ) |
14 | simp3 1132 | . . . 4 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → 𝐹 ∈ CMetSp) | |
15 | cncms 23343 | . . . . 5 ⊢ ℂfld ∈ CMetSp | |
16 | cnfldbas 19944 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
17 | 16 | subrgss 18975 | . . . . . 6 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 ⊆ ℂ) |
18 | 17 | 3ad2ant1 1127 | . . . . 5 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → 𝐾 ⊆ ℂ) |
19 | resscdrg.1 | . . . . . 6 ⊢ 𝐹 = (ℂfld ↾s 𝐾) | |
20 | 19, 16, 1 | cmsss 23339 | . . . . 5 ⊢ ((ℂfld ∈ CMetSp ∧ 𝐾 ⊆ ℂ) → (𝐹 ∈ CMetSp ↔ 𝐾 ∈ (Clsd‘(TopOpen‘ℂfld)))) |
21 | 15, 18, 20 | sylancr 698 | . . . 4 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → (𝐹 ∈ CMetSp ↔ 𝐾 ∈ (Clsd‘(TopOpen‘ℂfld)))) |
22 | 14, 21 | mpbid 222 | . . 3 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → 𝐾 ∈ (Clsd‘(TopOpen‘ℂfld))) |
23 | 19 | eleq1i 2822 | . . . . 5 ⊢ (𝐹 ∈ DivRing ↔ (ℂfld ↾s 𝐾) ∈ DivRing) |
24 | qsssubdrg 19999 | . . . . 5 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐾) ∈ DivRing) → ℚ ⊆ 𝐾) | |
25 | 23, 24 | sylan2b 493 | . . . 4 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing) → ℚ ⊆ 𝐾) |
26 | 25 | 3adant3 1126 | . . 3 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → ℚ ⊆ 𝐾) |
27 | 6 | clsss2 21070 | . . 3 ⊢ ((𝐾 ∈ (Clsd‘(TopOpen‘ℂfld)) ∧ ℚ ⊆ 𝐾) → ((cls‘(TopOpen‘ℂfld))‘ℚ) ⊆ 𝐾) |
28 | 22, 26, 27 | syl2anc 696 | . 2 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → ((cls‘(TopOpen‘ℂfld))‘ℚ) ⊆ 𝐾) |
29 | 13, 28 | syl5ss 3747 | 1 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → ℝ ⊆ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1072 = wceq 1624 ∈ wcel 2131 ∩ cin 3706 ⊆ wss 3707 ran crn 5259 ‘cfv 6041 (class class class)co 6805 ℂcc 10118 ℝcr 10119 ℚcq 11973 (,)cioo 12360 ↾s cress 16052 TopOpenctopn 16276 topGenctg 16292 DivRingcdr 18941 SubRingcsubrg 18970 ℂfldccnfld 19940 Topctop 20892 Clsdccld 21014 clsccl 21016 CMetSpccms 23321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-inf2 8703 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 ax-pre-sup 10198 ax-addf 10199 ax-mulf 10200 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-iin 4667 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-se 5218 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-isom 6050 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-of 7054 df-om 7223 df-1st 7325 df-2nd 7326 df-supp 7456 df-tpos 7513 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-2o 7722 df-oadd 7725 df-er 7903 df-map 8017 df-ixp 8067 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fsupp 8433 df-fi 8474 df-sup 8505 df-inf 8506 df-oi 8572 df-card 8947 df-cda 9174 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-div 10869 df-nn 11205 df-2 11263 df-3 11264 df-4 11265 df-5 11266 df-6 11267 df-7 11268 df-8 11269 df-9 11270 df-n0 11477 df-z 11562 df-dec 11678 df-uz 11872 df-q 11974 df-rp 12018 df-xneg 12131 df-xadd 12132 df-xmul 12133 df-ioo 12364 df-ico 12366 df-icc 12367 df-fz 12512 df-fzo 12652 df-seq 12988 df-exp 13047 df-hash 13304 df-cj 14030 df-re 14031 df-im 14032 df-sqrt 14166 df-abs 14167 df-struct 16053 df-ndx 16054 df-slot 16055 df-base 16057 df-sets 16058 df-ress 16059 df-plusg 16148 df-mulr 16149 df-starv 16150 df-sca 16151 df-vsca 16152 df-ip 16153 df-tset 16154 df-ple 16155 df-ds 16158 df-unif 16159 df-hom 16160 df-cco 16161 df-rest 16277 df-topn 16278 df-0g 16296 df-gsum 16297 df-topgen 16298 df-pt 16299 df-prds 16302 df-xrs 16356 df-qtop 16361 df-imas 16362 df-xps 16364 df-mre 16440 df-mrc 16441 df-acs 16443 df-mgm 17435 df-sgrp 17477 df-mnd 17488 df-submnd 17529 df-grp 17618 df-minusg 17619 df-mulg 17734 df-subg 17784 df-cntz 17942 df-cmn 18387 df-mgp 18682 df-ur 18694 df-ring 18741 df-cring 18742 df-oppr 18815 df-dvdsr 18833 df-unit 18834 df-invr 18864 df-dvr 18875 df-drng 18943 df-subrg 18972 df-psmet 19932 df-xmet 19933 df-met 19934 df-bl 19935 df-mopn 19936 df-fbas 19937 df-fg 19938 df-cnfld 19941 df-top 20893 df-topon 20910 df-topsp 20931 df-bases 20944 df-cld 21017 df-ntr 21018 df-cls 21019 df-nei 21096 df-cn 21225 df-cnp 21226 df-haus 21313 df-cmp 21384 df-tx 21559 df-hmeo 21752 df-fil 21843 df-flim 21936 df-fcls 21938 df-xms 22318 df-ms 22319 df-tms 22320 df-cncf 22874 df-cfil 23245 df-cmet 23247 df-cms 23324 |
This theorem is referenced by: cncdrg 23347 hlress 23356 |
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