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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 2778 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
| 2 | 1 | cnfldtop 23006 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ Top |
| 3 | ax-resscn 10331 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
| 4 | qssre 12111 | . . . . 5 ⊢ ℚ ⊆ ℝ | |
| 5 | 1 | cnfldtopon 23005 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
| 6 | 5 | toponunii 21139 | . . . . . 6 ⊢ ℂ = ∪ (TopOpen‘ℂfld) |
| 7 | 1 | tgioo2 23025 | . . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
| 8 | 6, 7 | restcls 21404 | . . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ ℝ ⊆ ℂ ∧ ℚ ⊆ ℝ) → ((cls‘(topGen‘ran (,)))‘ℚ) = (((cls‘(TopOpen‘ℂfld))‘ℚ) ∩ ℝ)) |
| 9 | 2, 3, 4, 8 | mp3an 1534 | . . . 4 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = (((cls‘(TopOpen‘ℂfld))‘ℚ) ∩ ℝ) |
| 10 | qdensere 22992 | . . . 4 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ | |
| 11 | 9, 10 | eqtr3i 2804 | . . 3 ⊢ (((cls‘(TopOpen‘ℂfld))‘ℚ) ∩ ℝ) = ℝ |
| 12 | sseqin2 4040 | . . 3 ⊢ (ℝ ⊆ ((cls‘(TopOpen‘ℂfld))‘ℚ) ↔ (((cls‘(TopOpen‘ℂfld))‘ℚ) ∩ ℝ) = ℝ) | |
| 13 | 11, 12 | mpbir 223 | . 2 ⊢ ℝ ⊆ ((cls‘(TopOpen‘ℂfld))‘ℚ) |
| 14 | simp3 1129 | . . . 4 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → 𝐹 ∈ CMetSp) | |
| 15 | cncms 23572 | . . . . 5 ⊢ ℂfld ∈ CMetSp | |
| 16 | cnfldbas 20157 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
| 17 | 16 | subrgss 19184 | . . . . . 6 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 ⊆ ℂ) |
| 18 | 17 | 3ad2ant1 1124 | . . . . 5 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → 𝐾 ⊆ ℂ) |
| 19 | resscdrg.1 | . . . . . 6 ⊢ 𝐹 = (ℂfld ↾s 𝐾) | |
| 20 | 19, 16, 1 | cmsss 23568 | . . . . 5 ⊢ ((ℂfld ∈ CMetSp ∧ 𝐾 ⊆ ℂ) → (𝐹 ∈ CMetSp ↔ 𝐾 ∈ (Clsd‘(TopOpen‘ℂfld)))) |
| 21 | 15, 18, 20 | sylancr 581 | . . . 4 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → (𝐹 ∈ CMetSp ↔ 𝐾 ∈ (Clsd‘(TopOpen‘ℂfld)))) |
| 22 | 14, 21 | mpbid 224 | . . 3 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → 𝐾 ∈ (Clsd‘(TopOpen‘ℂfld))) |
| 23 | 19 | eleq1i 2850 | . . . . 5 ⊢ (𝐹 ∈ DivRing ↔ (ℂfld ↾s 𝐾) ∈ DivRing) |
| 24 | qsssubdrg 20212 | . . . . 5 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐾) ∈ DivRing) → ℚ ⊆ 𝐾) | |
| 25 | 23, 24 | sylan2b 587 | . . . 4 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing) → ℚ ⊆ 𝐾) |
| 26 | 25 | 3adant3 1123 | . . 3 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → ℚ ⊆ 𝐾) |
| 27 | 6 | clsss2 21295 | . . 3 ⊢ ((𝐾 ∈ (Clsd‘(TopOpen‘ℂfld)) ∧ ℚ ⊆ 𝐾) → ((cls‘(TopOpen‘ℂfld))‘ℚ) ⊆ 𝐾) |
| 28 | 22, 26, 27 | syl2anc 579 | . 2 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → ((cls‘(TopOpen‘ℂfld))‘ℚ) ⊆ 𝐾) |
| 29 | 13, 28 | syl5ss 3832 | 1 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → ℝ ⊆ 𝐾) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∩ cin 3791 ⊆ wss 3792 ran crn 5358 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 ℝcr 10273 ℚcq 12100 (,)cioo 12492 ↾s cress 16267 TopOpenctopn 16479 topGenctg 16495 DivRingcdr 19150 SubRingcsubrg 19179 ℂfldccnfld 20153 Topctop 21116 Clsdccld 21239 clsccl 21241 CMetSpccms 23549 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 |
| This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-q 12101 df-rp 12143 df-xneg 12262 df-xadd 12263 df-xmul 12264 df-ioo 12496 df-ico 12498 df-icc 12499 df-fz 12649 df-fzo 12790 df-seq 13125 df-exp 13184 df-hash 13442 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-hom 16373 df-cco 16374 df-rest 16480 df-topn 16481 df-0g 16499 df-gsum 16500 df-topgen 16501 df-pt 16502 df-prds 16505 df-xrs 16559 df-qtop 16564 df-imas 16565 df-xps 16567 df-mre 16643 df-mrc 16644 df-acs 16646 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-submnd 17733 df-grp 17823 df-minusg 17824 df-mulg 17939 df-subg 17986 df-cntz 18144 df-cmn 18592 df-mgp 18888 df-ur 18900 df-ring 18947 df-cring 18948 df-oppr 19021 df-dvdsr 19039 df-unit 19040 df-invr 19070 df-dvr 19081 df-drng 19152 df-subrg 19181 df-psmet 20145 df-xmet 20146 df-met 20147 df-bl 20148 df-mopn 20149 df-fbas 20150 df-fg 20151 df-cnfld 20154 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-cld 21242 df-ntr 21243 df-cls 21244 df-nei 21321 df-cn 21450 df-cnp 21451 df-haus 21538 df-cmp 21610 df-tx 21785 df-hmeo 21978 df-fil 22069 df-flim 22162 df-fcls 22164 df-xms 22544 df-ms 22545 df-tms 22546 df-cncf 23100 df-cfil 23472 df-cmet 23474 df-cms 23552 |
| This theorem is referenced by: cncdrg 23576 hlress 23585 |
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