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Mirrors > Home > MPE Home > Th. List > cnfldstr | Structured version Visualization version GIF version |
Description: The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) |
Ref | Expression |
---|---|
cnfldstr | ⊢ ℂfld Struct 〈1, ;13〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnfld 20107 | . 2 ⊢ ℂfld = (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) | |
2 | eqid 2825 | . . . 4 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
3 | 2 | srngstr 16367 | . . 3 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) Struct 〈1, 4〉 |
4 | 9nn 11455 | . . . . 5 ⊢ 9 ∈ ℕ | |
5 | tsetndx 16399 | . . . . 5 ⊢ (TopSet‘ndx) = 9 | |
6 | 9lt10 11954 | . . . . 5 ⊢ 9 < ;10 | |
7 | 10nn 11837 | . . . . 5 ⊢ ;10 ∈ ℕ | |
8 | plendx 16406 | . . . . 5 ⊢ (le‘ndx) = ;10 | |
9 | 1nn0 11636 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
10 | 0nn0 11635 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
11 | 2nn 11424 | . . . . . 6 ⊢ 2 ∈ ℕ | |
12 | 2pos 11461 | . . . . . 6 ⊢ 0 < 2 | |
13 | 9, 10, 11, 12 | declt 11850 | . . . . 5 ⊢ ;10 < ;12 |
14 | 9, 11 | decnncl 11842 | . . . . 5 ⊢ ;12 ∈ ℕ |
15 | dsndx 16415 | . . . . 5 ⊢ (dist‘ndx) = ;12 | |
16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 16334 | . . . 4 ⊢ {〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} Struct 〈9, ;12〉 |
17 | 3nn 11430 | . . . . . 6 ⊢ 3 ∈ ℕ | |
18 | 9, 17 | decnncl 11842 | . . . . 5 ⊢ ;13 ∈ ℕ |
19 | unifndx 16417 | . . . . 5 ⊢ (UnifSet‘ndx) = ;13 | |
20 | 18, 19 | strle1 16332 | . . . 4 ⊢ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉} Struct 〈;13, ;13〉 |
21 | 2nn0 11637 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
22 | 2lt3 11530 | . . . . 5 ⊢ 2 < 3 | |
23 | 9, 21, 17, 22 | declt 11850 | . . . 4 ⊢ ;12 < ;13 |
24 | 16, 20, 23 | strleun 16331 | . . 3 ⊢ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}) Struct 〈9, ;13〉 |
25 | 4lt9 11561 | . . 3 ⊢ 4 < 9 | |
26 | 3, 24, 25 | strleun 16331 | . 2 ⊢ (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) Struct 〈1, ;13〉 |
27 | 1, 26 | eqbrtri 4894 | 1 ⊢ ℂfld Struct 〈1, ;13〉 |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3796 {csn 4397 {ctp 4401 〈cop 4403 class class class wbr 4873 ∘ ccom 5346 ‘cfv 6123 ℂcc 10250 0cc0 10252 1c1 10253 + caddc 10255 · cmul 10257 ≤ cle 10392 − cmin 10585 2c2 11406 3c3 11407 4c4 11408 9c9 11413 ;cdc 11821 ∗ccj 14213 abscabs 14351 Struct cstr 16218 ndxcnx 16219 Basecbs 16222 +gcplusg 16305 .rcmulr 16306 *𝑟cstv 16307 TopSetcts 16311 lecple 16312 distcds 16314 UnifSetcunif 16315 MetOpencmopn 20096 metUnifcmetu 20097 ℂfldccnfld 20106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-plusg 16318 df-mulr 16319 df-starv 16320 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-cnfld 20107 |
This theorem is referenced by: cnfldex 20109 cnfldbas 20110 cnfldadd 20111 cnfldmul 20112 cnfldcj 20113 cnfldtset 20114 cnfldle 20115 cnfldds 20116 cnfldunif 20117 cnfldfunALT 20119 cffldtocusgr 26745 |
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