Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 20474 | . 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 2818 | . . . 4 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
3 | 2 | srngstr 16615 | . . 3 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) Struct 〈1, 4〉 |
4 | 9nn 11723 | . . . . 5 ⊢ 9 ∈ ℕ | |
5 | tsetndx 16647 | . . . . 5 ⊢ (TopSet‘ndx) = 9 | |
6 | 9lt10 12217 | . . . . 5 ⊢ 9 < ;10 | |
7 | 10nn 12102 | . . . . 5 ⊢ ;10 ∈ ℕ | |
8 | plendx 16654 | . . . . 5 ⊢ (le‘ndx) = ;10 | |
9 | 1nn0 11901 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
10 | 0nn0 11900 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
11 | 2nn 11698 | . . . . . 6 ⊢ 2 ∈ ℕ | |
12 | 2pos 11728 | . . . . . 6 ⊢ 0 < 2 | |
13 | 9, 10, 11, 12 | declt 12114 | . . . . 5 ⊢ ;10 < ;12 |
14 | 9, 11 | decnncl 12106 | . . . . 5 ⊢ ;12 ∈ ℕ |
15 | dsndx 16663 | . . . . 5 ⊢ (dist‘ndx) = ;12 | |
16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 16582 | . . . 4 ⊢ {〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} Struct 〈9, ;12〉 |
17 | 3nn 11704 | . . . . . 6 ⊢ 3 ∈ ℕ | |
18 | 9, 17 | decnncl 12106 | . . . . 5 ⊢ ;13 ∈ ℕ |
19 | unifndx 16665 | . . . . 5 ⊢ (UnifSet‘ndx) = ;13 | |
20 | 18, 19 | strle1 16580 | . . . 4 ⊢ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉} Struct 〈;13, ;13〉 |
21 | 2nn0 11902 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
22 | 2lt3 11797 | . . . . 5 ⊢ 2 < 3 | |
23 | 9, 21, 17, 22 | declt 12114 | . . . 4 ⊢ ;12 < ;13 |
24 | 16, 20, 23 | strleun 16579 | . . 3 ⊢ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}) Struct 〈9, ;13〉 |
25 | 4lt9 11828 | . . 3 ⊢ 4 < 9 | |
26 | 3, 24, 25 | strleun 16579 | . 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 5078 | 1 ⊢ ℂfld Struct 〈1, ;13〉 |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3931 {csn 4557 {ctp 4561 〈cop 4563 class class class wbr 5057 ∘ ccom 5552 ‘cfv 6348 ℂcc 10523 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 ≤ cle 10664 − cmin 10858 2c2 11680 3c3 11681 4c4 11682 9c9 11687 ;cdc 12086 ∗ccj 14443 abscabs 14581 Struct cstr 16467 ndxcnx 16468 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 *𝑟cstv 16555 TopSetcts 16559 lecple 16560 distcds 16562 UnifSetcunif 16563 MetOpencmopn 20463 metUnifcmetu 20464 ℂfldccnfld 20473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-cnfld 20474 |
This theorem is referenced by: cnfldex 20476 cnfldbas 20477 cnfldadd 20478 cnfldmul 20479 cnfldcj 20480 cnfldtset 20481 cnfldle 20482 cnfldds 20483 cnfldunif 20484 cnfldfunALT 20486 cffldtocusgr 27156 |
Copyright terms: Public domain | W3C validator |