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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 19568 | . 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 2610 | . . . 4 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
3 | 2 | srngfn 15831 | . . 3 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) Struct 〈1, 4〉 |
4 | 9nn 11069 | . . . . 5 ⊢ 9 ∈ ℕ | |
5 | tsetndx 15863 | . . . . 5 ⊢ (TopSet‘ndx) = 9 | |
6 | 9lt10 11549 | . . . . 5 ⊢ 9 < ;10 | |
7 | 10nn 11390 | . . . . 5 ⊢ ;10 ∈ ℕ | |
8 | plendx 15870 | . . . . 5 ⊢ (le‘ndx) = ;10 | |
9 | 1nn0 11185 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
10 | 0nn0 11184 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
11 | 2nn 11062 | . . . . . 6 ⊢ 2 ∈ ℕ | |
12 | 2pos 10989 | . . . . . 6 ⊢ 0 < 2 | |
13 | 9, 10, 11, 12 | declt 11406 | . . . . 5 ⊢ ;10 < ;12 |
14 | 9, 11 | decnncl 11394 | . . . . 5 ⊢ ;12 ∈ ℕ |
15 | dsndx 15885 | . . . . 5 ⊢ (dist‘ndx) = ;12 | |
16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 15802 | . . . 4 ⊢ {〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} Struct 〈9, ;12〉 |
17 | 3nn 11063 | . . . . . 6 ⊢ 3 ∈ ℕ | |
18 | 9, 17 | decnncl 11394 | . . . . 5 ⊢ ;13 ∈ ℕ |
19 | unifndx 15887 | . . . . 5 ⊢ (UnifSet‘ndx) = ;13 | |
20 | 18, 19 | strle1 15800 | . . . 4 ⊢ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉} Struct 〈;13, ;13〉 |
21 | 2nn0 11186 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
22 | 2lt3 11072 | . . . . 5 ⊢ 2 < 3 | |
23 | 9, 21, 17, 22 | declt 11406 | . . . 4 ⊢ ;12 < ;13 |
24 | 16, 20, 23 | strleun 15799 | . . 3 ⊢ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}) Struct 〈9, ;13〉 |
25 | 4lt9 11103 | . . 3 ⊢ 4 < 9 | |
26 | 3, 24, 25 | strleun 15799 | . 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 4604 | 1 ⊢ ℂfld Struct 〈1, ;13〉 |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3538 {csn 4125 {ctp 4129 〈cop 4131 class class class wbr 4583 ∘ ccom 5042 ‘cfv 5804 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ≤ cle 9954 − cmin 10145 2c2 10947 3c3 10948 4c4 10949 9c9 10954 ;cdc 11369 ∗ccj 13684 abscabs 13822 Struct cstr 15691 ndxcnx 15692 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 *𝑟cstv 15770 TopSetcts 15774 lecple 15775 distcds 15777 UnifSetcunif 15778 MetOpencmopn 19557 metUnifcmetu 19558 ℂfldccnfld 19567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-cnfld 19568 |
This theorem is referenced by: cnfldex 19570 cnfldbas 19571 cnfldadd 19572 cnfldmul 19573 cnfldcj 19574 cnfldtset 19575 cnfldle 19576 cnfldds 19577 cnfldunif 19578 |
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