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Mirrors > Home > MPE Home > Th. List > cnfld0 | Structured version Visualization version GIF version |
Description: The zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
cnfld0 | ⊢ 0 = (0g‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 00id 10249 | . . 3 ⊢ (0 + 0) = 0 | |
2 | cnring 19816 | . . . . 5 ⊢ ℂfld ∈ Ring | |
3 | ringgrp 18598 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Grp) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ℂfld ∈ Grp |
5 | 0cn 10070 | . . . 4 ⊢ 0 ∈ ℂ | |
6 | cnfldbas 19798 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
7 | cnfldadd 19799 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
8 | eqid 2651 | . . . . 5 ⊢ (0g‘ℂfld) = (0g‘ℂfld) | |
9 | 6, 7, 8 | grpid 17504 | . . . 4 ⊢ ((ℂfld ∈ Grp ∧ 0 ∈ ℂ) → ((0 + 0) = 0 ↔ (0g‘ℂfld) = 0)) |
10 | 4, 5, 9 | mp2an 708 | . . 3 ⊢ ((0 + 0) = 0 ↔ (0g‘ℂfld) = 0) |
11 | 1, 10 | mpbi 220 | . 2 ⊢ (0g‘ℂfld) = 0 |
12 | 11 | eqcomi 2660 | 1 ⊢ 0 = (0g‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 0cc0 9974 + caddc 9977 0gc0g 16147 Grpcgrp 17469 Ringcrg 18593 ℂfldccnfld 19794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-addf 10053 ax-mulf 10054 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-mulr 16002 df-starv 16003 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-cmn 18241 df-mgp 18536 df-ring 18595 df-cring 18596 df-cnfld 19795 |
This theorem is referenced by: cnfldneg 19820 cndrng 19823 cnflddiv 19824 cnfldinv 19825 cnfldmulg 19826 cnsubmlem 19842 cnsubdrglem 19845 absabv 19851 qsssubdrg 19853 cnmgpabl 19855 cnmsubglem 19857 gzrngunitlem 19859 gzrngunit 19860 gsumfsum 19861 expmhm 19863 nn0srg 19864 rge0srg 19865 zring0 19876 zringunit 19884 expghm 19892 psgninv 19976 zrhpsgnmhm 19978 re0g 20006 regsumsupp 20016 cnfldnm 22629 clm0 22918 cphsubrglem 23023 cphreccllem 23024 tdeglem1 23863 tdeglem3 23864 tdeglem4 23865 plypf1 24013 dvply2g 24085 tayl0 24161 taylpfval 24164 efsubm 24342 jensenlem2 24759 jensen 24760 amgmlem 24761 amgm 24762 dchrghm 25026 dchrabs 25030 sum2dchr 25044 lgseisenlem4 25148 qrng0 25355 xrge0slmod 29972 zringnm 30132 rezh 30143 fsumcnsrcl 38053 cnsrplycl 38054 rngunsnply 38060 proot1ex 38096 deg1mhm 38102 2zrng0 42263 amgmwlem 42876 amgmlemALT 42877 |
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