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Mirrors > Home > MPE Home > Th. List > cnfld0 | Structured version Visualization version GIF version |
Description: Zero is 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 10804 | . . 3 ⊢ (0 + 0) = 0 | |
2 | cnring 20113 | . . . . 5 ⊢ ℂfld ∈ Ring | |
3 | ringgrp 19295 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Grp) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ℂfld ∈ Grp |
5 | 0cn 10622 | . . . 4 ⊢ 0 ∈ ℂ | |
6 | cnfldbas 20095 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
7 | cnfldadd 20096 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
8 | eqid 2798 | . . . . 5 ⊢ (0g‘ℂfld) = (0g‘ℂfld) | |
9 | 6, 7, 8 | grpid 18131 | . . . 4 ⊢ ((ℂfld ∈ Grp ∧ 0 ∈ ℂ) → ((0 + 0) = 0 ↔ (0g‘ℂfld) = 0)) |
10 | 4, 5, 9 | mp2an 691 | . . 3 ⊢ ((0 + 0) = 0 ↔ (0g‘ℂfld) = 0) |
11 | 1, 10 | mpbi 233 | . 2 ⊢ (0g‘ℂfld) = 0 |
12 | 11 | eqcomi 2807 | 1 ⊢ 0 = (0g‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 + caddc 10529 0gc0g 16705 Grpcgrp 18095 Ringcrg 19290 ℂfldccnfld 20091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-cmn 18900 df-mgp 19233 df-ring 19292 df-cring 19293 df-cnfld 20092 |
This theorem is referenced by: cnfldneg 20117 cndrng 20120 cnflddiv 20121 cnfldinv 20122 cnfldmulg 20123 cnsubmlem 20139 cnsubdrglem 20142 absabv 20148 qsssubdrg 20150 cnmgpabl 20152 cnmsubglem 20154 gzrngunitlem 20156 gzrngunit 20157 gsumfsum 20158 expmhm 20160 nn0srg 20161 rge0srg 20162 zring0 20173 zringunit 20181 expghm 20189 psgninv 20271 zrhpsgnmhm 20273 re0g 20301 regsumsupp 20311 mhpvarcl 20798 cnfldnm 23384 clm0 23677 cphsubrglem 23782 cphreccllem 23783 tdeglem1 24659 tdeglem3 24660 tdeglem4 24661 plypf1 24809 dvply2g 24881 tayl0 24957 taylpfval 24960 efsubm 25143 jensenlem2 25573 jensen 25574 amgmlem 25575 amgm 25576 dchrghm 25840 dchrabs 25844 sum2dchr 25858 lgseisenlem4 25962 qrng0 26205 xrge0slmod 30968 ccfldextdgrr 31145 zringnm 31311 rezh 31322 fsumcnsrcl 40110 cnsrplycl 40111 rngunsnply 40117 proot1ex 40145 deg1mhm 40151 2zrng0 44562 amgmwlem 45330 amgmlemALT 45331 |
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