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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 10809 | . . 3 ⊢ (0 + 0) = 0 | |
2 | cnring 20561 | . . . . 5 ⊢ ℂfld ∈ Ring | |
3 | ringgrp 19296 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Grp) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ℂfld ∈ Grp |
5 | 0cn 10627 | . . . 4 ⊢ 0 ∈ ℂ | |
6 | cnfldbas 20543 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
7 | cnfldadd 20544 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
8 | eqid 2821 | . . . . 5 ⊢ (0g‘ℂfld) = (0g‘ℂfld) | |
9 | 6, 7, 8 | grpid 18133 | . . . 4 ⊢ ((ℂfld ∈ Grp ∧ 0 ∈ ℂ) → ((0 + 0) = 0 ↔ (0g‘ℂfld) = 0)) |
10 | 4, 5, 9 | mp2an 690 | . . 3 ⊢ ((0 + 0) = 0 ↔ (0g‘ℂfld) = 0) |
11 | 1, 10 | mpbi 232 | . 2 ⊢ (0g‘ℂfld) = 0 |
12 | 11 | eqcomi 2830 | 1 ⊢ 0 = (0g‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 0cc0 10531 + caddc 10534 0gc0g 16707 Grpcgrp 18097 Ringcrg 19291 ℂfldccnfld 20539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-cmn 18902 df-mgp 19234 df-ring 19293 df-cring 19294 df-cnfld 20540 |
This theorem is referenced by: cnfldneg 20565 cndrng 20568 cnflddiv 20569 cnfldinv 20570 cnfldmulg 20571 cnsubmlem 20587 cnsubdrglem 20590 absabv 20596 qsssubdrg 20598 cnmgpabl 20600 cnmsubglem 20602 gzrngunitlem 20604 gzrngunit 20605 gsumfsum 20606 expmhm 20608 nn0srg 20609 rge0srg 20610 zring0 20621 zringunit 20629 expghm 20637 psgninv 20720 zrhpsgnmhm 20722 re0g 20750 regsumsupp 20760 cnfldnm 23381 clm0 23670 cphsubrglem 23775 cphreccllem 23776 tdeglem1 24646 tdeglem3 24647 tdeglem4 24648 plypf1 24796 dvply2g 24868 tayl0 24944 taylpfval 24947 efsubm 25129 jensenlem2 25559 jensen 25560 amgmlem 25561 amgm 25562 dchrghm 25826 dchrabs 25830 sum2dchr 25844 lgseisenlem4 25948 qrng0 26191 xrge0slmod 30912 ccfldextdgrr 31052 zringnm 31196 rezh 31207 fsumcnsrcl 39759 cnsrplycl 39760 rngunsnply 39766 proot1ex 39794 deg1mhm 39800 2zrng0 44203 amgmwlem 44897 amgmlemALT 44898 |
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