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Mirrors > Home > MPE Home > Th. List > abs00bd | Structured version Visualization version GIF version |
Description: If a complex number is zero, its absolute value is zero. Converse of abs00d 14563. One-way deduction form of abs00 14407. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
abs00bd.1 | ⊢ (𝜑 → 𝐴 = 0) |
Ref | Expression |
---|---|
abs00bd | ⊢ (𝜑 → (abs‘𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abs00bd.1 | . 2 ⊢ (𝜑 → 𝐴 = 0) | |
2 | 0cn 10349 | . . . 4 ⊢ 0 ∈ ℂ | |
3 | 1, 2 | syl6eqel 2915 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
4 | 3 | abs00ad 14408 | . 2 ⊢ (𝜑 → ((abs‘𝐴) = 0 ↔ 𝐴 = 0)) |
5 | 1, 4 | mpbird 249 | 1 ⊢ (𝜑 → (abs‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ‘cfv 6124 ℂcc 10251 0cc0 10253 abscabs 14352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-sup 8618 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-n0 11620 df-z 11706 df-uz 11970 df-rp 12114 df-seq 13097 df-exp 13156 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 |
This theorem is referenced by: lcmgcd 15694 blcvx 22972 mulc1cncf 23079 rrxdstprj1 23578 dvlip 24156 c1lip1 24160 dveq0 24163 dv11cn 24164 ftc1lem5 24203 dvradcnv 24575 abelthlem2 24586 abelthlem8 24593 abscxp2 24839 cxpcn3lem 24891 abscxpbnd 24897 chordthmlem3 24975 rlimcnp 25106 dchrabs2 25401 dchrisumlem3 25594 pntrsumbnd2 25670 siii 28264 nmbdfnlbi 29464 nmcfnlbi 29467 knoppndvlem13 33048 poimirlem29 33983 ftc1cnnc 34028 pellexlem6 38243 congabseq 38385 dvconstbi 39374 binomcxplemnn0 39389 dvdivbd 40934 dvbdfbdioolem2 40940 ioodvbdlimc1lem1 40942 |
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