![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > abs0 | Structured version Visualization version GIF version |
Description: The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
abs0 | ⊢ (abs‘0) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 11247 | . . 3 ⊢ 0 ∈ ℂ | |
2 | absval 15238 | . . 3 ⊢ (0 ∈ ℂ → (abs‘0) = (√‘(0 · (∗‘0)))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (abs‘0) = (√‘(0 · (∗‘0))) |
4 | 1 | cjcli 15169 | . . . 4 ⊢ (∗‘0) ∈ ℂ |
5 | 4 | mul02i 11444 | . . 3 ⊢ (0 · (∗‘0)) = 0 |
6 | 5 | fveq2i 6896 | . 2 ⊢ (√‘(0 · (∗‘0))) = (√‘0) |
7 | sqrt0 15241 | . 2 ⊢ (√‘0) = 0 | |
8 | 3, 6, 7 | 3eqtri 2758 | 1 ⊢ (abs‘0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ‘cfv 6546 (class class class)co 7416 ℂcc 11147 0cc0 11149 · cmul 11154 ∗ccj 15096 √csqrt 15233 abscabs 15234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-n0 12519 df-z 12605 df-uz 12869 df-rp 13023 df-seq 14016 df-exp 14076 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 |
This theorem is referenced by: abs00 15289 abs1m 15335 climconst 15540 rlimconst 15541 fsumabs 15800 georeclim 15871 geoisumr 15877 dvdsabseq 16310 gcd0id 16514 lcmid 16605 4sqlem19 16960 absabv 21417 gzrngunit 21426 zringunit 21452 aannenlem2 26354 aalioulem3 26359 tanabsge 26531 sinkpi 26546 sineq0 26548 isosctrlem2 26844 lgamgulmlem1 27054 ftalem3 27100 mule1 27173 zabsle1 27322 lgslem2 27324 lgsfcl2 27329 bcsiALT 31109 0cnfn 31910 nmfn0 31917 nmophmi 31961 nmcfnexi 31981 dnizeq0 36191 unbdqndv2lem2 36226 mblfinlem2 37372 ftc1anclem7 37413 ftc1anclem8 37414 ftc1anc 37415 dvgrat 44023 radcnvrat 44025 sineq0ALT 44650 constlimc 45281 0cnv 45399 |
Copyright terms: Public domain | W3C validator |