Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cnncvsabsnegdemo | Structured version Visualization version GIF version |
Description: Derive the absolute value of a negative complex number absneg 14622 to demonstrate the use of the properties of a normed subcomplex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
cnncvsabsnegdemo | ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldnm 23370 | . . . 4 ⊢ abs = (norm‘ℂfld) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → abs = (norm‘ℂfld)) |
3 | cnfldneg 20554 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((invg‘ℂfld)‘𝐴) = -𝐴) | |
4 | 3 | eqcomd 2827 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 = ((invg‘ℂfld)‘𝐴)) |
5 | 2, 4 | fveq12d 6663 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = ((norm‘ℂfld)‘((invg‘ℂfld)‘𝐴))) |
6 | cnngp 23371 | . . 3 ⊢ ℂfld ∈ NrmGrp | |
7 | cnfldbas 20532 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
8 | eqid 2821 | . . . 4 ⊢ (norm‘ℂfld) = (norm‘ℂfld) | |
9 | eqid 2821 | . . . 4 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
10 | 7, 8, 9 | nminv 23213 | . . 3 ⊢ ((ℂfld ∈ NrmGrp ∧ 𝐴 ∈ ℂ) → ((norm‘ℂfld)‘((invg‘ℂfld)‘𝐴)) = ((norm‘ℂfld)‘𝐴)) |
11 | 6, 10 | mpan 688 | . 2 ⊢ (𝐴 ∈ ℂ → ((norm‘ℂfld)‘((invg‘ℂfld)‘𝐴)) = ((norm‘ℂfld)‘𝐴)) |
12 | 1 | eqcomi 2830 | . . . 4 ⊢ (norm‘ℂfld) = abs |
13 | 12 | fveq1i 6657 | . . 3 ⊢ ((norm‘ℂfld)‘𝐴) = (abs‘𝐴) |
14 | 13 | a1i 11 | . 2 ⊢ (𝐴 ∈ ℂ → ((norm‘ℂfld)‘𝐴) = (abs‘𝐴)) |
15 | 5, 11, 14 | 3eqtrd 2860 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6341 ℂcc 10521 -cneg 10857 abscabs 14578 invgcminusg 18087 ℂfldccnfld 20528 normcnm 23169 NrmGrpcngp 23170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 ax-addf 10602 ax-mulf 10603 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-map 8394 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-sup 8892 df-inf 8893 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-q 12336 df-rp 12377 df-xneg 12494 df-xadd 12495 df-xmul 12496 df-fz 12883 df-seq 13360 df-exp 13420 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-plusg 16561 df-mulr 16562 df-starv 16563 df-tset 16567 df-ple 16568 df-ds 16570 df-unif 16571 df-rest 16679 df-topn 16680 df-0g 16698 df-topgen 16700 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-grp 18089 df-minusg 18090 df-sbg 18091 df-cmn 18891 df-mgp 19223 df-ring 19282 df-cring 19283 df-psmet 20520 df-xmet 20521 df-met 20522 df-bl 20523 df-mopn 20524 df-cnfld 20529 df-top 21485 df-topon 21502 df-topsp 21524 df-bases 21537 df-xms 22913 df-ms 22914 df-nm 23175 df-ngp 23176 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |