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Theorem pythag 23825
Description: Pythagorean theorem. Given three distinct points A, B, and C that form a right triangle (with the right angle at C), prove a relationship between their segment lengths. This theorem is expressed using the complex number plane as a plane, where  F is the signed angle construct (as used in ang180 23822),  X is the distance of line segment BC,  Y is the distance of line segment AC,  Z is the distance of line segment AB (the hypotenuse), and  O is the signed right angle m/_ BCA. We use the law of cosines lawcos 23824 to prove this, along with simple trigonometry facts like coshalfpi 23503 and cosneg 14278. (Contributed by David A. Wheeler, 13-Jun-2015.)
Hypotheses
Ref Expression
lawcos.1  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
lawcos.2  |-  X  =  ( abs `  ( B  -  C )
)
lawcos.3  |-  Y  =  ( abs `  ( A  -  C )
)
lawcos.4  |-  Z  =  ( abs `  ( A  -  B )
)
lawcos.5  |-  O  =  ( ( B  -  C ) F ( A  -  C ) )
Assertion
Ref Expression
pythag  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  ( Z ^ 2 )  =  ( ( X ^
2 )  +  ( Y ^ 2 ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y
Allowed substitution hints:    F( x, y)    O( x, y)    X( x, y)    Y( x, y)    Z( x, y)

Proof of Theorem pythag
StepHypRef Expression
1 lawcos.1 . . . 4  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
2 lawcos.2 . . . 4  |-  X  =  ( abs `  ( B  -  C )
)
3 lawcos.3 . . . 4  |-  Y  =  ( abs `  ( A  -  C )
)
4 lawcos.4 . . . 4  |-  Z  =  ( abs `  ( A  -  B )
)
5 lawcos.5 . . . 4  |-  O  =  ( ( B  -  C ) F ( A  -  C ) )
61, 2, 3, 4, 5lawcos 23824 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( Z ^ 2 )  =  ( ( ( X ^ 2 )  +  ( Y ^ 2 ) )  -  (
2  x.  ( ( X  x.  Y )  x.  ( cos `  O
) ) ) ) )
763adant3 1050 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  ( Z ^ 2 )  =  ( ( ( X ^ 2 )  +  ( Y ^ 2 ) )  -  (
2  x.  ( ( X  x.  Y )  x.  ( cos `  O
) ) ) ) )
8 elpri 3976 . . . . . . . . 9  |-  ( O  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) }  ->  ( O  =  ( pi 
/  2 )  \/  O  =  -u (
pi  /  2 ) ) )
9 fveq2 5879 . . . . . . . . . . 11  |-  ( O  =  ( pi  / 
2 )  ->  ( cos `  O )  =  ( cos `  (
pi  /  2 ) ) )
10 coshalfpi 23503 . . . . . . . . . . 11  |-  ( cos `  ( pi  /  2
) )  =  0
119, 10syl6eq 2521 . . . . . . . . . 10  |-  ( O  =  ( pi  / 
2 )  ->  ( cos `  O )  =  0 )
12 fveq2 5879 . . . . . . . . . . 11  |-  ( O  =  -u ( pi  / 
2 )  ->  ( cos `  O )  =  ( cos `  -u (
pi  /  2 ) ) )
13 cosneghalfpi 23504 . . . . . . . . . . 11  |-  ( cos `  -u ( pi  / 
2 ) )  =  0
1412, 13syl6eq 2521 . . . . . . . . . 10  |-  ( O  =  -u ( pi  / 
2 )  ->  ( cos `  O )  =  0 )
1511, 14jaoi 386 . . . . . . . . 9  |-  ( ( O  =  ( pi 
/  2 )  \/  O  =  -u (
pi  /  2 ) )  ->  ( cos `  O )  =  0 )
168, 15syl 17 . . . . . . . 8  |-  ( O  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) }  ->  ( cos `  O )  =  0 )
17163ad2ant3 1053 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  ( cos `  O )  =  0 )
1817oveq2d 6324 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  (
( X  x.  Y
)  x.  ( cos `  O ) )  =  ( ( X  x.  Y )  x.  0 ) )
19 subcl 9894 . . . . . . . . . . . . 13  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
20193adant1 1048 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
21203ad2ant1 1051 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  ( B  -  C )  e.  CC )
2221abscld 13575 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  ( abs `  ( B  -  C ) )  e.  RR )
2322recnd 9687 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  ( abs `  ( B  -  C ) )  e.  CC )
242, 23syl5eqel 2553 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  X  e.  CC )
25 subcl 9894 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  -  C
)  e.  CC )
26253adant2 1049 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  C )  e.  CC )
27263ad2ant1 1051 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  ( A  -  C )  e.  CC )
2827abscld 13575 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  ( abs `  ( A  -  C ) )  e.  RR )
2928recnd 9687 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  ( abs `  ( A  -  C ) )  e.  CC )
303, 29syl5eqel 2553 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  Y  e.  CC )
3124, 30mulcld 9681 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  ( X  x.  Y )  e.  CC )
3231mul01d 9850 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  (
( X  x.  Y
)  x.  0 )  =  0 )
3318, 32eqtrd 2505 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  (
( X  x.  Y
)  x.  ( cos `  O ) )  =  0 )
3433oveq2d 6324 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  (
2  x.  ( ( X  x.  Y )  x.  ( cos `  O
) ) )  =  ( 2  x.  0 ) )
35 2t0e0 10788 . . . 4  |-  ( 2  x.  0 )  =  0
3634, 35syl6eq 2521 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  (
2  x.  ( ( X  x.  Y )  x.  ( cos `  O
) ) )  =  0 )
3736oveq2d 6324 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  (
( ( X ^
2 )  +  ( Y ^ 2 ) )  -  ( 2  x.  ( ( X  x.  Y )  x.  ( cos `  O
) ) ) )  =  ( ( ( X ^ 2 )  +  ( Y ^
2 ) )  - 
0 ) )
3824sqcld 12452 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  ( X ^ 2 )  e.  CC )
3930sqcld 12452 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  ( Y ^ 2 )  e.  CC )
4038, 39addcld 9680 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  (
( X ^ 2 )  +  ( Y ^ 2 ) )  e.  CC )
4140subid1d 9994 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  (
( ( X ^
2 )  +  ( Y ^ 2 ) )  -  0 )  =  ( ( X ^ 2 )  +  ( Y ^ 2 ) ) )
427, 37, 413eqtrd 2509 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
)  /\  O  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )  ->  ( Z ^ 2 )  =  ( ( X ^
2 )  +  ( Y ^ 2 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641    \ cdif 3387   {csn 3959   {cpr 3961   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   CCcc 9555   0cc0 9557    + caddc 9560    x. cmul 9562    - cmin 9880   -ucneg 9881    / cdiv 10291   2c2 10681   ^cexp 12310   Imcim 13238   abscabs 13374   cosccos 14194   picpi 14196   logclog 23583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585
This theorem is referenced by:  chordthmlem3  23839
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