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Theorem isosctr 22911
Description: Isosceles triangle theorem. This is Metamath 100 proof #65. (Contributed by Saveliy Skresanov, 1-Jan-2017.)
Hypothesis
Ref Expression
isosctrlem3.1  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
Assertion
Ref Expression
isosctr  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( ( C  -  A ) F ( B  -  A ) )  =  ( ( A  -  B ) F ( C  -  B ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y
Allowed substitution hints:    F( x, y)

Proof of Theorem isosctr
StepHypRef Expression
1 simp11 1026 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  A  e.  CC )
2 simp13 1028 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  C  e.  CC )
31, 2subcld 9930 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( A  -  C )  e.  CC )
4 simp12 1027 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  B  e.  CC )
54, 2subcld 9930 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( B  -  C )  e.  CC )
6 simp21 1029 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  A  =/=  C )
71, 2, 6subne0d 9939 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( A  -  C )  =/=  0
)
8 simp22 1030 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  B  =/=  C )
94, 2, 8subne0d 9939 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( B  -  C )  =/=  0
)
10 simp23 1031 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  A  =/=  B )
11 subcan2 9844 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
)  =  ( B  -  C )  <->  A  =  B ) )
12113ad2ant1 1017 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( ( A  -  C )  =  ( B  -  C )  <->  A  =  B ) )
1312necon3bid 2725 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( ( A  -  C )  =/=  ( B  -  C
)  <->  A  =/=  B
) )
1410, 13mpbird 232 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( A  -  C )  =/=  ( B  -  C )
)
15 simp3 998 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )
16 isosctrlem3.1 . . . 4  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
1716isosctrlem3 22910 . . 3  |-  ( ( ( ( A  -  C )  e.  CC  /\  ( B  -  C
)  e.  CC )  /\  ( ( A  -  C )  =/=  0  /\  ( B  -  C )  =/=  0  /\  ( A  -  C )  =/=  ( B  -  C
) )  /\  ( abs `  ( A  -  C ) )  =  ( abs `  ( B  -  C )
) )  ->  ( -u ( A  -  C
) F ( ( B  -  C )  -  ( A  -  C ) ) )  =  ( ( ( A  -  C )  -  ( B  -  C ) ) F
-u ( B  -  C ) ) )
183, 5, 7, 9, 14, 15, 17syl231anc 1248 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( -u ( A  -  C ) F ( ( B  -  C )  -  ( A  -  C
) ) )  =  ( ( ( A  -  C )  -  ( B  -  C
) ) F -u ( B  -  C
) ) )
191, 2negsubdi2d 9946 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  -u ( A  -  C )  =  ( C  -  A
) )
204, 1, 2nnncan2d 9965 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( ( B  -  C )  -  ( A  -  C ) )  =  ( B  -  A
) )
2119, 20oveq12d 6302 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( -u ( A  -  C ) F ( ( B  -  C )  -  ( A  -  C
) ) )  =  ( ( C  -  A ) F ( B  -  A ) ) )
221, 4, 2nnncan2d 9965 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( ( A  -  C )  -  ( B  -  C ) )  =  ( A  -  B
) )
234, 2negsubdi2d 9946 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  -u ( B  -  C )  =  ( C  -  B
) )
2422, 23oveq12d 6302 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( (
( A  -  C
)  -  ( B  -  C ) ) F -u ( B  -  C ) )  =  ( ( A  -  B ) F ( C  -  B
) ) )
2518, 21, 243eqtr3d 2516 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B
)  /\  ( abs `  ( A  -  C
) )  =  ( abs `  ( B  -  C ) ) )  ->  ( ( C  -  A ) F ( B  -  A ) )  =  ( ( A  -  B ) F ( C  -  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473   {csn 4027   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   CCcc 9490   0cc0 9492    - cmin 9805   -ucneg 9806    / cdiv 10206   Imcim 12894   abscabs 13030   logclog 22698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-ef 13665  df-sin 13667  df-cos 13668  df-pi 13670  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-haus 19610  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145  df-limc 22033  df-dv 22034  df-log 22700
This theorem is referenced by: (None)
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