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Theorem cevathlem1 27960
Description: Ceva's theorem first lemma. Multiplies three identities and divides by the common factors. (Contributed by Saveliy Skresanov, 24-Sep-2017.)
Hypotheses
Ref Expression
cevathlem1.a  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
cevathlem1.b  |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC )
)
cevathlem1.c  |-  ( ph  ->  ( G  e.  CC  /\  H  e.  CC  /\  K  e.  CC )
)
cevathlem1.d  |-  ( ph  ->  ( A  =/=  0  /\  E  =/=  0  /\  C  =/=  0
) )
cevathlem1.e  |-  ( ph  ->  ( ( A  x.  B )  =  ( C  x.  D )  /\  ( E  x.  F )  =  ( A  x.  G )  /\  ( C  x.  H )  =  ( E  x.  K ) ) )
Assertion
Ref Expression
cevathlem1  |-  ( ph  ->  ( ( B  x.  F )  x.  H
)  =  ( ( D  x.  G )  x.  K ) )

Proof of Theorem cevathlem1
StepHypRef Expression
1 cevathlem1.a . . . . 5  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
21simp2d 968 . . . 4  |-  ( ph  ->  B  e.  CC )
3 cevathlem1.b . . . . 5  |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC )
)
43simp3d 969 . . . 4  |-  ( ph  ->  F  e.  CC )
52, 4mulcld 8871 . . 3  |-  ( ph  ->  ( B  x.  F
)  e.  CC )
6 cevathlem1.c . . . 4  |-  ( ph  ->  ( G  e.  CC  /\  H  e.  CC  /\  K  e.  CC )
)
76simp2d 968 . . 3  |-  ( ph  ->  H  e.  CC )
85, 7mulcld 8871 . 2  |-  ( ph  ->  ( ( B  x.  F )  x.  H
)  e.  CC )
93simp1d 967 . . . 4  |-  ( ph  ->  D  e.  CC )
106simp1d 967 . . . 4  |-  ( ph  ->  G  e.  CC )
119, 10mulcld 8871 . . 3  |-  ( ph  ->  ( D  x.  G
)  e.  CC )
126simp3d 969 . . 3  |-  ( ph  ->  K  e.  CC )
1311, 12mulcld 8871 . 2  |-  ( ph  ->  ( ( D  x.  G )  x.  K
)  e.  CC )
141simp1d 967 . . . 4  |-  ( ph  ->  A  e.  CC )
153simp2d 968 . . . 4  |-  ( ph  ->  E  e.  CC )
1614, 15mulcld 8871 . . 3  |-  ( ph  ->  ( A  x.  E
)  e.  CC )
171simp3d 969 . . 3  |-  ( ph  ->  C  e.  CC )
1816, 17mulcld 8871 . 2  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  e.  CC )
19 cevathlem1.d . . . . 5  |-  ( ph  ->  ( A  =/=  0  /\  E  =/=  0  /\  C  =/=  0
) )
2019simp1d 967 . . . 4  |-  ( ph  ->  A  =/=  0 )
2119simp2d 968 . . . 4  |-  ( ph  ->  E  =/=  0 )
2214, 15, 20, 21mulne0d 9436 . . 3  |-  ( ph  ->  ( A  x.  E
)  =/=  0 )
2319simp3d 969 . . 3  |-  ( ph  ->  C  =/=  0 )
2416, 17, 22, 23mulne0d 9436 . 2  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  =/=  0 )
25 cevathlem1.e . . . . . . . 8  |-  ( ph  ->  ( ( A  x.  B )  =  ( C  x.  D )  /\  ( E  x.  F )  =  ( A  x.  G )  /\  ( C  x.  H )  =  ( E  x.  K ) ) )
2625simp1d 967 . . . . . . 7  |-  ( ph  ->  ( A  x.  B
)  =  ( C  x.  D ) )
2725simp2d 968 . . . . . . 7  |-  ( ph  ->  ( E  x.  F
)  =  ( A  x.  G ) )
2826, 27oveq12d 5892 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  x.  ( E  x.  F )
)  =  ( ( C  x.  D )  x.  ( A  x.  G ) ) )
2914, 2, 15, 4mul4d 9040 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  x.  ( E  x.  F )
)  =  ( ( A  x.  E )  x.  ( B  x.  F ) ) )
3017, 9, 14, 10mul4d 9040 . . . . . 6  |-  ( ph  ->  ( ( C  x.  D )  x.  ( A  x.  G )
)  =  ( ( C  x.  A )  x.  ( D  x.  G ) ) )
3128, 29, 303eqtr3d 2336 . . . . 5  |-  ( ph  ->  ( ( A  x.  E )  x.  ( B  x.  F )
)  =  ( ( C  x.  A )  x.  ( D  x.  G ) ) )
3225simp3d 969 . . . . 5  |-  ( ph  ->  ( C  x.  H
)  =  ( E  x.  K ) )
3331, 32oveq12d 5892 . . . 4  |-  ( ph  ->  ( ( ( A  x.  E )  x.  ( B  x.  F
) )  x.  ( C  x.  H )
)  =  ( ( ( C  x.  A
)  x.  ( D  x.  G ) )  x.  ( E  x.  K ) ) )
3416, 5, 17, 7mul4d 9040 . . . 4  |-  ( ph  ->  ( ( ( A  x.  E )  x.  ( B  x.  F
) )  x.  ( C  x.  H )
)  =  ( ( ( A  x.  E
)  x.  C )  x.  ( ( B  x.  F )  x.  H ) ) )
3517, 14mulcld 8871 . . . . 5  |-  ( ph  ->  ( C  x.  A
)  e.  CC )
3635, 11, 15, 12mul4d 9040 . . . 4  |-  ( ph  ->  ( ( ( C  x.  A )  x.  ( D  x.  G
) )  x.  ( E  x.  K )
)  =  ( ( ( C  x.  A
)  x.  E )  x.  ( ( D  x.  G )  x.  K ) ) )
3733, 34, 363eqtr3d 2336 . . 3  |-  ( ph  ->  ( ( ( A  x.  E )  x.  C )  x.  (
( B  x.  F
)  x.  H ) )  =  ( ( ( C  x.  A
)  x.  E )  x.  ( ( D  x.  G )  x.  K ) ) )
3814, 15, 17mul32d 9038 . . . . 5  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  =  ( ( A  x.  C )  x.  E ) )
3914, 17mulcomd 8872 . . . . . 6  |-  ( ph  ->  ( A  x.  C
)  =  ( C  x.  A ) )
4039oveq1d 5889 . . . . 5  |-  ( ph  ->  ( ( A  x.  C )  x.  E
)  =  ( ( C  x.  A )  x.  E ) )
4138, 40eqtrd 2328 . . . 4  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  =  ( ( C  x.  A )  x.  E ) )
4241oveq1d 5889 . . 3  |-  ( ph  ->  ( ( ( A  x.  E )  x.  C )  x.  (
( D  x.  G
)  x.  K ) )  =  ( ( ( C  x.  A
)  x.  E )  x.  ( ( D  x.  G )  x.  K ) ) )
4337, 42eqtr4d 2331 . 2  |-  ( ph  ->  ( ( ( A  x.  E )  x.  C )  x.  (
( B  x.  F
)  x.  H ) )  =  ( ( ( A  x.  E
)  x.  C )  x.  ( ( D  x.  G )  x.  K ) ) )
448, 13, 18, 24, 43mulcanad 9419 1  |-  ( ph  ->  ( ( B  x.  F )  x.  H
)  =  ( ( D  x.  G )  x.  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459  (class class class)co 5874   CCcc 8751   0cc0 8753    x. cmul 8758
This theorem is referenced by:  cevath  27962
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056
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