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Theorem cevathlem1 27824
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 970 . . . 4  |-  ( ph  ->  B  e.  CC )
3 cevathlem1.b . . . . 5  |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC )
)
43simp3d 971 . . . 4  |-  ( ph  ->  F  e.  CC )
52, 4mulcld 9100 . . 3  |-  ( ph  ->  ( B  x.  F
)  e.  CC )
6 cevathlem1.c . . . 4  |-  ( ph  ->  ( G  e.  CC  /\  H  e.  CC  /\  K  e.  CC )
)
76simp2d 970 . . 3  |-  ( ph  ->  H  e.  CC )
85, 7mulcld 9100 . 2  |-  ( ph  ->  ( ( B  x.  F )  x.  H
)  e.  CC )
93simp1d 969 . . . 4  |-  ( ph  ->  D  e.  CC )
106simp1d 969 . . . 4  |-  ( ph  ->  G  e.  CC )
119, 10mulcld 9100 . . 3  |-  ( ph  ->  ( D  x.  G
)  e.  CC )
126simp3d 971 . . 3  |-  ( ph  ->  K  e.  CC )
1311, 12mulcld 9100 . 2  |-  ( ph  ->  ( ( D  x.  G )  x.  K
)  e.  CC )
141simp1d 969 . . . 4  |-  ( ph  ->  A  e.  CC )
153simp2d 970 . . . 4  |-  ( ph  ->  E  e.  CC )
1614, 15mulcld 9100 . . 3  |-  ( ph  ->  ( A  x.  E
)  e.  CC )
171simp3d 971 . . 3  |-  ( ph  ->  C  e.  CC )
1816, 17mulcld 9100 . 2  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  e.  CC )
19 cevathlem1.d . . . . 5  |-  ( ph  ->  ( A  =/=  0  /\  E  =/=  0  /\  C  =/=  0
) )
2019simp1d 969 . . . 4  |-  ( ph  ->  A  =/=  0 )
2119simp2d 970 . . . 4  |-  ( ph  ->  E  =/=  0 )
2214, 15, 20, 21mulne0d 9666 . . 3  |-  ( ph  ->  ( A  x.  E
)  =/=  0 )
2319simp3d 971 . . 3  |-  ( ph  ->  C  =/=  0 )
2416, 17, 22, 23mulne0d 9666 . 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 969 . . . . . . 7  |-  ( ph  ->  ( A  x.  B
)  =  ( C  x.  D ) )
2725simp2d 970 . . . . . . 7  |-  ( ph  ->  ( E  x.  F
)  =  ( A  x.  G ) )
2826, 27oveq12d 6091 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  x.  ( E  x.  F )
)  =  ( ( C  x.  D )  x.  ( A  x.  G ) ) )
2914, 2, 15, 4mul4d 9270 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  x.  ( E  x.  F )
)  =  ( ( A  x.  E )  x.  ( B  x.  F ) ) )
3017, 9, 14, 10mul4d 9270 . . . . . 6  |-  ( ph  ->  ( ( C  x.  D )  x.  ( A  x.  G )
)  =  ( ( C  x.  A )  x.  ( D  x.  G ) ) )
3128, 29, 303eqtr3d 2475 . . . . 5  |-  ( ph  ->  ( ( A  x.  E )  x.  ( B  x.  F )
)  =  ( ( C  x.  A )  x.  ( D  x.  G ) ) )
3225simp3d 971 . . . . 5  |-  ( ph  ->  ( C  x.  H
)  =  ( E  x.  K ) )
3331, 32oveq12d 6091 . . . 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 9270 . . . 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 9100 . . . . 5  |-  ( ph  ->  ( C  x.  A
)  e.  CC )
3635, 11, 15, 12mul4d 9270 . . . 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 2475 . . 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 9268 . . . . 5  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  =  ( ( A  x.  C )  x.  E ) )
3914, 17mulcomd 9101 . . . . . 6  |-  ( ph  ->  ( A  x.  C
)  =  ( C  x.  A ) )
4039oveq1d 6088 . . . . 5  |-  ( ph  ->  ( ( A  x.  C )  x.  E
)  =  ( ( C  x.  A )  x.  E ) )
4138, 40eqtrd 2467 . . . 4  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  =  ( ( C  x.  A )  x.  E ) )
4241oveq1d 6088 . . 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 2470 . 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 9649 1  |-  ( ph  ->  ( ( B  x.  F )  x.  H
)  =  ( ( D  x.  G )  x.  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598  (class class class)co 6073   CCcc 8980   0cc0 8982    x. cmul 8987
This theorem is referenced by:  cevath  27826
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286
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