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Theorem cevathlem1 27518
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 9034 . . 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 9034 . 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 9034 . . 3  |-  ( ph  ->  ( D  x.  G
)  e.  CC )
126simp3d 971 . . 3  |-  ( ph  ->  K  e.  CC )
1311, 12mulcld 9034 . 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 9034 . . 3  |-  ( ph  ->  ( A  x.  E
)  e.  CC )
171simp3d 971 . . 3  |-  ( ph  ->  C  e.  CC )
1816, 17mulcld 9034 . 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 9599 . . 3  |-  ( ph  ->  ( A  x.  E
)  =/=  0 )
2319simp3d 971 . . 3  |-  ( ph  ->  C  =/=  0 )
2416, 17, 22, 23mulne0d 9599 . 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 6031 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  x.  ( E  x.  F )
)  =  ( ( C  x.  D )  x.  ( A  x.  G ) ) )
2914, 2, 15, 4mul4d 9203 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  x.  ( E  x.  F )
)  =  ( ( A  x.  E )  x.  ( B  x.  F ) ) )
3017, 9, 14, 10mul4d 9203 . . . . . 6  |-  ( ph  ->  ( ( C  x.  D )  x.  ( A  x.  G )
)  =  ( ( C  x.  A )  x.  ( D  x.  G ) ) )
3128, 29, 303eqtr3d 2420 . . . . 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 6031 . . . 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 9203 . . . 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 9034 . . . . 5  |-  ( ph  ->  ( C  x.  A
)  e.  CC )
3635, 11, 15, 12mul4d 9203 . . . 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 2420 . . 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 9201 . . . . 5  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  =  ( ( A  x.  C )  x.  E ) )
3914, 17mulcomd 9035 . . . . . 6  |-  ( ph  ->  ( A  x.  C
)  =  ( C  x.  A ) )
4039oveq1d 6028 . . . . 5  |-  ( ph  ->  ( ( A  x.  C )  x.  E
)  =  ( ( C  x.  A )  x.  E ) )
4138, 40eqtrd 2412 . . . 4  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  =  ( ( C  x.  A )  x.  E ) )
4241oveq1d 6028 . . 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 2415 . 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 9582 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 1649    e. wcel 1717    =/= wne 2543  (class class class)co 6013   CCcc 8914   0cc0 8916    x. cmul 8921
This theorem is referenced by:  cevath  27520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219
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