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Theorem cevathlem1 27857
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 8855 . . 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 8855 . 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 8855 . . 3  |-  ( ph  ->  ( D  x.  G
)  e.  CC )
126simp3d 969 . . 3  |-  ( ph  ->  K  e.  CC )
1311, 12mulcld 8855 . 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 8855 . . 3  |-  ( ph  ->  ( A  x.  E
)  e.  CC )
171simp3d 969 . . 3  |-  ( ph  ->  C  e.  CC )
1816, 17mulcld 8855 . 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 9420 . . 3  |-  ( ph  ->  ( A  x.  E
)  =/=  0 )
2319simp3d 969 . . 3  |-  ( ph  ->  C  =/=  0 )
2416, 17, 22, 23mulne0d 9420 . 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 5876 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  x.  ( E  x.  F )
)  =  ( ( C  x.  D )  x.  ( A  x.  G ) ) )
2914, 2, 15, 4mul4d 9024 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  x.  ( E  x.  F )
)  =  ( ( A  x.  E )  x.  ( B  x.  F ) ) )
3017, 9, 14, 10mul4d 9024 . . . . . 6  |-  ( ph  ->  ( ( C  x.  D )  x.  ( A  x.  G )
)  =  ( ( C  x.  A )  x.  ( D  x.  G ) ) )
3128, 29, 303eqtr3d 2323 . . . . 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 5876 . . . 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 9024 . . . 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 8855 . . . . 5  |-  ( ph  ->  ( C  x.  A
)  e.  CC )
3635, 11, 15, 12mul4d 9024 . . . 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 2323 . . 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 9022 . . . . 5  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  =  ( ( A  x.  C )  x.  E ) )
3914, 17mulcomd 8856 . . . . . 6  |-  ( ph  ->  ( A  x.  C
)  =  ( C  x.  A ) )
4039oveq1d 5873 . . . . 5  |-  ( ph  ->  ( ( A  x.  C )  x.  E
)  =  ( ( C  x.  A )  x.  E ) )
4138, 40eqtrd 2315 . . . 4  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  =  ( ( C  x.  A )  x.  E ) )
4241oveq1d 5873 . . 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 2318 . 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 9403 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 1623    e. wcel 1684    =/= wne 2446  (class class class)co 5858   CCcc 8735   0cc0 8737    x. cmul 8742
This theorem is referenced by:  cevath  27859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040
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