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Theorem cevathlem1 29908
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 1001 . . . 4  |-  ( ph  ->  B  e.  CC )
3 cevathlem1.b . . . . 5  |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC )
)
43simp3d 1002 . . . 4  |-  ( ph  ->  F  e.  CC )
52, 4mulcld 9411 . . 3  |-  ( ph  ->  ( B  x.  F
)  e.  CC )
6 cevathlem1.c . . . 4  |-  ( ph  ->  ( G  e.  CC  /\  H  e.  CC  /\  K  e.  CC )
)
76simp2d 1001 . . 3  |-  ( ph  ->  H  e.  CC )
85, 7mulcld 9411 . 2  |-  ( ph  ->  ( ( B  x.  F )  x.  H
)  e.  CC )
93simp1d 1000 . . . 4  |-  ( ph  ->  D  e.  CC )
106simp1d 1000 . . . 4  |-  ( ph  ->  G  e.  CC )
119, 10mulcld 9411 . . 3  |-  ( ph  ->  ( D  x.  G
)  e.  CC )
126simp3d 1002 . . 3  |-  ( ph  ->  K  e.  CC )
1311, 12mulcld 9411 . 2  |-  ( ph  ->  ( ( D  x.  G )  x.  K
)  e.  CC )
141simp1d 1000 . . . 4  |-  ( ph  ->  A  e.  CC )
153simp2d 1001 . . . 4  |-  ( ph  ->  E  e.  CC )
1614, 15mulcld 9411 . . 3  |-  ( ph  ->  ( A  x.  E
)  e.  CC )
171simp3d 1002 . . 3  |-  ( ph  ->  C  e.  CC )
1816, 17mulcld 9411 . 2  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  e.  CC )
19 cevathlem1.d . . . . 5  |-  ( ph  ->  ( A  =/=  0  /\  E  =/=  0  /\  C  =/=  0
) )
2019simp1d 1000 . . . 4  |-  ( ph  ->  A  =/=  0 )
2119simp2d 1001 . . . 4  |-  ( ph  ->  E  =/=  0 )
2214, 15, 20, 21mulne0d 9993 . . 3  |-  ( ph  ->  ( A  x.  E
)  =/=  0 )
2319simp3d 1002 . . 3  |-  ( ph  ->  C  =/=  0 )
2416, 17, 22, 23mulne0d 9993 . 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 1000 . . . . . . 7  |-  ( ph  ->  ( A  x.  B
)  =  ( C  x.  D ) )
2725simp2d 1001 . . . . . . 7  |-  ( ph  ->  ( E  x.  F
)  =  ( A  x.  G ) )
2826, 27oveq12d 6114 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  x.  ( E  x.  F )
)  =  ( ( C  x.  D )  x.  ( A  x.  G ) ) )
2914, 2, 15, 4mul4d 9586 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  x.  ( E  x.  F )
)  =  ( ( A  x.  E )  x.  ( B  x.  F ) ) )
3017, 9, 14, 10mul4d 9586 . . . . . 6  |-  ( ph  ->  ( ( C  x.  D )  x.  ( A  x.  G )
)  =  ( ( C  x.  A )  x.  ( D  x.  G ) ) )
3128, 29, 303eqtr3d 2483 . . . . 5  |-  ( ph  ->  ( ( A  x.  E )  x.  ( B  x.  F )
)  =  ( ( C  x.  A )  x.  ( D  x.  G ) ) )
3225simp3d 1002 . . . . 5  |-  ( ph  ->  ( C  x.  H
)  =  ( E  x.  K ) )
3331, 32oveq12d 6114 . . . 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 9586 . . . 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 9411 . . . . 5  |-  ( ph  ->  ( C  x.  A
)  e.  CC )
3635, 11, 15, 12mul4d 9586 . . . 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 2483 . . 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 9584 . . . . 5  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  =  ( ( A  x.  C )  x.  E ) )
3914, 17mulcomd 9412 . . . . . 6  |-  ( ph  ->  ( A  x.  C
)  =  ( C  x.  A ) )
4039oveq1d 6111 . . . . 5  |-  ( ph  ->  ( ( A  x.  C )  x.  E
)  =  ( ( C  x.  A )  x.  E ) )
4138, 40eqtrd 2475 . . . 4  |-  ( ph  ->  ( ( A  x.  E )  x.  C
)  =  ( ( C  x.  A )  x.  E ) )
4241oveq1d 6111 . . 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 2478 . 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 9976 1  |-  ( ph  ->  ( ( B  x.  F )  x.  H
)  =  ( ( D  x.  G )  x.  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611  (class class class)co 6096   CCcc 9285   0cc0 9287    x. cmul 9292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603
This theorem is referenced by:  cevath  29910
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