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Theorem i2linesd 31131
Description: Solve for the intersection of two lines expressed in Y = MX+B form (note that the lines cannot be vertical). Here we use deduction form. We just solve for X, since Y can be trivially found by using X. This is an example of how to use the algebra helpers. Notice that because this proof uses algebra helpers, the main steps of the proof are higher level and easier to follow by a human reader. (Contributed by David A. Wheeler, 15-Oct-2018.)
Hypotheses
Ref Expression
i2linesd.1  |-  ( ph  ->  A  e.  CC )
i2linesd.2  |-  ( ph  ->  B  e.  CC )
i2linesd.3  |-  ( ph  ->  C  e.  CC )
i2linesd.4  |-  ( ph  ->  D  e.  CC )
i2linesd.5  |-  ( ph  ->  X  e.  CC )
i2linesd.6  |-  ( ph  ->  Y  =  ( ( A  x.  X )  +  B ) )
i2linesd.7  |-  ( ph  ->  Y  =  ( ( C  x.  X )  +  D ) )
i2linesd.8  |-  ( ph  ->  ( A  -  C
)  =/=  0 )
Assertion
Ref Expression
i2linesd  |-  ( ph  ->  X  =  ( ( D  -  B )  /  ( A  -  C ) ) )

Proof of Theorem i2linesd
StepHypRef Expression
1 i2linesd.1 . . 3  |-  ( ph  ->  A  e.  CC )
2 i2linesd.3 . . 3  |-  ( ph  ->  C  e.  CC )
31, 2subcld 9719 . 2  |-  ( ph  ->  ( A  -  C
)  e.  CC )
4 i2linesd.5 . 2  |-  ( ph  ->  X  e.  CC )
5 i2linesd.8 . 2  |-  ( ph  ->  ( A  -  C
)  =/=  0 )
62, 4mulcld 9406 . . . 4  |-  ( ph  ->  ( C  x.  X
)  e.  CC )
7 i2linesd.4 . . . . 5  |-  ( ph  ->  D  e.  CC )
8 i2linesd.2 . . . . 5  |-  ( ph  ->  B  e.  CC )
97, 8subcld 9719 . . . 4  |-  ( ph  ->  ( D  -  B
)  e.  CC )
101, 4mulcld 9406 . . . . . 6  |-  ( ph  ->  ( A  x.  X
)  e.  CC )
11 i2linesd.6 . . . . . . 7  |-  ( ph  ->  Y  =  ( ( A  x.  X )  +  B ) )
12 i2linesd.7 . . . . . . 7  |-  ( ph  ->  Y  =  ( ( C  x.  X )  +  D ) )
1311, 12eqtr3d 2477 . . . . . 6  |-  ( ph  ->  ( ( A  x.  X )  +  B
)  =  ( ( C  x.  X )  +  D ) )
1410, 8, 13mvlraddd 31116 . . . . 5  |-  ( ph  ->  ( A  x.  X
)  =  ( ( ( C  x.  X
)  +  D )  -  B ) )
156, 7, 8, 14assraddsubd 31122 . . . 4  |-  ( ph  ->  ( A  x.  X
)  =  ( ( C  x.  X )  +  ( D  -  B ) ) )
166, 9, 15mvrladdd 31118 . . 3  |-  ( ph  ->  ( ( A  x.  X )  -  ( C  x.  X )
)  =  ( D  -  B ) )
171, 4, 2, 16joinlmulsubmuld 31126 . 2  |-  ( ph  ->  ( ( A  -  C )  x.  X
)  =  ( D  -  B ) )
183, 4, 5, 17mvllmuld 10163 1  |-  ( ph  ->  X  =  ( ( D  -  B )  /  ( A  -  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    =/= wne 2606  (class class class)co 6091   CCcc 9280   0cc0 9282    + caddc 9285    x. cmul 9287    - cmin 9595    / cdiv 9993
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 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994
This theorem is referenced by: (None)
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