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Theorem i2linesi 44807
Description: Solve for the intersection of two lines expressed in Y = MX+B form (note that the lines cannot be vertical). Here we use inference 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, 11-Oct-2018.)
Hypotheses
Ref Expression
i2linesi.1 𝐴 ∈ ℂ
i2linesi.2 𝐵 ∈ ℂ
i2linesi.3 𝐶 ∈ ℂ
i2linesi.4 𝐷 ∈ ℂ
i2linesi.5 𝑋 ∈ ℂ
i2linesi.6 𝑌 = ((𝐴 · 𝑋) + 𝐵)
i2linesi.7 𝑌 = ((𝐶 · 𝑋) + 𝐷)
i2linesi.8 (𝐴𝐶) ≠ 0
Assertion
Ref Expression
i2linesi 𝑋 = ((𝐷𝐵) / (𝐴𝐶))

Proof of Theorem i2linesi
StepHypRef Expression
1 i2linesi.1 . . 3 𝐴 ∈ ℂ
2 i2linesi.3 . . 3 𝐶 ∈ ℂ
31, 2subcli 10950 . 2 (𝐴𝐶) ∈ ℂ
4 i2linesi.5 . 2 𝑋 ∈ ℂ
5 i2linesi.8 . 2 (𝐴𝐶) ≠ 0
62, 4mulcli 10636 . . . 4 (𝐶 · 𝑋) ∈ ℂ
7 i2linesi.4 . . . . 5 𝐷 ∈ ℂ
8 i2linesi.2 . . . . 5 𝐵 ∈ ℂ
97, 8subcli 10950 . . . 4 (𝐷𝐵) ∈ ℂ
101, 4mulcli 10636 . . . . . 6 (𝐴 · 𝑋) ∈ ℂ
11 i2linesi.6 . . . . . . 7 𝑌 = ((𝐴 · 𝑋) + 𝐵)
12 i2linesi.7 . . . . . . 7 𝑌 = ((𝐶 · 𝑋) + 𝐷)
1311, 12eqtr3i 2843 . . . . . 6 ((𝐴 · 𝑋) + 𝐵) = ((𝐶 · 𝑋) + 𝐷)
1410, 8, 13mvlraddi 44799 . . . . 5 (𝐴 · 𝑋) = (((𝐶 · 𝑋) + 𝐷) − 𝐵)
156, 7, 8, 14assraddsubi 44801 . . . 4 (𝐴 · 𝑋) = ((𝐶 · 𝑋) + (𝐷𝐵))
166, 9, 15mvrladdi 44800 . . 3 ((𝐴 · 𝑋) − (𝐶 · 𝑋)) = (𝐷𝐵)
171, 4, 2, 16joinlmulsubmuli 44804 . 2 ((𝐴𝐶) · 𝑋) = (𝐷𝐵)
183, 4, 5, 17mvllmuli 11461 1 𝑋 = ((𝐷𝐵) / (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  wne 3013  (class class class)co 7145  cc 10523  0cc0 10525   + caddc 10528   · cmul 10530  cmin 10858   / cdiv 11285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286
This theorem is referenced by: (None)
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