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Theorem i2linesi 45245
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 10951 . 2 (𝐴𝐶) ∈ ℂ
4 i2linesi.5 . 2 𝑋 ∈ ℂ
5 i2linesi.8 . 2 (𝐴𝐶) ≠ 0
62, 4mulcli 10637 . . . 4 (𝐶 · 𝑋) ∈ ℂ
7 i2linesi.4 . . . . 5 𝐷 ∈ ℂ
8 i2linesi.2 . . . . 5 𝐵 ∈ ℂ
97, 8subcli 10951 . . . 4 (𝐷𝐵) ∈ ℂ
101, 4mulcli 10637 . . . . . 6 (𝐴 · 𝑋) ∈ ℂ
11 i2linesi.6 . . . . . . 7 𝑌 = ((𝐴 · 𝑋) + 𝐵)
12 i2linesi.7 . . . . . . 7 𝑌 = ((𝐶 · 𝑋) + 𝐷)
1311, 12eqtr3i 2847 . . . . . 6 ((𝐴 · 𝑋) + 𝐵) = ((𝐶 · 𝑋) + 𝐷)
1410, 8, 13mvlraddi 45237 . . . . 5 (𝐴 · 𝑋) = (((𝐶 · 𝑋) + 𝐷) − 𝐵)
156, 7, 8, 14assraddsubi 45239 . . . 4 (𝐴 · 𝑋) = ((𝐶 · 𝑋) + (𝐷𝐵))
166, 9, 15mvrladdi 45238 . . 3 ((𝐴 · 𝑋) − (𝐶 · 𝑋)) = (𝐷𝐵)
171, 4, 2, 16joinlmulsubmuli 45242 . 2 ((𝐴𝐶) · 𝑋) = (𝐷𝐵)
183, 4, 5, 17mvllmuli 11462 1 𝑋 = ((𝐷𝐵) / (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2114  wne 3011  (class class class)co 7140  cc 10524  0cc0 10526   + caddc 10529   · cmul 10531  cmin 10859   / cdiv 11286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-po 5451  df-so 5452  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287
This theorem is referenced by: (None)
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