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Theorem i2linesd 45302
 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 (𝜑𝐴 ∈ ℂ)
i2linesd.2 (𝜑𝐵 ∈ ℂ)
i2linesd.3 (𝜑𝐶 ∈ ℂ)
i2linesd.4 (𝜑𝐷 ∈ ℂ)
i2linesd.5 (𝜑𝑋 ∈ ℂ)
i2linesd.6 (𝜑𝑌 = ((𝐴 · 𝑋) + 𝐵))
i2linesd.7 (𝜑𝑌 = ((𝐶 · 𝑋) + 𝐷))
i2linesd.8 (𝜑 → (𝐴𝐶) ≠ 0)
Assertion
Ref Expression
i2linesd (𝜑𝑋 = ((𝐷𝐵) / (𝐴𝐶)))

Proof of Theorem i2linesd
StepHypRef Expression
1 i2linesd.1 . . 3 (𝜑𝐴 ∈ ℂ)
2 i2linesd.3 . . 3 (𝜑𝐶 ∈ ℂ)
31, 2subcld 10986 . 2 (𝜑 → (𝐴𝐶) ∈ ℂ)
4 i2linesd.5 . 2 (𝜑𝑋 ∈ ℂ)
5 i2linesd.8 . 2 (𝜑 → (𝐴𝐶) ≠ 0)
62, 4mulcld 10650 . . . 4 (𝜑 → (𝐶 · 𝑋) ∈ ℂ)
7 i2linesd.4 . . . . 5 (𝜑𝐷 ∈ ℂ)
8 i2linesd.2 . . . . 5 (𝜑𝐵 ∈ ℂ)
97, 8subcld 10986 . . . 4 (𝜑 → (𝐷𝐵) ∈ ℂ)
101, 4mulcld 10650 . . . . . 6 (𝜑 → (𝐴 · 𝑋) ∈ ℂ)
11 i2linesd.6 . . . . . . 7 (𝜑𝑌 = ((𝐴 · 𝑋) + 𝐵))
12 i2linesd.7 . . . . . . 7 (𝜑𝑌 = ((𝐶 · 𝑋) + 𝐷))
1311, 12eqtr3d 2835 . . . . . 6 (𝜑 → ((𝐴 · 𝑋) + 𝐵) = ((𝐶 · 𝑋) + 𝐷))
1410, 8, 13mvlraddd 11039 . . . . 5 (𝜑 → (𝐴 · 𝑋) = (((𝐶 · 𝑋) + 𝐷) − 𝐵))
156, 7, 8, 14assraddsubd 11043 . . . 4 (𝜑 → (𝐴 · 𝑋) = ((𝐶 · 𝑋) + (𝐷𝐵)))
166, 9, 15mvrladdd 11042 . . 3 (𝜑 → ((𝐴 · 𝑋) − (𝐶 · 𝑋)) = (𝐷𝐵))
171, 4, 2, 16joinlmulsubmuld 45297 . 2 (𝜑 → ((𝐴𝐶) · 𝑋) = (𝐷𝐵))
183, 4, 5, 17mvllmuld 11461 1 (𝜑𝑋 = ((𝐷𝐵) / (𝐴𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  (class class class)co 7135  ℂcc 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  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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