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Theorem i2linesi 42401
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 10106 . 2 (𝐴𝐶) ∈ ℂ
4 i2linesi.5 . 2 𝑋 ∈ ℂ
5 i2linesi.8 . 2 (𝐴𝐶) ≠ 0
62, 4mulcli 9798 . . . 4 (𝐶 · 𝑋) ∈ ℂ
7 i2linesi.4 . . . . 5 𝐷 ∈ ℂ
8 i2linesi.2 . . . . 5 𝐵 ∈ ℂ
97, 8subcli 10106 . . . 4 (𝐷𝐵) ∈ ℂ
101, 4mulcli 9798 . . . . . 6 (𝐴 · 𝑋) ∈ ℂ
11 i2linesi.6 . . . . . . 7 𝑌 = ((𝐴 · 𝑋) + 𝐵)
12 i2linesi.7 . . . . . . 7 𝑌 = ((𝐶 · 𝑋) + 𝐷)
1311, 12eqtr3i 2538 . . . . . 6 ((𝐴 · 𝑋) + 𝐵) = ((𝐶 · 𝑋) + 𝐷)
1410, 8, 13mvlraddi 42391 . . . . 5 (𝐴 · 𝑋) = (((𝐶 · 𝑋) + 𝐷) − 𝐵)
156, 7, 8, 14assraddsubi 42395 . . . 4 (𝐴 · 𝑋) = ((𝐶 · 𝑋) + (𝐷𝐵))
166, 9, 15mvrladdi 42393 . . 3 ((𝐴 · 𝑋) − (𝐶 · 𝑋)) = (𝐷𝐵)
171, 4, 2, 16joinlmulsubmuli 42398 . 2 ((𝐴𝐶) · 𝑋) = (𝐷𝐵)
183, 4, 5, 17mvllmuli 10605 1 𝑋 = ((𝐷𝐵) / (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wcel 1938  wne 2684  (class class class)co 6425  cc 9687  0cc0 9689   + caddc 9692   · cmul 9694  cmin 10015   / cdiv 10431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721  ax-resscn 9746  ax-1cn 9747  ax-icn 9748  ax-addcl 9749  ax-addrcl 9750  ax-mulcl 9751  ax-mulrcl 9752  ax-mulcom 9753  ax-addass 9754  ax-mulass 9755  ax-distr 9756  ax-i2m1 9757  ax-1ne0 9758  ax-1rid 9759  ax-rnegex 9760  ax-rrecex 9761  ax-cnre 9762  ax-pre-lttri 9763  ax-pre-lttrn 9764  ax-pre-ltadd 9765  ax-pre-mulgt0 9766
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-po 4853  df-so 4854  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6387  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-er 7503  df-en 7716  df-dom 7717  df-sdom 7718  df-pnf 9829  df-mnf 9830  df-xr 9831  df-ltxr 9832  df-le 9833  df-sub 10017  df-neg 10018  df-div 10432
This theorem is referenced by: (None)
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