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Theorem i2linesd 47728
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 11567 . 2 (𝜑 → (𝐴𝐶) ∈ ℂ)
4 i2linesd.5 . 2 (𝜑𝑋 ∈ ℂ)
5 i2linesd.8 . 2 (𝜑 → (𝐴𝐶) ≠ 0)
62, 4mulcld 11230 . . . 4 (𝜑 → (𝐶 · 𝑋) ∈ ℂ)
7 i2linesd.4 . . . . 5 (𝜑𝐷 ∈ ℂ)
8 i2linesd.2 . . . . 5 (𝜑𝐵 ∈ ℂ)
97, 8subcld 11567 . . . 4 (𝜑 → (𝐷𝐵) ∈ ℂ)
101, 4mulcld 11230 . . . . . 6 (𝜑 → (𝐴 · 𝑋) ∈ ℂ)
11 i2linesd.6 . . . . . . 7 (𝜑𝑌 = ((𝐴 · 𝑋) + 𝐵))
12 i2linesd.7 . . . . . . 7 (𝜑𝑌 = ((𝐶 · 𝑋) + 𝐷))
1311, 12eqtr3d 2775 . . . . . 6 (𝜑 → ((𝐴 · 𝑋) + 𝐵) = ((𝐶 · 𝑋) + 𝐷))
1410, 8, 13mvlraddd 11620 . . . . 5 (𝜑 → (𝐴 · 𝑋) = (((𝐶 · 𝑋) + 𝐷) − 𝐵))
156, 7, 8, 14assraddsubd 11624 . . . 4 (𝜑 → (𝐴 · 𝑋) = ((𝐶 · 𝑋) + (𝐷𝐵)))
166, 9, 15mvrladdd 11623 . . 3 (𝜑 → ((𝐴 · 𝑋) − (𝐶 · 𝑋)) = (𝐷𝐵))
171, 4, 2, 16joinlmulsubmuld 47723 . 2 (𝜑 → ((𝐴𝐶) · 𝑋) = (𝐷𝐵))
183, 4, 5, 17mvllmuld 12042 1 (𝜑𝑋 = ((𝐷𝐵) / (𝐴𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wne 2941  (class class class)co 7404  cc 11104  0cc0 11106   + caddc 11109   · cmul 11111  cmin 11440   / cdiv 11867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868
This theorem is referenced by: (None)
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