Theorem i2linesi 43612

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	⊢ (𝐴 − 𝐶) ≠ 0

Assertion

Ref	Expression
i2linesi	⊢ 𝑋 = ((𝐷 − 𝐵) / (𝐴 − 𝐶))

Proof of Theorem i2linesi

Step	Hyp	Ref	Expression
1		i2linesi.1	. . 3 ⊢ 𝐴 ∈ ℂ
2		i2linesi.3	. . 3 ⊢ 𝐶 ∈ ℂ
3	1, 2	subcli 10699	. 2 ⊢ (𝐴 − 𝐶) ∈ ℂ
4		i2linesi.5	. 2 ⊢ 𝑋 ∈ ℂ
5		i2linesi.8	. 2 ⊢ (𝐴 − 𝐶) ≠ 0
6	2, 4	mulcli 10384	. . . 4 ⊢ (𝐶 · 𝑋) ∈ ℂ
7		i2linesi.4	. . . . 5 ⊢ 𝐷 ∈ ℂ
8		i2linesi.2	. . . . 5 ⊢ 𝐵 ∈ ℂ
9	7, 8	subcli 10699	. . . 4 ⊢ (𝐷 − 𝐵) ∈ ℂ
10	1, 4	mulcli 10384	. . . . . 6 ⊢ (𝐴 · 𝑋) ∈ ℂ
11		i2linesi.6	. . . . . . 7 ⊢ 𝑌 = ((𝐴 · 𝑋) + 𝐵)
12		i2linesi.7	. . . . . . 7 ⊢ 𝑌 = ((𝐶 · 𝑋) + 𝐷)
13	11, 12	eqtr3i 2803	. . . . . 6 ⊢ ((𝐴 · 𝑋) + 𝐵) = ((𝐶 · 𝑋) + 𝐷)
14	10, 8, 13	mvlraddi 43604	. . . . 5 ⊢ (𝐴 · 𝑋) = (((𝐶 · 𝑋) + 𝐷) − 𝐵)
15	6, 7, 8, 14	assraddsubi 43606	. . . 4 ⊢ (𝐴 · 𝑋) = ((𝐶 · 𝑋) + (𝐷 − 𝐵))
16	6, 9, 15	mvrladdi 43605	. . 3 ⊢ ((𝐴 · 𝑋) − (𝐶 · 𝑋)) = (𝐷 − 𝐵)
17	1, 4, 2, 16	joinlmulsubmuli 43609	. 2 ⊢ ((𝐴 − 𝐶) · 𝑋) = (𝐷 − 𝐵)
18	3, 4, 5, 17	mvllmuli 11208	1 ⊢ 𝑋 = ((𝐷 − 𝐵) / (𝐴 − 𝐶))

Syntax hints:   = wceq 1601   ∈ wcel 2106   ≠ wne 2968  (class class class)co 6922  ℂcc 10270  0cc0 10272   + caddc 10275   · cmul 10277   − cmin 10606   / cdiv 11032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-po 5274  df-so 5275  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033

This theorem is referenced by: (None)