Theorem i2linesi 49884

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 11443	. 2 ⊢ (𝐴 − 𝐶) ∈ ℂ
4		i2linesi.5	. 2 ⊢ 𝑋 ∈ ℂ
5		i2linesi.8	. 2 ⊢ (𝐴 − 𝐶) ≠ 0
6	2, 4	mulcli 11125	. . . 4 ⊢ (𝐶 · 𝑋) ∈ ℂ
7		i2linesi.4	. . . . 5 ⊢ 𝐷 ∈ ℂ
8		i2linesi.2	. . . . 5 ⊢ 𝐵 ∈ ℂ
9	7, 8	subcli 11443	. . . 4 ⊢ (𝐷 − 𝐵) ∈ ℂ
10	1, 4	mulcli 11125	. . . . . 6 ⊢ (𝐴 · 𝑋) ∈ ℂ
11		i2linesi.6	. . . . . . 7 ⊢ 𝑌 = ((𝐴 · 𝑋) + 𝐵)
12		i2linesi.7	. . . . . . 7 ⊢ 𝑌 = ((𝐶 · 𝑋) + 𝐷)
13	11, 12	eqtr3i 2756	. . . . . 6 ⊢ ((𝐴 · 𝑋) + 𝐵) = ((𝐶 · 𝑋) + 𝐷)
14	10, 8, 13	mvlraddi 49877	. . . . 5 ⊢ (𝐴 · 𝑋) = (((𝐶 · 𝑋) + 𝐷) − 𝐵)
15	6, 7, 8, 14	assraddsubi 49878	. . . 4 ⊢ (𝐴 · 𝑋) = ((𝐶 · 𝑋) + (𝐷 − 𝐵))
16	6, 9, 15	mvrladdi 11384	. . 3 ⊢ ((𝐴 · 𝑋) − (𝐶 · 𝑋)) = (𝐷 − 𝐵)
17	1, 4, 2, 16	joinlmulsubmuli 49881	. 2 ⊢ ((𝐴 − 𝐶) · 𝑋) = (𝐷 − 𝐵)
18	3, 4, 5, 17	mvllmuli 11960	1 ⊢ 𝑋 = ((𝐷 − 𝐵) / (𝐴 − 𝐶))

Syntax hints:   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  (class class class)co 7352  ℂcc 11010  0cc0 11012   + caddc 11015   · cmul 11017   − cmin 11350   / cdiv 11780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-resscn 11069  ax-1cn 11070  ax-icn 11071  ax-addcl 11072  ax-addrcl 11073  ax-mulcl 11074  ax-mulrcl 11075  ax-mulcom 11076  ax-addass 11077  ax-mulass 11078  ax-distr 11079  ax-i2m1 11080  ax-1ne0 11081  ax-1rid 11082  ax-rnegex 11083  ax-rrecex 11084  ax-cnre 11085  ax-pre-lttri 11086  ax-pre-lttrn 11087  ax-pre-ltadd 11088  ax-pre-mulgt0 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-sub 11352  df-neg 11353  df-div 11781

This theorem is referenced by: (None)