Theorem i2linesi 48399

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 11573	. 2 ⊢ (𝐴 − 𝐶) ∈ ℂ
4		i2linesi.5	. 2 ⊢ 𝑋 ∈ ℂ
5		i2linesi.8	. 2 ⊢ (𝐴 − 𝐶) ≠ 0
6	2, 4	mulcli 11258	. . . 4 ⊢ (𝐶 · 𝑋) ∈ ℂ
7		i2linesi.4	. . . . 5 ⊢ 𝐷 ∈ ℂ
8		i2linesi.2	. . . . 5 ⊢ 𝐵 ∈ ℂ
9	7, 8	subcli 11573	. . . 4 ⊢ (𝐷 − 𝐵) ∈ ℂ
10	1, 4	mulcli 11258	. . . . . 6 ⊢ (𝐴 · 𝑋) ∈ ℂ
11		i2linesi.6	. . . . . . 7 ⊢ 𝑌 = ((𝐴 · 𝑋) + 𝐵)
12		i2linesi.7	. . . . . . 7 ⊢ 𝑌 = ((𝐶 · 𝑋) + 𝐷)
13	11, 12	eqtr3i 2755	. . . . . 6 ⊢ ((𝐴 · 𝑋) + 𝐵) = ((𝐶 · 𝑋) + 𝐷)
14	10, 8, 13	mvlraddi 48391	. . . . 5 ⊢ (𝐴 · 𝑋) = (((𝐶 · 𝑋) + 𝐷) − 𝐵)
15	6, 7, 8, 14	assraddsubi 48393	. . . 4 ⊢ (𝐴 · 𝑋) = ((𝐶 · 𝑋) + (𝐷 − 𝐵))
16	6, 9, 15	mvrladdi 48392	. . 3 ⊢ ((𝐴 · 𝑋) − (𝐶 · 𝑋)) = (𝐷 − 𝐵)
17	1, 4, 2, 16	joinlmulsubmuli 48396	. 2 ⊢ ((𝐴 − 𝐶) · 𝑋) = (𝐷 − 𝐵)
18	3, 4, 5, 17	mvllmuli 12085	1 ⊢ 𝑋 = ((𝐷 − 𝐵) / (𝐴 − 𝐶))

Syntax hints:   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  (class class class)co 7419  ℂcc 11143  0cc0 11145   + caddc 11148   · cmul 11150   − cmin 11481   / cdiv 11908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-div 11909

This theorem is referenced by: (None)