Theorem i2linesi 47825

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 11536	. 2 ⊢ (𝐴 − 𝐶) ∈ ℂ
4		i2linesi.5	. 2 ⊢ 𝑋 ∈ ℂ
5		i2linesi.8	. 2 ⊢ (𝐴 − 𝐶) ≠ 0
6	2, 4	mulcli 11221	. . . 4 ⊢ (𝐶 · 𝑋) ∈ ℂ
7		i2linesi.4	. . . . 5 ⊢ 𝐷 ∈ ℂ
8		i2linesi.2	. . . . 5 ⊢ 𝐵 ∈ ℂ
9	7, 8	subcli 11536	. . . 4 ⊢ (𝐷 − 𝐵) ∈ ℂ
10	1, 4	mulcli 11221	. . . . . 6 ⊢ (𝐴 · 𝑋) ∈ ℂ
11		i2linesi.6	. . . . . . 7 ⊢ 𝑌 = ((𝐴 · 𝑋) + 𝐵)
12		i2linesi.7	. . . . . . 7 ⊢ 𝑌 = ((𝐶 · 𝑋) + 𝐷)
13	11, 12	eqtr3i 2763	. . . . . 6 ⊢ ((𝐴 · 𝑋) + 𝐵) = ((𝐶 · 𝑋) + 𝐷)
14	10, 8, 13	mvlraddi 47817	. . . . 5 ⊢ (𝐴 · 𝑋) = (((𝐶 · 𝑋) + 𝐷) − 𝐵)
15	6, 7, 8, 14	assraddsubi 47819	. . . 4 ⊢ (𝐴 · 𝑋) = ((𝐶 · 𝑋) + (𝐷 − 𝐵))
16	6, 9, 15	mvrladdi 47818	. . 3 ⊢ ((𝐴 · 𝑋) − (𝐶 · 𝑋)) = (𝐷 − 𝐵)
17	1, 4, 2, 16	joinlmulsubmuli 47822	. 2 ⊢ ((𝐴 − 𝐶) · 𝑋) = (𝐷 − 𝐵)
18	3, 4, 5, 17	mvllmuli 12047	1 ⊢ 𝑋 = ((𝐷 − 𝐵) / (𝐴 − 𝐶))

Syntax hints:   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  (class class class)co 7409  ℂcc 11108  0cc0 11110   + caddc 11113   · cmul 11115   − cmin 11444   / cdiv 11871

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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872

This theorem is referenced by: (None)