Theorem i2linesd 44954

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	⊢ (𝜑 → 𝐴 ∈ ℂ)
i2linesd.2	⊢ (𝜑 → 𝐵 ∈ ℂ)
i2linesd.3	⊢ (𝜑 → 𝐶 ∈ ℂ)
i2linesd.4	⊢ (𝜑 → 𝐷 ∈ ℂ)
i2linesd.5	⊢ (𝜑 → 𝑋 ∈ ℂ)
i2linesd.6	⊢ (𝜑 → 𝑌 = ((𝐴 · 𝑋) + 𝐵))
i2linesd.7	⊢ (𝜑 → 𝑌 = ((𝐶 · 𝑋) + 𝐷))
i2linesd.8	⊢ (𝜑 → (𝐴 − 𝐶) ≠ 0)

Assertion

Ref	Expression
i2linesd	⊢ (𝜑 → 𝑋 = ((𝐷 − 𝐵) / (𝐴 − 𝐶)))

Proof of Theorem i2linesd

Step	Hyp	Ref	Expression
1		i2linesd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		i2linesd.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
3	1, 2	subcld 10990	. 2 ⊢ (𝜑 → (𝐴 − 𝐶) ∈ ℂ)
4		i2linesd.5	. 2 ⊢ (𝜑 → 𝑋 ∈ ℂ)
5		i2linesd.8	. 2 ⊢ (𝜑 → (𝐴 − 𝐶) ≠ 0)
6	2, 4	mulcld 10654	. . . 4 ⊢ (𝜑 → (𝐶 · 𝑋) ∈ ℂ)
7		i2linesd.4	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ)
8		i2linesd.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ)
9	7, 8	subcld 10990	. . . 4 ⊢ (𝜑 → (𝐷 − 𝐵) ∈ ℂ)
10	1, 4	mulcld 10654	. . . . . 6 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ ℂ)
11		i2linesd.6	. . . . . . 7 ⊢ (𝜑 → 𝑌 = ((𝐴 · 𝑋) + 𝐵))
12		i2linesd.7	. . . . . . 7 ⊢ (𝜑 → 𝑌 = ((𝐶 · 𝑋) + 𝐷))
13	11, 12	eqtr3d 2857	. . . . . 6 ⊢ (𝜑 → ((𝐴 · 𝑋) + 𝐵) = ((𝐶 · 𝑋) + 𝐷))
14	10, 8, 13	mvlraddd 11043	. . . . 5 ⊢ (𝜑 → (𝐴 · 𝑋) = (((𝐶 · 𝑋) + 𝐷) − 𝐵))
15	6, 7, 8, 14	assraddsubd 11047	. . . 4 ⊢ (𝜑 → (𝐴 · 𝑋) = ((𝐶 · 𝑋) + (𝐷 − 𝐵)))
16	6, 9, 15	mvrladdd 11046	. . 3 ⊢ (𝜑 → ((𝐴 · 𝑋) − (𝐶 · 𝑋)) = (𝐷 − 𝐵))
17	1, 4, 2, 16	joinlmulsubmuld 44949	. 2 ⊢ (𝜑 → ((𝐴 − 𝐶) · 𝑋) = (𝐷 − 𝐵))
18	3, 4, 5, 17	mvllmuld 11465	1 ⊢ (𝜑 → 𝑋 = ((𝐷 − 𝐵) / (𝐴 − 𝐶)))

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113   ≠ wne 3015  (class class class)co 7149  ℂcc 10528  0cc0 10530   + caddc 10533   · cmul 10535   − cmin 10863   / cdiv 11290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-er 8282  df-en 8503  df-dom 8504  df-sdom 8505  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291

This theorem is referenced by: (None)