Theorem line 45966

Description: The line passing through the two different points 𝑋 and 𝑌 in a left module (or any extended structure having a base set, an addition, and a scalar multiplication). (Contributed by AV, 14-Jan-2023.)

Hypotheses

Ref	Expression
lines.b	⊢ 𝐵 = (Base‘𝑊)
lines.l	⊢ 𝐿 = (LineM‘𝑊)
lines.s	⊢ 𝑆 = (Scalar‘𝑊)
lines.k	⊢ 𝐾 = (Base‘𝑆)
lines.p	⊢ · = ( ·𝑠 ‘𝑊)
lines.a	⊢ + = (+g‘𝑊)
lines.m	⊢ − = (-g‘𝑆)
lines.1	⊢ 1 = (1r‘𝑆)

Assertion

Ref	Expression
line	⊢ ((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})

Distinct variable groups:   𝐵,𝑝   𝑡,𝐾   𝑡,𝑆   𝑊,𝑝,𝑡   𝑋,𝑝,𝑡   𝑌,𝑝,𝑡

Allowed substitution hints:   𝐵(𝑡)   + (𝑡,𝑝)   𝑆(𝑝)   · (𝑡,𝑝)   1 (𝑡,𝑝)   𝐾(𝑝)   𝐿(𝑡,𝑝)   − (𝑡,𝑝)   𝑉(𝑡,𝑝)

Proof of Theorem line

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lines.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
2		lines.l	. . . . 5 ⊢ 𝐿 = (LineM‘𝑊)
3		lines.s	. . . . 5 ⊢ 𝑆 = (Scalar‘𝑊)
4		lines.k	. . . . 5 ⊢ 𝐾 = (Base‘𝑆)
5		lines.p	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
6		lines.a	. . . . 5 ⊢ + = (+g‘𝑊)
7		lines.m	. . . . 5 ⊢ − = (-g‘𝑆)
8		lines.1	. . . . 5 ⊢ 1 = (1r‘𝑆)
9	1, 2, 3, 4, 5, 6, 7, 8	lines 45965	. . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐿 = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
10	9	oveqd 7272	. . 3 ⊢ (𝑊 ∈ 𝑉 → (𝑋𝐿𝑌) = (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌))
11	10	adantr 480	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌))
12		eqidd 2739	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
13		oveq2 7263	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (( 1 − 𝑡) · 𝑥) = (( 1 − 𝑡) · 𝑋))
14		oveq2 7263	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑡 · 𝑦) = (𝑡 · 𝑌))
15	13, 14	oveqan12d 7274	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌)))
16	15	eqeq2d 2749	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ 𝑝 = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))))
17	16	rexbidv 3225	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))))
18	17	rabbidv 3404	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})
19	18	adantl 481	. . 3 ⊢ (((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})
20		sneq 4568	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
21	20	difeq2d 4053	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
22	21	adantl 481	. . 3 ⊢ (((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑥 = 𝑋) → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
23		simpr1 1192	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ 𝐵)
24		id 22	. . . . . . . 8 ⊢ (𝑋 ≠ 𝑌 → 𝑋 ≠ 𝑌)
25	24	necomd 2998	. . . . . . 7 ⊢ (𝑋 ≠ 𝑌 → 𝑌 ≠ 𝑋)
26	25	anim2i 616	. . . . . 6 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 𝑋))
27	26	3adant1 1128	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 𝑋))
28		eldifsn 4717	. . . . 5 ⊢ (𝑌 ∈ (𝐵 ∖ {𝑋}) ↔ (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 𝑋))
29	27, 28	sylibr 233	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
30	29	adantl 481	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
31	1	fvexi 6770	. . . . 5 ⊢ 𝐵 ∈ V
32	31	rabex 5251	. . . 4 ⊢ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))} ∈ V
33	32	a1i 11	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))} ∈ V)
34	12, 19, 22, 23, 30, 33	ovmpodx 7402	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌) = {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})
35	11, 34	eqtrd 2778	1 ⊢ ((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064  {crab 3067  Vcvv 3422   ∖ cdif 3880  {csn 4558  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  Basecbs 16840  +_gcplusg 16888  Scalarcsca 16891   ·_𝑠 cvsca 16892  -_gcsg 18494  1_rcur 19652  Line_Mcline 45961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-line 45963

This theorem is referenced by: (None)