Theorem line 49315

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 49314	. . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐿 = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
10	9	oveqd 7408	. . 3 ⊢ (𝑊 ∈ 𝑉 → (𝑋𝐿𝑌) = (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌))
11	10	adantr 484	. 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 2762	. . 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 7399	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (( 1 − 𝑡) · 𝑥) = (( 1 − 𝑡) · 𝑋))
14		oveq2 7399	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑡 · 𝑦) = (𝑡 · 𝑌))
15	13, 14	oveqan12d 7410	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌)))
16	15	eqeq2d 2772	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ 𝑝 = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))))
17	16	rexbidv 3185	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))))
18	17	rabbidv 3420	. . . 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 485	. . 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 4589	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
21	20	difeq2d 4078	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
22	21	adantl 485	. . 3 ⊢ (((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑥 = 𝑋) → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
23		simpr1 1207	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ 𝐵)
24		id 22	. . . . . . . 8 ⊢ (𝑋 ≠ 𝑌 → 𝑋 ≠ 𝑌)
25	24	necomd 3011	. . . . . . 7 ⊢ (𝑋 ≠ 𝑌 → 𝑌 ≠ 𝑋)
26	25	anim2i 626	. . . . . 6 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 𝑋))
27	26	3adant1 1142	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 𝑋))
28		eldifsn 4743	. . . . 5 ⊢ (𝑌 ∈ (𝐵 ∖ {𝑋}) ↔ (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 𝑋))
29	27, 28	sylibr 236	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
30	29	adantl 485	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
31	1	fvexi 6876	. . . . 5 ⊢ 𝐵 ∈ V
32	31	rabex 5292	. . . 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 7542	. 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 2796	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 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085  {crab 3413  Vcvv 3453   ∖ cdif 3899  {csn 4579  ‘cfv 6516  (class class class)co 7391   ∈ cmpo 7393  Basecbs 17236  +_gcplusg 17277  Scalarcsca 17280   ·_𝑠 cvsca 17281  -_gcsg 18968  1_rcur 20218  Line_Mcline 49310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-line 49312

This theorem is referenced by: (None)