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Theorem line 48653
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 = (LineM𝑊)
lines.s 𝑆 = (Scalar‘𝑊)
lines.k 𝐾 = (Base‘𝑆)
lines.p · = ( ·𝑠𝑊)
lines.a + = (+g𝑊)
lines.m = (-g𝑆)
lines.1 1 = (1r𝑆)
Assertion
Ref Expression
line ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → (𝑋𝐿𝑌) = {𝑝𝐵 ∣ ∃𝑡𝐾 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.
StepHypRef Expression
1 lines.b . . . . 5 𝐵 = (Base‘𝑊)
2 lines.l . . . . 5 𝐿 = (LineM𝑊)
3 lines.s . . . . 5 𝑆 = (Scalar‘𝑊)
4 lines.k . . . . 5 𝐾 = (Base‘𝑆)
5 lines.p . . . . 5 · = ( ·𝑠𝑊)
6 lines.a . . . . 5 + = (+g𝑊)
7 lines.m . . . . 5 = (-g𝑆)
8 lines.1 . . . . 5 1 = (1r𝑆)
91, 2, 3, 4, 5, 6, 7, 8lines 48652 . . . 4 (𝑊𝑉𝐿 = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
109oveqd 7448 . . 3 (𝑊𝑉 → (𝑋𝐿𝑌) = (𝑋(𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌))
1110adantr 480 . 2 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → (𝑋𝐿𝑌) = (𝑋(𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌))
12 eqidd 2738 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
13 oveq2 7439 . . . . . . . 8 (𝑥 = 𝑋 → (( 1 𝑡) · 𝑥) = (( 1 𝑡) · 𝑋))
14 oveq2 7439 . . . . . . . 8 (𝑦 = 𝑌 → (𝑡 · 𝑦) = (𝑡 · 𝑌))
1513, 14oveqan12d 7450 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌)))
1615eqeq2d 2748 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))))
1716rexbidv 3179 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))))
1817rabbidv 3444 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
1918adantl 481 . . 3 (((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
20 sneq 4636 . . . . 5 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2120difeq2d 4126 . . . 4 (𝑥 = 𝑋 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
2221adantl 481 . . 3 (((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) ∧ 𝑥 = 𝑋) → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
23 simpr1 1195 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → 𝑋𝐵)
24 id 22 . . . . . . . 8 (𝑋𝑌𝑋𝑌)
2524necomd 2996 . . . . . . 7 (𝑋𝑌𝑌𝑋)
2625anim2i 617 . . . . . 6 ((𝑌𝐵𝑋𝑌) → (𝑌𝐵𝑌𝑋))
27263adant1 1131 . . . . 5 ((𝑋𝐵𝑌𝐵𝑋𝑌) → (𝑌𝐵𝑌𝑋))
28 eldifsn 4786 . . . . 5 (𝑌 ∈ (𝐵 ∖ {𝑋}) ↔ (𝑌𝐵𝑌𝑋))
2927, 28sylibr 234 . . . 4 ((𝑋𝐵𝑌𝐵𝑋𝑌) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
3029adantl 481 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
311fvexi 6920 . . . . 5 𝐵 ∈ V
3231rabex 5339 . . . 4 {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))} ∈ V
3332a1i 11 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))} ∈ V)
3412, 19, 22, 23, 30, 33ovmpodx 7584 . 2 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → (𝑋(𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌) = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
3511, 34eqtrd 2777 1 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → (𝑋𝐿𝑌) = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  {csn 4626  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  +gcplusg 17297  Scalarcsca 17300   ·𝑠 cvsca 17301  -gcsg 18953  1rcur 20178  LineMcline 48648
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-line 48650
This theorem is referenced by: (None)
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