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Theorem line 45754
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 45753 . . . 4 (𝑊𝑉𝐿 = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
109oveqd 7230 . . 3 (𝑊𝑉 → (𝑋𝐿𝑌) = (𝑋(𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌))
1110adantr 484 . 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 7221 . . . . . . . 8 (𝑥 = 𝑋 → (( 1 𝑡) · 𝑥) = (( 1 𝑡) · 𝑋))
14 oveq2 7221 . . . . . . . 8 (𝑦 = 𝑌 → (𝑡 · 𝑦) = (𝑡 · 𝑌))
1513, 14oveqan12d 7232 . . . . . . 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 3216 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))))
1817rabbidv 3390 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
1918adantl 485 . . 3 (((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
20 sneq 4551 . . . . 5 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2120difeq2d 4037 . . . 4 (𝑥 = 𝑋 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
2221adantl 485 . . 3 (((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) ∧ 𝑥 = 𝑋) → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
23 simpr1 1196 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → 𝑋𝐵)
24 id 22 . . . . . . . 8 (𝑋𝑌𝑋𝑌)
2524necomd 2996 . . . . . . 7 (𝑋𝑌𝑌𝑋)
2625anim2i 620 . . . . . 6 ((𝑌𝐵𝑋𝑌) → (𝑌𝐵𝑌𝑋))
27263adant1 1132 . . . . 5 ((𝑋𝐵𝑌𝐵𝑋𝑌) → (𝑌𝐵𝑌𝑋))
28 eldifsn 4700 . . . . 5 (𝑌 ∈ (𝐵 ∖ {𝑋}) ↔ (𝑌𝐵𝑌𝑋))
2927, 28sylibr 237 . . . 4 ((𝑋𝐵𝑌𝐵𝑋𝑌) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
3029adantl 485 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
311fvexi 6731 . . . . 5 𝐵 ∈ V
3231rabex 5225 . . . 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 7360 . 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 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940  wrex 3062  {crab 3065  Vcvv 3408  cdif 3863  {csn 4541  cfv 6380  (class class class)co 7213  cmpo 7215  Basecbs 16760  +gcplusg 16802  Scalarcsca 16805   ·𝑠 cvsca 16806  -gcsg 18367  1rcur 19516  LineMcline 45749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-line 45751
This theorem is referenced by: (None)
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