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Theorem line 48582
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 48581 . . . 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 2736 . . 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 2746 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))))
1716rexbidv 3177 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))))
1817rabbidv 3441 . . . 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 4641 . . . . 5 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2120difeq2d 4136 . . . 4 (𝑥 = 𝑋 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
2221adantl 481 . . 3 (((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) ∧ 𝑥 = 𝑋) → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
23 simpr1 1193 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → 𝑋𝐵)
24 id 22 . . . . . . . 8 (𝑋𝑌𝑋𝑌)
2524necomd 2994 . . . . . . 7 (𝑋𝑌𝑌𝑋)
2625anim2i 617 . . . . . 6 ((𝑌𝐵𝑋𝑌) → (𝑌𝐵𝑌𝑋))
27263adant1 1129 . . . . 5 ((𝑋𝐵𝑌𝐵𝑋𝑌) → (𝑌𝐵𝑌𝑋))
28 eldifsn 4791 . . . . 5 (𝑌 ∈ (𝐵 ∖ {𝑋}) ↔ (𝑌𝐵𝑌𝑋))
2927, 28sylibr 234 . . . 4 ((𝑋𝐵𝑌𝐵𝑋𝑌) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
3029adantl 481 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
311fvexi 6921 . . . . 5 𝐵 ∈ V
3231rabex 5345 . . . 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 2775 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 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  {csn 4631  cfv 6563  (class class class)co 7431  cmpo 7433  Basecbs 17245  +gcplusg 17298  Scalarcsca 17301   ·𝑠 cvsca 17302  -gcsg 18966  1rcur 20199  LineMcline 48577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-line 48579
This theorem is referenced by: (None)
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