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Theorem line 47796
Description: The line passing through the two different points 𝑋 and π‘Œ 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 𝐡 = (Baseβ€˜π‘Š)
lines.l 𝐿 = (LineMβ€˜π‘Š)
lines.s 𝑆 = (Scalarβ€˜π‘Š)
lines.k 𝐾 = (Baseβ€˜π‘†)
lines.p Β· = ( ·𝑠 β€˜π‘Š)
lines.a + = (+gβ€˜π‘Š)
lines.m βˆ’ = (-gβ€˜π‘†)
lines.1 1 = (1rβ€˜π‘†)
Assertion
Ref Expression
line ((π‘Š ∈ 𝑉 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ)) β†’ (π‘‹πΏπ‘Œ) = {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· 𝑋) + (𝑑 Β· π‘Œ))})
Distinct variable groups:   𝐡,𝑝   𝑑,𝐾   𝑑,𝑆   π‘Š,𝑝,𝑑   𝑋,𝑝,𝑑   π‘Œ,𝑝,𝑑
Allowed substitution hints:   𝐡(𝑑)   + (𝑑,𝑝)   𝑆(𝑝)   Β· (𝑑,𝑝)   1 (𝑑,𝑝)   𝐾(𝑝)   𝐿(𝑑,𝑝)   βˆ’ (𝑑,𝑝)   𝑉(𝑑,𝑝)

Proof of Theorem line
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lines.b . . . . 5 𝐡 = (Baseβ€˜π‘Š)
2 lines.l . . . . 5 𝐿 = (LineMβ€˜π‘Š)
3 lines.s . . . . 5 𝑆 = (Scalarβ€˜π‘Š)
4 lines.k . . . . 5 𝐾 = (Baseβ€˜π‘†)
5 lines.p . . . . 5 Β· = ( ·𝑠 β€˜π‘Š)
6 lines.a . . . . 5 + = (+gβ€˜π‘Š)
7 lines.m . . . . 5 βˆ’ = (-gβ€˜π‘†)
8 lines.1 . . . . 5 1 = (1rβ€˜π‘†)
91, 2, 3, 4, 5, 6, 7, 8lines 47795 . . . 4 (π‘Š ∈ 𝑉 β†’ 𝐿 = (π‘₯ ∈ 𝐡, 𝑦 ∈ (𝐡 βˆ– {π‘₯}) ↦ {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· π‘₯) + (𝑑 Β· 𝑦))}))
109oveqd 7432 . . 3 (π‘Š ∈ 𝑉 β†’ (π‘‹πΏπ‘Œ) = (𝑋(π‘₯ ∈ 𝐡, 𝑦 ∈ (𝐡 βˆ– {π‘₯}) ↦ {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· π‘₯) + (𝑑 Β· 𝑦))})π‘Œ))
1110adantr 480 . 2 ((π‘Š ∈ 𝑉 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ)) β†’ (π‘‹πΏπ‘Œ) = (𝑋(π‘₯ ∈ 𝐡, 𝑦 ∈ (𝐡 βˆ– {π‘₯}) ↦ {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· π‘₯) + (𝑑 Β· 𝑦))})π‘Œ))
12 eqidd 2729 . . 3 ((π‘Š ∈ 𝑉 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ)) β†’ (π‘₯ ∈ 𝐡, 𝑦 ∈ (𝐡 βˆ– {π‘₯}) ↦ {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· π‘₯) + (𝑑 Β· 𝑦))}) = (π‘₯ ∈ 𝐡, 𝑦 ∈ (𝐡 βˆ– {π‘₯}) ↦ {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· π‘₯) + (𝑑 Β· 𝑦))}))
13 oveq2 7423 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (( 1 βˆ’ 𝑑) Β· π‘₯) = (( 1 βˆ’ 𝑑) Β· 𝑋))
14 oveq2 7423 . . . . . . . 8 (𝑦 = π‘Œ β†’ (𝑑 Β· 𝑦) = (𝑑 Β· π‘Œ))
1513, 14oveqan12d 7434 . . . . . . 7 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ ((( 1 βˆ’ 𝑑) Β· π‘₯) + (𝑑 Β· 𝑦)) = ((( 1 βˆ’ 𝑑) Β· 𝑋) + (𝑑 Β· π‘Œ)))
1615eqeq2d 2739 . . . . . 6 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (𝑝 = ((( 1 βˆ’ 𝑑) Β· π‘₯) + (𝑑 Β· 𝑦)) ↔ 𝑝 = ((( 1 βˆ’ 𝑑) Β· 𝑋) + (𝑑 Β· π‘Œ))))
1716rexbidv 3174 . . . . 5 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· π‘₯) + (𝑑 Β· 𝑦)) ↔ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· 𝑋) + (𝑑 Β· π‘Œ))))
1817rabbidv 3436 . . . 4 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· π‘₯) + (𝑑 Β· 𝑦))} = {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· 𝑋) + (𝑑 Β· π‘Œ))})
1918adantl 481 . . 3 (((π‘Š ∈ 𝑉 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ)) ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) β†’ {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· π‘₯) + (𝑑 Β· 𝑦))} = {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· 𝑋) + (𝑑 Β· π‘Œ))})
20 sneq 4635 . . . . 5 (π‘₯ = 𝑋 β†’ {π‘₯} = {𝑋})
2120difeq2d 4119 . . . 4 (π‘₯ = 𝑋 β†’ (𝐡 βˆ– {π‘₯}) = (𝐡 βˆ– {𝑋}))
2221adantl 481 . . 3 (((π‘Š ∈ 𝑉 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ)) ∧ π‘₯ = 𝑋) β†’ (𝐡 βˆ– {π‘₯}) = (𝐡 βˆ– {𝑋}))
23 simpr1 1192 . . 3 ((π‘Š ∈ 𝑉 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ)) β†’ 𝑋 ∈ 𝐡)
24 id 22 . . . . . . . 8 (𝑋 β‰  π‘Œ β†’ 𝑋 β‰  π‘Œ)
2524necomd 2992 . . . . . . 7 (𝑋 β‰  π‘Œ β†’ π‘Œ β‰  𝑋)
2625anim2i 616 . . . . . 6 ((π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ) β†’ (π‘Œ ∈ 𝐡 ∧ π‘Œ β‰  𝑋))
27263adant1 1128 . . . . 5 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ) β†’ (π‘Œ ∈ 𝐡 ∧ π‘Œ β‰  𝑋))
28 eldifsn 4787 . . . . 5 (π‘Œ ∈ (𝐡 βˆ– {𝑋}) ↔ (π‘Œ ∈ 𝐡 ∧ π‘Œ β‰  𝑋))
2927, 28sylibr 233 . . . 4 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ) β†’ π‘Œ ∈ (𝐡 βˆ– {𝑋}))
3029adantl 481 . . 3 ((π‘Š ∈ 𝑉 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ)) β†’ π‘Œ ∈ (𝐡 βˆ– {𝑋}))
311fvexi 6906 . . . . 5 𝐡 ∈ V
3231rabex 5329 . . . 4 {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· 𝑋) + (𝑑 Β· π‘Œ))} ∈ V
3332a1i 11 . . 3 ((π‘Š ∈ 𝑉 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ)) β†’ {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· 𝑋) + (𝑑 Β· π‘Œ))} ∈ V)
3412, 19, 22, 23, 30, 33ovmpodx 7567 . 2 ((π‘Š ∈ 𝑉 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ)) β†’ (𝑋(π‘₯ ∈ 𝐡, 𝑦 ∈ (𝐡 βˆ– {π‘₯}) ↦ {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· π‘₯) + (𝑑 Β· 𝑦))})π‘Œ) = {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· 𝑋) + (𝑑 Β· π‘Œ))})
3511, 34eqtrd 2768 1 ((π‘Š ∈ 𝑉 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ)) β†’ (π‘‹πΏπ‘Œ) = {𝑝 ∈ 𝐡 ∣ βˆƒπ‘‘ ∈ 𝐾 𝑝 = ((( 1 βˆ’ 𝑑) Β· 𝑋) + (𝑑 Β· π‘Œ))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2936  βˆƒwrex 3066  {crab 3428  Vcvv 3470   βˆ– cdif 3942  {csn 4625  β€˜cfv 6543  (class class class)co 7415   ∈ cmpo 7417  Basecbs 17174  +gcplusg 17227  Scalarcsca 17230   ·𝑠 cvsca 17231  -gcsg 18886  1rcur 20115  LineMcline 47791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7988  df-2nd 7989  df-line 47793
This theorem is referenced by: (None)
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