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Theorem line 46892
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 46891 . . . 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 7379 . . 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 482 . 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 2738 . . 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 7370 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (( 1 βˆ’ 𝑑) Β· π‘₯) = (( 1 βˆ’ 𝑑) Β· 𝑋))
14 oveq2 7370 . . . . . . . 8 (𝑦 = π‘Œ β†’ (𝑑 Β· 𝑦) = (𝑑 Β· π‘Œ))
1513, 14oveqan12d 7381 . . . . . . 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 2748 . . . . . 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 3176 . . . . 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 3418 . . . 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 483 . . 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 4601 . . . . 5 (π‘₯ = 𝑋 β†’ {π‘₯} = {𝑋})
2120difeq2d 4087 . . . 4 (π‘₯ = 𝑋 β†’ (𝐡 βˆ– {π‘₯}) = (𝐡 βˆ– {𝑋}))
2221adantl 483 . . 3 (((π‘Š ∈ 𝑉 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ)) ∧ π‘₯ = 𝑋) β†’ (𝐡 βˆ– {π‘₯}) = (𝐡 βˆ– {𝑋}))
23 simpr1 1195 . . 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 3000 . . . . . . 7 (𝑋 β‰  π‘Œ β†’ π‘Œ β‰  𝑋)
2625anim2i 618 . . . . . 6 ((π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ) β†’ (π‘Œ ∈ 𝐡 ∧ π‘Œ β‰  𝑋))
27263adant1 1131 . . . . 5 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ) β†’ (π‘Œ ∈ 𝐡 ∧ π‘Œ β‰  𝑋))
28 eldifsn 4752 . . . . 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 483 . . 3 ((π‘Š ∈ 𝑉 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ)) β†’ π‘Œ ∈ (𝐡 βˆ– {𝑋}))
311fvexi 6861 . . . . 5 𝐡 ∈ V
3231rabex 5294 . . . 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 7511 . 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 2777 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074  {crab 3410  Vcvv 3448   βˆ– cdif 3912  {csn 4591  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364  Basecbs 17090  +gcplusg 17140  Scalarcsca 17143   ·𝑠 cvsca 17144  -gcsg 18757  1rcur 19920  LineMcline 46887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-line 46889
This theorem is referenced by: (None)
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