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Theorem lines 48652
Description: The lines passing through two different points 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
lines (𝑊𝑉𝐿 = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
Distinct variable groups:   𝐵,𝑝,𝑥,𝑦   𝑡,𝐾   𝑡,𝑆   𝑊,𝑝,𝑡,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑡)   + (𝑥,𝑦,𝑡,𝑝)   𝑆(𝑥,𝑦,𝑝)   · (𝑥,𝑦,𝑡,𝑝)   1 (𝑥,𝑦,𝑡,𝑝)   𝐾(𝑥,𝑦,𝑝)   𝐿(𝑥,𝑦,𝑡,𝑝)   (𝑥,𝑦,𝑡,𝑝)   𝑉(𝑥,𝑦,𝑡,𝑝)

Proof of Theorem lines
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 lines.l . 2 𝐿 = (LineM𝑊)
2 df-line 48650 . . 3 LineM = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ ((Base‘𝑤) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥)(+g𝑤)(𝑡( ·𝑠𝑤)𝑦))}))
3 lines.b . . . . . . 7 𝐵 = (Base‘𝑊)
4 fveq2 6906 . . . . . . 7 (𝑊 = 𝑤 → (Base‘𝑊) = (Base‘𝑤))
53, 4eqtrid 2789 . . . . . 6 (𝑊 = 𝑤𝐵 = (Base‘𝑤))
65difeq1d 4125 . . . . . 6 (𝑊 = 𝑤 → (𝐵 ∖ {𝑥}) = ((Base‘𝑤) ∖ {𝑥}))
7 lines.k . . . . . . . . 9 𝐾 = (Base‘𝑆)
8 lines.s . . . . . . . . . . 11 𝑆 = (Scalar‘𝑊)
9 fveq2 6906 . . . . . . . . . . 11 (𝑊 = 𝑤 → (Scalar‘𝑊) = (Scalar‘𝑤))
108, 9eqtrid 2789 . . . . . . . . . 10 (𝑊 = 𝑤𝑆 = (Scalar‘𝑤))
1110fveq2d 6910 . . . . . . . . 9 (𝑊 = 𝑤 → (Base‘𝑆) = (Base‘(Scalar‘𝑤)))
127, 11eqtrid 2789 . . . . . . . 8 (𝑊 = 𝑤𝐾 = (Base‘(Scalar‘𝑤)))
13 lines.a . . . . . . . . . . 11 + = (+g𝑊)
14 fveq2 6906 . . . . . . . . . . 11 (𝑊 = 𝑤 → (+g𝑊) = (+g𝑤))
1513, 14eqtrid 2789 . . . . . . . . . 10 (𝑊 = 𝑤+ = (+g𝑤))
16 lines.p . . . . . . . . . . . 12 · = ( ·𝑠𝑊)
17 fveq2 6906 . . . . . . . . . . . 12 (𝑊 = 𝑤 → ( ·𝑠𝑊) = ( ·𝑠𝑤))
1816, 17eqtrid 2789 . . . . . . . . . . 11 (𝑊 = 𝑤· = ( ·𝑠𝑤))
19 lines.m . . . . . . . . . . . . . 14 = (-g𝑆)
208fveq2i 6909 . . . . . . . . . . . . . 14 (-g𝑆) = (-g‘(Scalar‘𝑊))
2119, 20eqtri 2765 . . . . . . . . . . . . 13 = (-g‘(Scalar‘𝑊))
22 2fveq3 6911 . . . . . . . . . . . . 13 (𝑊 = 𝑤 → (-g‘(Scalar‘𝑊)) = (-g‘(Scalar‘𝑤)))
2321, 22eqtrid 2789 . . . . . . . . . . . 12 (𝑊 = 𝑤 = (-g‘(Scalar‘𝑤)))
24 lines.1 . . . . . . . . . . . . . 14 1 = (1r𝑆)
258fveq2i 6909 . . . . . . . . . . . . . 14 (1r𝑆) = (1r‘(Scalar‘𝑊))
2624, 25eqtri 2765 . . . . . . . . . . . . 13 1 = (1r‘(Scalar‘𝑊))
27 2fveq3 6911 . . . . . . . . . . . . 13 (𝑊 = 𝑤 → (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑤)))
2826, 27eqtrid 2789 . . . . . . . . . . . 12 (𝑊 = 𝑤1 = (1r‘(Scalar‘𝑤)))
29 eqidd 2738 . . . . . . . . . . . 12 (𝑊 = 𝑤𝑡 = 𝑡)
3023, 28, 29oveq123d 7452 . . . . . . . . . . 11 (𝑊 = 𝑤 → ( 1 𝑡) = ((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡))
31 eqidd 2738 . . . . . . . . . . 11 (𝑊 = 𝑤𝑥 = 𝑥)
3218, 30, 31oveq123d 7452 . . . . . . . . . 10 (𝑊 = 𝑤 → (( 1 𝑡) · 𝑥) = (((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥))
3318oveqd 7448 . . . . . . . . . 10 (𝑊 = 𝑤 → (𝑡 · 𝑦) = (𝑡( ·𝑠𝑤)𝑦))
3415, 32, 33oveq123d 7452 . . . . . . . . 9 (𝑊 = 𝑤 → ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥)(+g𝑤)(𝑡( ·𝑠𝑤)𝑦)))
3534eqeq2d 2748 . . . . . . . 8 (𝑊 = 𝑤 → (𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ 𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥)(+g𝑤)(𝑡( ·𝑠𝑤)𝑦))))
3612, 35rexeqbidv 3347 . . . . . . 7 (𝑊 = 𝑤 → (∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥)(+g𝑤)(𝑡( ·𝑠𝑤)𝑦))))
375, 36rabeqbidv 3455 . . . . . 6 (𝑊 = 𝑤 → {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥)(+g𝑤)(𝑡( ·𝑠𝑤)𝑦))})
385, 6, 37mpoeq123dv 7508 . . . . 5 (𝑊 = 𝑤 → (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}) = (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ ((Base‘𝑤) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥)(+g𝑤)(𝑡( ·𝑠𝑤)𝑦))}))
3938eqcomd 2743 . . . 4 (𝑊 = 𝑤 → (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ ((Base‘𝑤) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥)(+g𝑤)(𝑡( ·𝑠𝑤)𝑦))}) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
4039eqcoms 2745 . . 3 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ ((Base‘𝑤) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥)(+g𝑤)(𝑡( ·𝑠𝑤)𝑦))}) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
41 elex 3501 . . 3 (𝑊𝑉𝑊 ∈ V)
423fvexi 6920 . . . . 5 𝐵 ∈ V
4342difexi 5330 . . . . 5 (𝐵 ∖ {𝑥}) ∈ V
4442, 43mpoex 8104 . . . 4 (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}) ∈ V
4544a1i 11 . . 3 (𝑊𝑉 → (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}) ∈ V)
462, 40, 41, 45fvmptd3 7039 . 2 (𝑊𝑉 → (LineM𝑊) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
471, 46eqtrid 2789 1 (𝑊𝑉𝐿 = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  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:  line  48653  rrxlines  48654
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