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Theorem lines 48581
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 48579 . . 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 6907 . . . . . . 7 (𝑊 = 𝑤 → (Base‘𝑊) = (Base‘𝑤))
53, 4eqtrid 2787 . . . . . 6 (𝑊 = 𝑤𝐵 = (Base‘𝑤))
65difeq1d 4135 . . . . . 6 (𝑊 = 𝑤 → (𝐵 ∖ {𝑥}) = ((Base‘𝑤) ∖ {𝑥}))
7 lines.k . . . . . . . . 9 𝐾 = (Base‘𝑆)
8 lines.s . . . . . . . . . . 11 𝑆 = (Scalar‘𝑊)
9 fveq2 6907 . . . . . . . . . . 11 (𝑊 = 𝑤 → (Scalar‘𝑊) = (Scalar‘𝑤))
108, 9eqtrid 2787 . . . . . . . . . 10 (𝑊 = 𝑤𝑆 = (Scalar‘𝑤))
1110fveq2d 6911 . . . . . . . . 9 (𝑊 = 𝑤 → (Base‘𝑆) = (Base‘(Scalar‘𝑤)))
127, 11eqtrid 2787 . . . . . . . 8 (𝑊 = 𝑤𝐾 = (Base‘(Scalar‘𝑤)))
13 lines.a . . . . . . . . . . 11 + = (+g𝑊)
14 fveq2 6907 . . . . . . . . . . 11 (𝑊 = 𝑤 → (+g𝑊) = (+g𝑤))
1513, 14eqtrid 2787 . . . . . . . . . 10 (𝑊 = 𝑤+ = (+g𝑤))
16 lines.p . . . . . . . . . . . 12 · = ( ·𝑠𝑊)
17 fveq2 6907 . . . . . . . . . . . 12 (𝑊 = 𝑤 → ( ·𝑠𝑊) = ( ·𝑠𝑤))
1816, 17eqtrid 2787 . . . . . . . . . . 11 (𝑊 = 𝑤· = ( ·𝑠𝑤))
19 lines.m . . . . . . . . . . . . . 14 = (-g𝑆)
208fveq2i 6910 . . . . . . . . . . . . . 14 (-g𝑆) = (-g‘(Scalar‘𝑊))
2119, 20eqtri 2763 . . . . . . . . . . . . 13 = (-g‘(Scalar‘𝑊))
22 2fveq3 6912 . . . . . . . . . . . . 13 (𝑊 = 𝑤 → (-g‘(Scalar‘𝑊)) = (-g‘(Scalar‘𝑤)))
2321, 22eqtrid 2787 . . . . . . . . . . . 12 (𝑊 = 𝑤 = (-g‘(Scalar‘𝑤)))
24 lines.1 . . . . . . . . . . . . . 14 1 = (1r𝑆)
258fveq2i 6910 . . . . . . . . . . . . . 14 (1r𝑆) = (1r‘(Scalar‘𝑊))
2624, 25eqtri 2763 . . . . . . . . . . . . 13 1 = (1r‘(Scalar‘𝑊))
27 2fveq3 6912 . . . . . . . . . . . . 13 (𝑊 = 𝑤 → (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑤)))
2826, 27eqtrid 2787 . . . . . . . . . . . 12 (𝑊 = 𝑤1 = (1r‘(Scalar‘𝑤)))
29 eqidd 2736 . . . . . . . . . . . 12 (𝑊 = 𝑤𝑡 = 𝑡)
3023, 28, 29oveq123d 7452 . . . . . . . . . . 11 (𝑊 = 𝑤 → ( 1 𝑡) = ((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡))
31 eqidd 2736 . . . . . . . . . . 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 2746 . . . . . . . 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 3345 . . . . . . 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 3452 . . . . . 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 2741 . . . 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 2743 . . 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 3499 . . 3 (𝑊𝑉𝑊 ∈ V)
423fvexi 6921 . . . . 5 𝐵 ∈ V
4342difexi 5336 . . . . 5 (𝐵 ∖ {𝑥}) ∈ V
4442, 43mpoex 8103 . . . 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 2787 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 1537  wcel 2106  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:  line  48582  rrxlines  48583
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