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Theorem lines 44999
 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 44997 . . 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 6659 . . . . . . 7 (𝑊 = 𝑤 → (Base‘𝑊) = (Base‘𝑤))
53, 4syl5eq 2871 . . . . . 6 (𝑊 = 𝑤𝐵 = (Base‘𝑤))
65difeq1d 4084 . . . . . 6 (𝑊 = 𝑤 → (𝐵 ∖ {𝑥}) = ((Base‘𝑤) ∖ {𝑥}))
7 lines.k . . . . . . . . 9 𝐾 = (Base‘𝑆)
8 lines.s . . . . . . . . . . 11 𝑆 = (Scalar‘𝑊)
9 fveq2 6659 . . . . . . . . . . 11 (𝑊 = 𝑤 → (Scalar‘𝑊) = (Scalar‘𝑤))
108, 9syl5eq 2871 . . . . . . . . . 10 (𝑊 = 𝑤𝑆 = (Scalar‘𝑤))
1110fveq2d 6663 . . . . . . . . 9 (𝑊 = 𝑤 → (Base‘𝑆) = (Base‘(Scalar‘𝑤)))
127, 11syl5eq 2871 . . . . . . . 8 (𝑊 = 𝑤𝐾 = (Base‘(Scalar‘𝑤)))
13 lines.a . . . . . . . . . . 11 + = (+g𝑊)
14 fveq2 6659 . . . . . . . . . . 11 (𝑊 = 𝑤 → (+g𝑊) = (+g𝑤))
1513, 14syl5eq 2871 . . . . . . . . . 10 (𝑊 = 𝑤+ = (+g𝑤))
16 lines.p . . . . . . . . . . . 12 · = ( ·𝑠𝑊)
17 fveq2 6659 . . . . . . . . . . . 12 (𝑊 = 𝑤 → ( ·𝑠𝑊) = ( ·𝑠𝑤))
1816, 17syl5eq 2871 . . . . . . . . . . 11 (𝑊 = 𝑤· = ( ·𝑠𝑤))
19 lines.m . . . . . . . . . . . . . 14 = (-g𝑆)
208fveq2i 6662 . . . . . . . . . . . . . 14 (-g𝑆) = (-g‘(Scalar‘𝑊))
2119, 20eqtri 2847 . . . . . . . . . . . . 13 = (-g‘(Scalar‘𝑊))
22 2fveq3 6664 . . . . . . . . . . . . 13 (𝑊 = 𝑤 → (-g‘(Scalar‘𝑊)) = (-g‘(Scalar‘𝑤)))
2321, 22syl5eq 2871 . . . . . . . . . . . 12 (𝑊 = 𝑤 = (-g‘(Scalar‘𝑤)))
24 lines.1 . . . . . . . . . . . . . 14 1 = (1r𝑆)
258fveq2i 6662 . . . . . . . . . . . . . 14 (1r𝑆) = (1r‘(Scalar‘𝑊))
2624, 25eqtri 2847 . . . . . . . . . . . . 13 1 = (1r‘(Scalar‘𝑊))
27 2fveq3 6664 . . . . . . . . . . . . 13 (𝑊 = 𝑤 → (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑤)))
2826, 27syl5eq 2871 . . . . . . . . . . . 12 (𝑊 = 𝑤1 = (1r‘(Scalar‘𝑤)))
29 eqidd 2825 . . . . . . . . . . . 12 (𝑊 = 𝑤𝑡 = 𝑡)
3023, 28, 29oveq123d 7167 . . . . . . . . . . 11 (𝑊 = 𝑤 → ( 1 𝑡) = ((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡))
31 eqidd 2825 . . . . . . . . . . 11 (𝑊 = 𝑤𝑥 = 𝑥)
3218, 30, 31oveq123d 7167 . . . . . . . . . 10 (𝑊 = 𝑤 → (( 1 𝑡) · 𝑥) = (((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥))
3318oveqd 7163 . . . . . . . . . 10 (𝑊 = 𝑤 → (𝑡 · 𝑦) = (𝑡( ·𝑠𝑤)𝑦))
3415, 32, 33oveq123d 7167 . . . . . . . . 9 (𝑊 = 𝑤 → ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥)(+g𝑤)(𝑡( ·𝑠𝑤)𝑦)))
3534eqeq2d 2835 . . . . . . . 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 3394 . . . . . . 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 3471 . . . . . 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 7219 . . . . 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 2830 . . . 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 2832 . . 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 3498 . . 3 (𝑊𝑉𝑊 ∈ V)
423fvexi 6673 . . . . 5 𝐵 ∈ V
4342difexi 5219 . . . . 5 (𝐵 ∖ {𝑥}) ∈ V
4442, 43mpoex 7769 . . . 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 6780 . 2 (𝑊𝑉 → (LineM𝑊) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
471, 46syl5eq 2871 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 1538   ∈ wcel 2115  ∃wrex 3134  {crab 3137  Vcvv 3480   ∖ cdif 3916  {csn 4550  ‘cfv 6344  (class class class)co 7146   ∈ cmpo 7148  Basecbs 16481  +gcplusg 16563  Scalarcsca 16566   ·𝑠 cvsca 16567  -gcsg 18103  1rcur 19249  LineMcline 44995 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-1st 7681  df-2nd 7682  df-line 44997 This theorem is referenced by:  line  45000  rrxlines  45001
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