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Theorem lines 47798
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 47796 . . 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 6891 . . . . . . 7 (𝑊 = 𝑤 → (Base‘𝑊) = (Base‘𝑤))
53, 4eqtrid 2780 . . . . . 6 (𝑊 = 𝑤𝐵 = (Base‘𝑤))
65difeq1d 4117 . . . . . 6 (𝑊 = 𝑤 → (𝐵 ∖ {𝑥}) = ((Base‘𝑤) ∖ {𝑥}))
7 lines.k . . . . . . . . 9 𝐾 = (Base‘𝑆)
8 lines.s . . . . . . . . . . 11 𝑆 = (Scalar‘𝑊)
9 fveq2 6891 . . . . . . . . . . 11 (𝑊 = 𝑤 → (Scalar‘𝑊) = (Scalar‘𝑤))
108, 9eqtrid 2780 . . . . . . . . . 10 (𝑊 = 𝑤𝑆 = (Scalar‘𝑤))
1110fveq2d 6895 . . . . . . . . 9 (𝑊 = 𝑤 → (Base‘𝑆) = (Base‘(Scalar‘𝑤)))
127, 11eqtrid 2780 . . . . . . . 8 (𝑊 = 𝑤𝐾 = (Base‘(Scalar‘𝑤)))
13 lines.a . . . . . . . . . . 11 + = (+g𝑊)
14 fveq2 6891 . . . . . . . . . . 11 (𝑊 = 𝑤 → (+g𝑊) = (+g𝑤))
1513, 14eqtrid 2780 . . . . . . . . . 10 (𝑊 = 𝑤+ = (+g𝑤))
16 lines.p . . . . . . . . . . . 12 · = ( ·𝑠𝑊)
17 fveq2 6891 . . . . . . . . . . . 12 (𝑊 = 𝑤 → ( ·𝑠𝑊) = ( ·𝑠𝑤))
1816, 17eqtrid 2780 . . . . . . . . . . 11 (𝑊 = 𝑤· = ( ·𝑠𝑤))
19 lines.m . . . . . . . . . . . . . 14 = (-g𝑆)
208fveq2i 6894 . . . . . . . . . . . . . 14 (-g𝑆) = (-g‘(Scalar‘𝑊))
2119, 20eqtri 2756 . . . . . . . . . . . . 13 = (-g‘(Scalar‘𝑊))
22 2fveq3 6896 . . . . . . . . . . . . 13 (𝑊 = 𝑤 → (-g‘(Scalar‘𝑊)) = (-g‘(Scalar‘𝑤)))
2321, 22eqtrid 2780 . . . . . . . . . . . 12 (𝑊 = 𝑤 = (-g‘(Scalar‘𝑤)))
24 lines.1 . . . . . . . . . . . . . 14 1 = (1r𝑆)
258fveq2i 6894 . . . . . . . . . . . . . 14 (1r𝑆) = (1r‘(Scalar‘𝑊))
2624, 25eqtri 2756 . . . . . . . . . . . . 13 1 = (1r‘(Scalar‘𝑊))
27 2fveq3 6896 . . . . . . . . . . . . 13 (𝑊 = 𝑤 → (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑤)))
2826, 27eqtrid 2780 . . . . . . . . . . . 12 (𝑊 = 𝑤1 = (1r‘(Scalar‘𝑤)))
29 eqidd 2729 . . . . . . . . . . . 12 (𝑊 = 𝑤𝑡 = 𝑡)
3023, 28, 29oveq123d 7435 . . . . . . . . . . 11 (𝑊 = 𝑤 → ( 1 𝑡) = ((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡))
31 eqidd 2729 . . . . . . . . . . 11 (𝑊 = 𝑤𝑥 = 𝑥)
3218, 30, 31oveq123d 7435 . . . . . . . . . 10 (𝑊 = 𝑤 → (( 1 𝑡) · 𝑥) = (((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥))
3318oveqd 7431 . . . . . . . . . 10 (𝑊 = 𝑤 → (𝑡 · 𝑦) = (𝑡( ·𝑠𝑤)𝑦))
3415, 32, 33oveq123d 7435 . . . . . . . . 9 (𝑊 = 𝑤 → ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥)(+g𝑤)(𝑡( ·𝑠𝑤)𝑦)))
3534eqeq2d 2739 . . . . . . . 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 3339 . . . . . . 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 3445 . . . . . 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 7489 . . . . 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 2734 . . . 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 2736 . . 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 3489 . . 3 (𝑊𝑉𝑊 ∈ V)
423fvexi 6905 . . . . 5 𝐵 ∈ V
4342difexi 5324 . . . . 5 (𝐵 ∖ {𝑥}) ∈ V
4442, 43mpoex 8078 . . . 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 7022 . 2 (𝑊𝑉 → (LineM𝑊) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
471, 46eqtrid 2780 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 1534  wcel 2099  wrex 3066  {crab 3428  Vcvv 3470  cdif 3942  {csn 4624  cfv 6542  (class class class)co 7414  cmpo 7416  Basecbs 17173  +gcplusg 17226  Scalarcsca 17229   ·𝑠 cvsca 17230  -gcsg 18885  1rcur 20114  LineMcline 47794
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 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
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 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-line 47796
This theorem is referenced by:  line  47799  rrxlines  47800
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