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Theorem line 46808
Description: The line passing through the two different points 𝑋 and 𝑌 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
line ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → (𝑋𝐿𝑌) = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
Distinct variable groups:   𝐵,𝑝   𝑡,𝐾   𝑡,𝑆   𝑊,𝑝,𝑡   𝑋,𝑝,𝑡   𝑌,𝑝,𝑡
Allowed substitution hints:   𝐵(𝑡)   + (𝑡,𝑝)   𝑆(𝑝)   · (𝑡,𝑝)   1 (𝑡,𝑝)   𝐾(𝑝)   𝐿(𝑡,𝑝)   (𝑡,𝑝)   𝑉(𝑡,𝑝)

Proof of Theorem line
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lines.b . . . . 5 𝐵 = (Base‘𝑊)
2 lines.l . . . . 5 𝐿 = (LineM𝑊)
3 lines.s . . . . 5 𝑆 = (Scalar‘𝑊)
4 lines.k . . . . 5 𝐾 = (Base‘𝑆)
5 lines.p . . . . 5 · = ( ·𝑠𝑊)
6 lines.a . . . . 5 + = (+g𝑊)
7 lines.m . . . . 5 = (-g𝑆)
8 lines.1 . . . . 5 1 = (1r𝑆)
91, 2, 3, 4, 5, 6, 7, 8lines 46807 . . . 4 (𝑊𝑉𝐿 = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
109oveqd 7374 . . 3 (𝑊𝑉 → (𝑋𝐿𝑌) = (𝑋(𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌))
1110adantr 481 . 2 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → (𝑋𝐿𝑌) = (𝑋(𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌))
12 eqidd 2737 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
13 oveq2 7365 . . . . . . . 8 (𝑥 = 𝑋 → (( 1 𝑡) · 𝑥) = (( 1 𝑡) · 𝑋))
14 oveq2 7365 . . . . . . . 8 (𝑦 = 𝑌 → (𝑡 · 𝑦) = (𝑡 · 𝑌))
1513, 14oveqan12d 7376 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌)))
1615eqeq2d 2747 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))))
1716rexbidv 3175 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))))
1817rabbidv 3415 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
1918adantl 482 . . 3 (((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
20 sneq 4596 . . . . 5 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2120difeq2d 4082 . . . 4 (𝑥 = 𝑋 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
2221adantl 482 . . 3 (((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) ∧ 𝑥 = 𝑋) → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
23 simpr1 1194 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → 𝑋𝐵)
24 id 22 . . . . . . . 8 (𝑋𝑌𝑋𝑌)
2524necomd 2999 . . . . . . 7 (𝑋𝑌𝑌𝑋)
2625anim2i 617 . . . . . 6 ((𝑌𝐵𝑋𝑌) → (𝑌𝐵𝑌𝑋))
27263adant1 1130 . . . . 5 ((𝑋𝐵𝑌𝐵𝑋𝑌) → (𝑌𝐵𝑌𝑋))
28 eldifsn 4747 . . . . 5 (𝑌 ∈ (𝐵 ∖ {𝑋}) ↔ (𝑌𝐵𝑌𝑋))
2927, 28sylibr 233 . . . 4 ((𝑋𝐵𝑌𝐵𝑋𝑌) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
3029adantl 482 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
311fvexi 6856 . . . . 5 𝐵 ∈ V
3231rabex 5289 . . . 4 {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))} ∈ V
3332a1i 11 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))} ∈ V)
3412, 19, 22, 23, 30, 33ovmpodx 7506 . 2 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → (𝑋(𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌) = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
3511, 34eqtrd 2776 1 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → (𝑋𝐿𝑌) = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  {csn 4586  cfv 6496  (class class class)co 7357  cmpo 7359  Basecbs 17083  +gcplusg 17133  Scalarcsca 17136   ·𝑠 cvsca 17137  -gcsg 18750  1rcur 19913  LineMcline 46803
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-line 46805
This theorem is referenced by: (None)
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