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Theorem line 47883
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 47882 . . . 4 (𝑊𝑉𝐿 = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
109oveqd 7443 . . 3 (𝑊𝑉 → (𝑋𝐿𝑌) = (𝑋(𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌))
1110adantr 479 . 2 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → (𝑋𝐿𝑌) = (𝑋(𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))})𝑌))
12 eqidd 2729 . . 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 7434 . . . . . . . 8 (𝑥 = 𝑋 → (( 1 𝑡) · 𝑥) = (( 1 𝑡) · 𝑋))
14 oveq2 7434 . . . . . . . 8 (𝑦 = 𝑌 → (𝑡 · 𝑦) = (𝑡 · 𝑌))
1513, 14oveqan12d 7445 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌)))
1615eqeq2d 2739 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))))
1716rexbidv 3176 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))))
1817rabbidv 3438 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
1918adantl 480 . . 3 (((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
20 sneq 4642 . . . . 5 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2120difeq2d 4122 . . . 4 (𝑥 = 𝑋 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
2221adantl 480 . . 3 (((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) ∧ 𝑥 = 𝑋) → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑋}))
23 simpr1 1191 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → 𝑋𝐵)
24 id 22 . . . . . . . 8 (𝑋𝑌𝑋𝑌)
2524necomd 2993 . . . . . . 7 (𝑋𝑌𝑌𝑋)
2625anim2i 615 . . . . . 6 ((𝑌𝐵𝑋𝑌) → (𝑌𝐵𝑌𝑋))
27263adant1 1127 . . . . 5 ((𝑋𝐵𝑌𝐵𝑋𝑌) → (𝑌𝐵𝑌𝑋))
28 eldifsn 4795 . . . . 5 (𝑌 ∈ (𝐵 ∖ {𝑋}) ↔ (𝑌𝐵𝑌𝑋))
2927, 28sylibr 233 . . . 4 ((𝑋𝐵𝑌𝐵𝑋𝑌) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
3029adantl 480 . . 3 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))
311fvexi 6916 . . . . 5 𝐵 ∈ V
3231rabex 5338 . . . 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 7578 . 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 2768 1 ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → (𝑋𝐿𝑌) = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2937  wrex 3067  {crab 3430  Vcvv 3473  cdif 3946  {csn 4632  cfv 6553  (class class class)co 7426  cmpo 7428  Basecbs 17187  +gcplusg 17240  Scalarcsca 17243   ·𝑠 cvsca 17244  -gcsg 18899  1rcur 20128  LineMcline 47878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-line 47880
This theorem is referenced by: (None)
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