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Theorem tgisline 25567
Description: The property of being a proper line, generated by two distinct points. (Contributed by Thierry Arnoux, 25-May-2019.)
Hypotheses
Ref Expression
tglineelsb2.p 𝐵 = (Base‘𝐺)
tglineelsb2.i 𝐼 = (Itv‘𝐺)
tglineelsb2.l 𝐿 = (LineG‘𝐺)
tglineelsb2.g (𝜑𝐺 ∈ TarskiG)
tgisline.1 (𝜑𝐴 ∈ ran 𝐿)
Assertion
Ref Expression
tgisline (𝜑 → ∃𝑥𝐵𝑦𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐺,𝑦   𝑥,𝐼,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐿(𝑥,𝑦)

Proof of Theorem tgisline
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tglineelsb2.p . . . . . 6 𝐵 = (Base‘𝐺)
2 tglineelsb2.l . . . . . 6 𝐿 = (LineG‘𝐺)
3 tglineelsb2.i . . . . . 6 𝐼 = (Itv‘𝐺)
4 tglineelsb2.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
54adantr 480 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝐺 ∈ TarskiG)
6 simprl 809 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑥𝐵)
7 simprr 811 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦 ∈ (𝐵 ∖ {𝑥}))
87eldifad 3619 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦𝐵)
9 eldifsn 4350 . . . . . . . . 9 (𝑦 ∈ (𝐵 ∖ {𝑥}) ↔ (𝑦𝐵𝑦𝑥))
107, 9sylib 208 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → (𝑦𝐵𝑦𝑥))
1110simprd 478 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦𝑥)
1211necomd 2878 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑥𝑦)
131, 2, 3, 5, 6, 8, 12tglngval 25491 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → (𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
1413, 12jca 553 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → ((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦))
1514ralrimivva 3000 . . 3 (𝜑 → ∀𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦))
16 tgisline.1 . . . . 5 (𝜑𝐴 ∈ ran 𝐿)
171, 2, 3tglng 25486 . . . . . . 7 (𝐺 ∈ TarskiG → 𝐿 = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
184, 17syl 17 . . . . . 6 (𝜑𝐿 = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
1918rneqd 5385 . . . . 5 (𝜑 → ran 𝐿 = ran (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
2016, 19eleqtrd 2732 . . . 4 (𝜑𝐴 ∈ ran (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
21 eqid 2651 . . . . . 6 (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
2221elrnmpt2g 6814 . . . . 5 (𝐴 ∈ ran 𝐿 → (𝐴 ∈ ran (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) ↔ ∃𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
2316, 22syl 17 . . . 4 (𝜑 → (𝐴 ∈ ran (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) ↔ ∃𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
2420, 23mpbid 222 . . 3 (𝜑 → ∃𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
2515, 24r19.29d2r 3109 . 2 (𝜑 → ∃𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})(((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
26 difss 3770 . . . 4 (𝐵 ∖ {𝑥}) ⊆ 𝐵
27 simpr 476 . . . . . . 7 ((((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
28 simpll 805 . . . . . . 7 ((((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → (𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
2927, 28eqtr4d 2688 . . . . . 6 ((((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝐴 = (𝑥𝐿𝑦))
30 simplr 807 . . . . . 6 ((((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝑥𝑦)
3129, 30jca 553 . . . . 5 ((((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
3231reximi 3040 . . . 4 (∃𝑦 ∈ (𝐵 ∖ {𝑥})(((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → ∃𝑦 ∈ (𝐵 ∖ {𝑥})(𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
33 ssrexv 3700 . . . 4 ((𝐵 ∖ {𝑥}) ⊆ 𝐵 → (∃𝑦 ∈ (𝐵 ∖ {𝑥})(𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦) → ∃𝑦𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)))
3426, 32, 33mpsyl 68 . . 3 (∃𝑦 ∈ (𝐵 ∖ {𝑥})(((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → ∃𝑦𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
3534reximi 3040 . 2 (∃𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})(((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → ∃𝑥𝐵𝑦𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
3625, 35syl 17 1 (𝜑 → ∃𝑥𝐵𝑦𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3o 1053   = wceq 1523  wcel 2030  wne 2823  wrex 2942  {crab 2945  cdif 3604  wss 3607  {csn 4210  ran crn 5144  cfv 5926  (class class class)co 6690  cmpt2 6692  Basecbs 15904  TarskiGcstrkg 25374  Itvcitv 25380  LineGclng 25381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-trkg 25397
This theorem is referenced by:  tglnne  25568  tglndim0  25569  tglinethru  25576  tglnne0  25580  tglnpt2  25581  footex  25658  opptgdim2  25682
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