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Theorem tgisline 28635
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 771 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑥𝐵)
7 simprr 773 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦 ∈ (𝐵 ∖ {𝑥}))
87eldifad 3963 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦𝐵)
9 eldifsn 4786 . . . . . . . . 9 (𝑦 ∈ (𝐵 ∖ {𝑥}) ↔ (𝑦𝐵𝑦𝑥))
107, 9sylib 218 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → (𝑦𝐵𝑦𝑥))
1110simprd 495 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦𝑥)
1211necomd 2996 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑥𝑦)
131, 2, 3, 5, 6, 8, 12tglngval 28559 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → (𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
1413, 12jca 511 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → ((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦))
1514ralrimivva 3202 . . 3 (𝜑 → ∀𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦))
16 tgisline.1 . . . . 5 (𝜑𝐴 ∈ ran 𝐿)
171, 2, 3tglng 28554 . . . . . . 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 5949 . . . . 5 (𝜑 → ran 𝐿 = ran (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
2016, 19eleqtrd 2843 . . . 4 (𝜑𝐴 ∈ ran (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
21 eqid 2737 . . . . . 6 (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
2221elrnmpog 7568 . . . . 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 232 . . 3 (𝜑 → ∃𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
2515, 24r19.29d2r 3140 . 2 (𝜑 → ∃𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})(((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
26 difss 4136 . . . 4 (𝐵 ∖ {𝑥}) ⊆ 𝐵
27 simpr 484 . . . . . . 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 767 . . . . . . 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 2780 . . . . . 6 ((((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝐴 = (𝑥𝐿𝑦))
30 simplr 769 . . . . . 6 ((((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝑥𝑦)
3129, 30jca 511 . . . . 5 ((((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
3231reximi 3084 . . . 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 4053 . . . 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 3084 . 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 206  wa 395  w3o 1086   = wceq 1540  wcel 2108  wne 2940  wrex 3070  {crab 3436  cdif 3948  wss 3951  {csn 4626  ran crn 5686  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-trkg 28461
This theorem is referenced by:  tglnne  28636  tglndim0  28637  tglinethru  28644  tglnne0  28648  tglnpt2  28649  footexALT  28726  footex  28729  opptgdim2  28753
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