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Theorem tgisline 26327
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 481 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝐺 ∈ TarskiG)
6 simprl 767 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑥𝐵)
7 simprr 769 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦 ∈ (𝐵 ∖ {𝑥}))
87eldifad 3952 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦𝐵)
9 eldifsn 4718 . . . . . . . . 9 (𝑦 ∈ (𝐵 ∖ {𝑥}) ↔ (𝑦𝐵𝑦𝑥))
107, 9sylib 219 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → (𝑦𝐵𝑦𝑥))
1110simprd 496 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦𝑥)
1211necomd 3076 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑥𝑦)
131, 2, 3, 5, 6, 8, 12tglngval 26251 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → (𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
1413, 12jca 512 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → ((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦))
1514ralrimivva 3196 . . 3 (𝜑 → ∀𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦))
16 tgisline.1 . . . . 5 (𝜑𝐴 ∈ ran 𝐿)
171, 2, 3tglng 26246 . . . . . . 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 5807 . . . . 5 (𝜑 → ran 𝐿 = ran (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
2016, 19eleqtrd 2920 . . . 4 (𝜑𝐴 ∈ ran (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
21 eqid 2826 . . . . . 6 (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
2221elrnmpog 7276 . . . . 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 233 . . 3 (𝜑 → ∃𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
2515, 24r19.29d2r 3340 . 2 (𝜑 → ∃𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})(((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
26 difss 4112 . . . 4 (𝐵 ∖ {𝑥}) ⊆ 𝐵
27 simpr 485 . . . . . . 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 763 . . . . . . 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 2864 . . . . . 6 ((((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝐴 = (𝑥𝐿𝑦))
30 simplr 765 . . . . . 6 ((((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝑥𝑦)
3129, 30jca 512 . . . . 5 ((((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
3231reximi 3248 . . . 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 4038 . . . 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 3248 . 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 207  wa 396  w3o 1080   = wceq 1530  wcel 2107  wne 3021  wrex 3144  {crab 3147  cdif 3937  wss 3940  {csn 4564  ran crn 5555  cfv 6352  (class class class)co 7148  cmpo 7150  Basecbs 16473  TarskiGcstrkg 26130  Itvcitv 26136  LineGclng 26137
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-trkg 26153
This theorem is referenced by:  tglnne  26328  tglndim0  26329  tglinethru  26336  tglnne0  26340  tglnpt2  26341  footexALT  26418  footex  26421  opptgdim2  26445
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