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Theorem tgisline 27569
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 769 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑥𝐵)
7 simprr 771 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦 ∈ (𝐵 ∖ {𝑥}))
87eldifad 3922 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦𝐵)
9 eldifsn 4747 . . . . . . . . 9 (𝑦 ∈ (𝐵 ∖ {𝑥}) ↔ (𝑦𝐵𝑦𝑥))
107, 9sylib 217 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → (𝑦𝐵𝑦𝑥))
1110simprd 496 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦𝑥)
1211necomd 2999 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑥𝑦)
131, 2, 3, 5, 6, 8, 12tglngval 27493 . . . . 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 3197 . . 3 (𝜑 → ∀𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦))
16 tgisline.1 . . . . 5 (𝜑𝐴 ∈ ran 𝐿)
171, 2, 3tglng 27488 . . . . . . 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 5893 . . . . 5 (𝜑 → ran 𝐿 = ran (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
2016, 19eleqtrd 2840 . . . 4 (𝜑𝐴 ∈ ran (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
21 eqid 2736 . . . . . 6 (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
2221elrnmpog 7491 . . . . 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 231 . . 3 (𝜑 → ∃𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
2515, 24r19.29d2r 3137 . 2 (𝜑 → ∃𝑥𝐵𝑦 ∈ (𝐵 ∖ {𝑥})(((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
26 difss 4091 . . . 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 765 . . . . . . 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 2779 . . . . . 6 ((((𝑥𝐿𝑦) = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥𝑦) ∧ 𝐴 = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝐴 = (𝑥𝐿𝑦))
30 simplr 767 . . . . . 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 3087 . . . 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 4011 . . . 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 3087 . 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 205  wa 396  w3o 1086   = wceq 1541  wcel 2106  wne 2943  wrex 3073  {crab 3407  cdif 3907  wss 3910  {csn 4586  ran crn 5634  cfv 6496  (class class class)co 7357  cmpo 7359  Basecbs 17083  TarskiGcstrkg 27369  Itvcitv 27375  LineGclng 27376
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-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-trkg 27395
This theorem is referenced by:  tglnne  27570  tglndim0  27571  tglinethru  27578  tglnne0  27582  tglnpt2  27583  footexALT  27660  footex  27663  opptgdim2  27687
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