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Theorem istrkgl 26250
Description: Building lines from the segment property. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
istrkg.p 𝑃 = (Base‘𝐺)
istrkg.d = (dist‘𝐺)
istrkg.i 𝐼 = (Itv‘𝐺)
Assertion
Ref Expression
istrkgl (𝐺 ∈ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})} ↔ (𝐺 ∈ V ∧ (LineG‘𝐺) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
Distinct variable groups:   𝑓,𝑖,𝑝,𝐺   𝑥,𝑓,𝑦,𝑧,𝐼,𝑖,𝑝   𝑃,𝑓,𝑖,𝑝,𝑥,𝑦,𝑧   ,𝑓,𝑖,𝑝,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem istrkgl
StepHypRef Expression
1 istrkg.p . . . 4 𝑃 = (Base‘𝐺)
2 istrkg.i . . . 4 𝐼 = (Itv‘𝐺)
3 simpl 486 . . . . . . 7 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑝 = 𝑃)
43eqcomd 2828 . . . . . 6 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑃 = 𝑝)
54adantr 484 . . . . . . 7 (((𝑝 = 𝑃𝑖 = 𝐼) ∧ 𝑥𝑃) → 𝑃 = 𝑝)
65difeq1d 4073 . . . . . 6 (((𝑝 = 𝑃𝑖 = 𝐼) ∧ 𝑥𝑃) → (𝑃 ∖ {𝑥}) = (𝑝 ∖ {𝑥}))
7 simpr 488 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑖 = 𝐼)
87eqcomd 2828 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝐼 = 𝑖)
98oveqd 7157 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑥𝐼𝑦) = (𝑥𝑖𝑦))
109eleq2d 2899 . . . . . . . . 9 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥𝑖𝑦)))
118oveqd 7157 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑧𝐼𝑦) = (𝑧𝑖𝑦))
1211eleq2d 2899 . . . . . . . . 9 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑥 ∈ (𝑧𝑖𝑦)))
138oveqd 7157 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑥𝐼𝑧) = (𝑥𝑖𝑧))
1413eleq2d 2899 . . . . . . . . 9 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑥𝑖𝑧)))
1510, 12, 143orbi123d 1432 . . . . . . . 8 ((𝑝 = 𝑃𝑖 = 𝐼) → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))
164, 15rabeqbidv 3461 . . . . . . 7 ((𝑝 = 𝑃𝑖 = 𝐼) → {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} = {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
1716adantr 484 . . . . . 6 (((𝑝 = 𝑃𝑖 = 𝐼) ∧ (𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥}))) → {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} = {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
184, 6, 17mpoeq123dva 7212 . . . . 5 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}))
1918eqeq2d 2833 . . . 4 ((𝑝 = 𝑃𝑖 = 𝐼) → ((LineG‘𝑓) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) ↔ (LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})))
201, 2, 19sbcie2s 16531 . . 3 (𝑓 = 𝐺 → ([(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}) ↔ (LineG‘𝑓) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
21 fveqeq2 6661 . . 3 (𝑓 = 𝐺 → ((LineG‘𝑓) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) ↔ (LineG‘𝐺) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
2220, 21bitrd 282 . 2 (𝑓 = 𝐺 → ([(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}) ↔ (LineG‘𝐺) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
23 eqid 2822 . 2 {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})} = {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}
2422, 23elab4g 3646 1 (𝐺 ∈ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})} ↔ (𝐺 ∈ V ∧ (LineG‘𝐺) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3o 1083   = wceq 1538  wcel 2114  {cab 2800  {crab 3134  Vcvv 3469  [wsbc 3747  cdif 3905  {csn 4539  cfv 6334  (class class class)co 7140  cmpo 7142  Basecbs 16474  distcds 16565  Itvcitv 26228  LineGclng 26229
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145
This theorem is referenced by:  tglng  26338  f1otrg  26663  eengtrkg  26778
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