Theorem istrkgl 28385

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) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})} ↔ (𝐺 ∈ V ∧ (LineG‘𝐺) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (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

Step	Hyp	Ref	Expression
1		istrkg.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		istrkg.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
3		simpl 482	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → 𝑝 = 𝑃)
4	3	difeq1d 4088	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑝 ∖ {𝑥}) = (𝑃 ∖ {𝑥}))
5		simpr 484	. . . . . . . . . 10 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼)
6	5	oveqd 7404	. . . . . . . . 9 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑥𝑖𝑦) = (𝑥𝐼𝑦))
7	6	eleq2d 2814	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑧 ∈ (𝑥𝑖𝑦) ↔ 𝑧 ∈ (𝑥𝐼𝑦)))
8	5	oveqd 7404	. . . . . . . . 9 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑧𝑖𝑦) = (𝑧𝐼𝑦))
9	8	eleq2d 2814	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑥 ∈ (𝑧𝑖𝑦) ↔ 𝑥 ∈ (𝑧𝐼𝑦)))
10	5	oveqd 7404	. . . . . . . . 9 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑥𝑖𝑧) = (𝑥𝐼𝑧))
11	10	eleq2d 2814	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑦 ∈ (𝑥𝑖𝑧) ↔ 𝑦 ∈ (𝑥𝐼𝑧)))
12	7, 9, 11	3orbi123d 1437	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → ((𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)) ↔ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))
13	3, 12	rabeqbidv 3424	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))} = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
14	3, 4, 13	mpoeq123dv 7464	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
15	14	eqeq2d 2740	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → ((LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}) ↔ (LineG‘𝑓) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
16	1, 2, 15	sbcie2s 17131	. . 3 ⊢ (𝑓 = 𝐺 → ([(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}) ↔ (LineG‘𝑓) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
17		fveqeq2 6867	. . 3 ⊢ (𝑓 = 𝐺 → ((LineG‘𝑓) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) ↔ (LineG‘𝐺) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
18	16, 17	bitrd 279	. 2 ⊢ (𝑓 = 𝐺 → ([(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}) ↔ (LineG‘𝐺) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
19		eqid 2729	. 2 ⊢ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})} = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}
20	18, 19	elab4g 3650	1 ⊢ (𝐺 ∈ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})} ↔ (𝐺 ∈ V ∧ (LineG‘𝐺) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109  {cab 2707  {crab 3405  Vcvv 3447  [wsbc 3753   ∖ cdif 3911  {csn 4589  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  Basecbs 17179  distcds 17229  Itvcitv 28360  LineGclng 28361

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392

This theorem is referenced by:  tglng  28473  f1otrg  28798  eengtrkg  28913