Theorem tglngval 28635

Description: The line going through points 𝑋 and 𝑌. (Contributed by Thierry Arnoux, 28-Mar-2019.)

Hypotheses

Ref	Expression
tglngval.p	⊢ 𝑃 = (Base‘𝐺)
tglngval.l	⊢ 𝐿 = (LineG‘𝐺)
tglngval.i	⊢ 𝐼 = (Itv‘𝐺)
tglngval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglngval.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tglngval.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tglngval.z	⊢ (𝜑 → 𝑋 ≠ 𝑌)

Assertion

Ref	Expression
tglngval	⊢ (𝜑 → (𝑋𝐿𝑌) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})

Distinct variable groups:   𝑧,𝐺   𝑧,𝐼   𝑧,𝑃   𝑧,𝑋   𝑧,𝑌   𝜑,𝑧

Allowed substitution hint:   𝐿(𝑧)

Proof of Theorem tglngval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tglngval.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
2		tglngval.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
3		tglngval.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
4		tglngval.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
5	2, 3, 4	tglng 28630	. . . 4 ⊢ (𝐺 ∈ TarskiG → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
6	1, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
7	6	oveqd 7385	. 2 ⊢ (𝜑 → (𝑋𝐿𝑌) = (𝑋(𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})𝑌))
8		tglngval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
9		tglngval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
10		tglngval.z	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
11	10	necomd 2988	. . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
12		eldifsn 4744	. . . 4 ⊢ (𝑌 ∈ (𝑃 ∖ {𝑋}) ↔ (𝑌 ∈ 𝑃 ∧ 𝑌 ≠ 𝑋))
13	9, 11, 12	sylanbrc 584	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))
14	2	fvexi 6856	. . . . 5 ⊢ 𝑃 ∈ V
15	14	rabex 5286	. . . 4 ⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ∈ V
16	15	a1i 11	. . 3 ⊢ (𝜑 → {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ∈ V)
17		oveq12 7377	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐼𝑦) = (𝑋𝐼𝑌))
18	17	eleq2d 2823	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑌)))
19		simpl 482	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
20		simpr 484	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
21	20	oveq2d 7384	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑧𝐼𝑦) = (𝑧𝐼𝑌))
22	19, 21	eleq12d 2831	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑌)))
23	19	oveq1d 7383	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
24	20, 23	eleq12d 2831	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
25	18, 22, 24	3orbi123d 1438	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))))
26	25	rabbidv 3408	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
27		sneq 4592	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
28	27	difeq2d 4080	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑃 ∖ {𝑥}) = (𝑃 ∖ {𝑋}))
29		eqid 2737	. . . 4 ⊢ (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))})
30	26, 28, 29	ovmpox 7521	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ (𝑃 ∖ {𝑋}) ∧ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ∈ V) → (𝑋(𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})𝑌) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
31	8, 13, 16, 30	syl3anc 1374	. 2 ⊢ (𝜑 → (𝑋(𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})𝑌) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
32	7, 31	eqtrd 2772	1 ⊢ (𝜑 → (𝑋𝐿𝑌) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})

Syntax hints:   → wi 4   ∧ wa 395   ∨ w3o 1086   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3401  Vcvv 3442   ∖ cdif 3900  {csn 4582  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  Basecbs 17148  TarskiGcstrkg 28511  Itvcitv 28517  LineGclng 28518

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-trkg 28537

This theorem is referenced by:  tglnssp  28636  tgellng  28637  tgisline  28711