Theorem tgelrnln 28575

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgelrnln.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
tgelrnln.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
tgelrnln.d	⊢ (𝜑 → 𝑋 ≠ 𝑌)

Assertion

Ref	Expression
tgelrnln	⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)

Proof of Theorem tgelrnln

Step	Hyp	Ref	Expression
1		df-ov 7352	. 2 ⊢ (𝑋𝐿𝑌) = (𝐿‘⟨𝑋, 𝑌⟩)
2		tglineelsb2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
3		tglineelsb2.p	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
4		tglineelsb2.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
5		tglineelsb2.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
6	3, 4, 5	tglnfn 28492	. . . 4 ⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝐵 × 𝐵) ∖ I ))
7	2, 6	syl 17	. . 3 ⊢ (𝜑 → 𝐿 Fn ((𝐵 × 𝐵) ∖ I ))
8		tgelrnln.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		tgelrnln.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10	8, 9	opelxpd 5658	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11		tgelrnln.d	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
12		df-br 5093	. . . . . . . 8 ⊢ (𝑋 I 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ I )
13		ideqg 5794	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌))
14	12, 13	bitr3id 285	. . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → (⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 = 𝑌))
15	14	necon3bbid 2962	. . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (¬ ⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 ≠ 𝑌))
16	15	biimpar 477	. . . . 5 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
17	9, 11, 16	syl2anc 584	. . . 4 ⊢ (𝜑 → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
18	10, 17	eldifd 3914	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I ))
19		fnfvelrn 7014	. . 3 ⊢ ((𝐿 Fn ((𝐵 × 𝐵) ∖ I ) ∧ ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I )) → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
20	7, 18, 19	syl2anc 584	. 2 ⊢ (𝜑 → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
21	1, 20	eqeltrid 2832	1 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∖ cdif 3900  ⟨cop 4583   class class class wbr 5092   I cid 5513   × cxp 5617  ran crn 5620   Fn wfn 6477  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  TarskiGcstrkg 28372  Itvcitv 28378  LineGclng 28379

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-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

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-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-trkg 28398

This theorem is referenced by:  tghilberti1  28582  tglineinteq  28590  colline  28594  tglowdim2ln  28596  footexALT  28663  footexlem2  28665  foot  28667  perprag  28671  colperpexlem3  28677  mideulem2  28679  midex  28682  outpasch  28700  lnopp2hpgb  28708  colopp  28714  lmieu  28729  lmimid  28739  hypcgrlem1  28744  hypcgrlem2  28745  lnperpex  28748  trgcopy  28749  trgcopyeulem  28750  acopy  28778  acopyeu  28779  tgasa1  28803