Theorem tgelrnln 28865

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 7414	. 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 28782	. . . 4 ⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝐵 × 𝐵) ∖ I ))
7	2, 6	syl 18	. . 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 5701	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11		tgelrnln.d	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
12		df-br 5114	. . . . . . . 8 ⊢ (𝑋 I 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ I )
13		ideqg 5838	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌))
14	12, 13	bitr3id 288	. . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → (⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 = 𝑌))
15	14	necon3bbid 3001	. . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (¬ ⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 ≠ 𝑌))
16	15	biimpar 482	. . . . 5 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
17	9, 11, 16	syl2anc 595	. . . 4 ⊢ (𝜑 → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
18	10, 17	eldifd 3924	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I ))
19		fnfvelrn 7076	. . 3 ⊢ ((𝐿 Fn ((𝐵 × 𝐵) ∖ I ) ∧ ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I )) → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
20	7, 18, 19	syl2anc 595	. 2 ⊢ (𝜑 → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
21	1, 20	eqeltrid 2873	1 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   ∖ cdif 3910  ⟨cop 4600   class class class wbr 5113   I cid 5556   × cxp 5660  ran crn 5663   Fn wfn 6532  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  TarskiGcstrkg 28662  Itvcitv 28668  LineGclng 28669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-trkg 28688

This theorem is referenced by:  tghilberti1  28872  tglineinteq  28881  colline  28885  tglowdim2ln  28887  footexALT  28957  footexlem2  28959  foot  28961  perprag  28966  colperpexlem3  28972  mideulem2  28974  midex  28977  outpasch  28996  lnopp2hpgb  29004  colopp  29010  plngrotlem1  29027  plngrotlem2  29028  lnssplnglem  29031  lnssplng  29032  mirplncl  29035  lmieu  29051  lmimid  29061  hypcgrlem1  29066  hypcgrlem2  29067  lnperpex  29070  trgcopy  29072  trgcopyeulem  29073  acopy  29101  acopyeu  29102  ragraghl  29104  tgasa1  29130  prlngex  29154  prlngmolem1  29155