Theorem tgelrnln 28801

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 7401	. 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 28718	. . . 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 5688	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11		tgelrnln.d	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
12		df-br 5103	. . . . . . . 8 ⊢ (𝑋 I 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ I )
13		ideqg 5825	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌))
14	12, 13	bitr3id 287	. . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → (⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 = 𝑌))
15	14	necon3bbid 2996	. . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (¬ ⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 ≠ 𝑌))
16	15	biimpar 481	. . . . 5 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
17	9, 11, 16	syl2anc 593	. . . 4 ⊢ (𝜑 → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
18	10, 17	eldifd 3917	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I ))
19		fnfvelrn 7063	. . 3 ⊢ ((𝐿 Fn ((𝐵 × 𝐵) ∖ I ) ∧ ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I )) → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
20	7, 18, 19	syl2anc 593	. 2 ⊢ (𝜑 → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
21	1, 20	eqeltrid 2868	1 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1562   ∈ wcel 2144   ≠ wne 2959   ∖ cdif 3903  ⟨cop 4590   class class class wbr 5102   I cid 5543   × cxp 5647  ran crn 5650   Fn wfn 6518  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  TarskiGcstrkg 28598  Itvcitv 28604  LineGclng 28605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-trkg 28624

This theorem is referenced by:  tghilberti1  28808  tglineinteq  28817  colline  28821  tglowdim2ln  28823  footexALT  28893  footexlem2  28895  foot  28897  perprag  28901  colperpexlem3  28907  mideulem2  28909  midex  28912  outpasch  28930  lnopp2hpgb  28938  colopp  28944  lmieu  28959  lmimid  28969  hypcgrlem1  28974  hypcgrlem2  28975  lnperpex  28978  trgcopy  28979  trgcopyeulem  28980  plngrotlem1  28996  plngrotlem2  28997  lnssplnglem  29000  lnssplng  29001  acopy  29029  acopyeu  29030  tgasa1  29054