Theorem tgelrnln 28720

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 7373	. 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 28637	. . . 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 5673	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11		tgelrnln.d	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
12		df-br 5101	. . . . . . . 8 ⊢ (𝑋 I 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ I )
13		ideqg 5810	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌))
14	12, 13	bitr3id 285	. . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → (⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 = 𝑌))
15	14	necon3bbid 2970	. . . . . 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 585	. . . 4 ⊢ (𝜑 → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
18	10, 17	eldifd 3914	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I ))
19		fnfvelrn 7036	. . 3 ⊢ ((𝐿 Fn ((𝐵 × 𝐵) ∖ I ) ∧ ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I )) → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
20	7, 18, 19	syl2anc 585	. 2 ⊢ (𝜑 → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
21	1, 20	eqeltrid 2841	1 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∖ cdif 3900  ⟨cop 4588   class class class wbr 5100   I cid 5528   × cxp 5632  ran crn 5635   Fn wfn 6497  ‘cfv 6502  (class class class)co 7370  Basecbs 17150  TarskiGcstrkg 28516  Itvcitv 28522  LineGclng 28523

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 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

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-csb 3852  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-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946  df-trkg 28542

This theorem is referenced by:  tghilberti1  28727  tglineinteq  28735  colline  28739  tglowdim2ln  28741  footexALT  28808  footexlem2  28810  foot  28812  perprag  28816  colperpexlem3  28822  mideulem2  28824  midex  28827  outpasch  28845  lnopp2hpgb  28853  colopp  28859  lmieu  28874  lmimid  28884  hypcgrlem1  28889  hypcgrlem2  28890  lnperpex  28893  trgcopy  28894  trgcopyeulem  28895  acopy  28923  acopyeu  28924  tgasa1  28948