Theorem tgelrnln 29113

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 7433	. 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 29025	. . . 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 5694	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11		tgelrnln.d	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
12		df-br 5107	. . . . . . . 8 ⊢ (𝑋 I 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ I )
13		ideqg 5834	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌))
14	12, 13	bitr3id 288	. . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → (⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 = 𝑌))
15	14	necon3bbid 2996	. . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (¬ ⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 ≠ 𝑌))
16	15	biimpar 483	. . . . 5 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
17	9, 11, 16	syl2anc 596	. . . 4 ⊢ (𝜑 → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
18	10, 17	eldifd 4053	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I ))
19		fnfvelrn 7091	. . 3 ⊢ ((𝐿 Fn ((𝐵 × 𝐵) ∖ I ) ∧ ⟨𝑋, 𝑌⟩ ∈ ((𝐵 × 𝐵) ∖ I )) → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
20	7, 18, 19	syl2anc 596	. 2 ⊢ (𝜑 → (𝐿‘⟨𝑋, 𝑌⟩) ∈ ran 𝐿)
21	1, 20	eqeltrid 2868	1 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2146   ≠ wne 2959   ∖ cdif 4048  ⟨cop 4838   class class class wbr 5106   I cid 5549   × cxp 5653  ran crn 5656   Fn wfn 6542  ‘cfv 6547  (class class class)co 7430  Basecbs 17402  TarskiGcstrkg 28904  Itvcitv 28910  LineGclng 28911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8011  df-2nd 8012  df-trkg 28930

This theorem is used by:  tghilberti1  29120  tglineinteq  29129  colline  29133  tglowdim2ln  29135  symquadprlnglem  29180  footexALT  29208  footexlem2  29210  foot  29212  perprag  29217  colperpexlem3  29223  mideulem2  29225  midex  29228  outpasch  29248  lnopp2hpgb  29256  colopp  29262  hlopp  29265  plngrotlem1  29280  plngrotlem2  29281  lnssplnglem  29284  lnssplng  29285  mirplncl  29288  lmieu  29304  lmimid  29314  symquadmid  29319  hypcgrlem1  29320  hypcgrlem2  29321  lnperpex  29324  trgcopy  29326  trgcopyeulem  29327  acopy  29356  acopyeu  29357  ragraghl  29361  tgaaddcpbllem1  29364  tgaaddcpbllem2  29365  tgaaddcpbllem3  29366  tgaaddcpbl  29367  angmgmaddeu1  29394  angmgmaddcpbl  29405  tgasa1  29418  dfprlng2  29440  prlngex  29444  prlngmolem1  29445  prlngmid2  29454  symquadprlng  29455  quadcgrprlng  29459  tgaltai  29460  tgextamirn  33082  tgextahpl  33084  tgextamirnhl  33087  tgextancgra  33089