Theorem tglinerflx2 29117

Description: Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-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)
tglineelsb2.1	⊢ (𝜑 → 𝑃 ∈ 𝐵)
tglineelsb2.2	⊢ (𝜑 → 𝑄 ∈ 𝐵)
tglineelsb2.4	⊢ (𝜑 → 𝑃 ≠ 𝑄)

Assertion

Ref	Expression
tglinerflx2	⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄))

Proof of Theorem tglinerflx2

Step	Hyp	Ref	Expression
1		tglineelsb2.p	. 2 ⊢ 𝐵 = (Base‘𝐺)
2		tglineelsb2.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
3		tglineelsb2.l	. 2 ⊢ 𝐿 = (LineG‘𝐺)
4		tglineelsb2.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tglineelsb2.1	. 2 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
6		tglineelsb2.2	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
7		tglineelsb2.4	. 2 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
8		eqid 2763	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
9	1, 8, 2, 4, 5, 6	tgbtwntriv2 28965	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐼𝑄))
10	1, 2, 3, 4, 5, 6, 6, 7, 9	btwnlng1 29102	1 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146   ≠ wne 2959  ‘cfv 6547  (class class class)co 7430  Basecbs 17402  distcds 17452  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

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-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-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6501  df-fun 6549  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-trkgc 28925  df-trkgcb 28927  df-trkg 28930

This theorem is used by:  tghilberti1  29120  tglinesseq  29123  colline  29133  tglnpt2  29136  footexALT  29208  footexlem2  29210  foot  29212  footne  29213  perprag  29217  colperpexlem3  29223  mideulem2  29225  opphllem  29226  opphllem5  29242  opphllem6  29243  opphl  29245  outpasch  29248  hlpasch  29249  lnopp2hpgb  29256  plngrotlem1  29280  plngrotlem2  29281  plngrot  29283  lnssplnglem  29284  lnssplng  29285  mirplncl  29288  plng3p  29290  hypcgrlem1  29320  hypcgrlem2  29321  trgcopyeulem  29327  acopy  29356  acopyeu  29357  ragraghl  29361  perpeqlem  29362  perpeq  29363  tgaaddcpbllem1  29364  tgaaddcpbllem2  29365  tgaaddcpbl  29367  angmgmaddlid  29407  tgasa1  29418  dfprlng2  29440  prlngex  29444  prlngmid2  29454  prlngsymquadlem  29456  prlngsymquadopp  29458  quadcgrprlng  29459  tgaltai  29460  tgextahpl  33084  tgextamirnhl  33087  tgextancgra  33089