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Theorem tglng 24059
 Description: Lines of a Tarski Geometry. This relates to both Definition 4.10 of [Schwabhauser] p. 36. and Definition 6.14 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 28-Mar-2019.)
Hypotheses
Ref Expression
tglng.p
tglng.l LineG
tglng.i Itv
Assertion
Ref Expression
tglng TarskiG
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem tglng
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-trkg 23976 . . . 4 TarskiG TarskiGC TarskiGB TarskiGCB Itv LineG
2 inss2 3715 . . . . 5 TarskiGC TarskiGB TarskiGCB Itv LineG TarskiGCB Itv LineG
3 inss2 3715 . . . . 5 TarskiGCB Itv LineG Itv LineG
42, 3sstri 3508 . . . 4 TarskiGC TarskiGB TarskiGCB Itv LineG Itv LineG
51, 4eqsstri 3529 . . 3 TarskiG Itv LineG
65sseli 3495 . 2 TarskiG Itv LineG
7 tglng.l . . 3 LineG
8 tglng.p . . . . 5
9 eqid 2457 . . . . 5
10 tglng.i . . . . 5 Itv
118, 9, 10istrkgl 23981 . . . 4 Itv LineG LineG
1211simprbi 464 . . 3 Itv LineG LineG
137, 12syl5eq 2510 . 2 Itv LineG
146, 13syl 16 1 TarskiG
 Colors of variables: wff setvar class Syntax hints:   wi 4   w3o 972   wceq 1395   wcel 1819  cab 2442  crab 2811  cvv 3109  wsbc 3327   cdif 3468   cin 3470  csn 4032  cfv 5594  (class class class)co 6296   cmpt2 6298  cbs 14644  cds 14721  TarskiGcstrkg 23951  TarskiGCcstrkgc 23952  TarskiGBcstrkgb 23953  TarskiGCBcstrkgcb 23954  Itvcitv 23958  LineGclng 23959 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-trkg 23976 This theorem is referenced by:  tglnfn  24060  tglnunirn  24061  tglngval  24064  tgisline  24133
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