Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgcgrextend Structured version   Visualization version   GIF version

Theorem tgcgrextend 25853
 Description: Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) (Shortened by David A. Wheeler and Thierry Arnoux, 22-Apr-2020.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgcgrextend.a (𝜑𝐴𝑃)
tgcgrextend.b (𝜑𝐵𝑃)
tgcgrextend.c (𝜑𝐶𝑃)
tgcgrextend.d (𝜑𝐷𝑃)
tgcgrextend.e (𝜑𝐸𝑃)
tgcgrextend.f (𝜑𝐹𝑃)
tgcgrextend.1 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgcgrextend.2 (𝜑𝐸 ∈ (𝐷𝐼𝐹))
tgcgrextend.3 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
tgcgrextend.4 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
Assertion
Ref Expression
tgcgrextend (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))

Proof of Theorem tgcgrextend
StepHypRef Expression
1 tgcgrextend.4 . . . 4 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
21adantr 474 . . 3 ((𝜑𝐴 = 𝐵) → (𝐵 𝐶) = (𝐸 𝐹))
3 simpr 479 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
43oveq1d 6939 . . 3 ((𝜑𝐴 = 𝐵) → (𝐴 𝐶) = (𝐵 𝐶))
5 tkgeom.p . . . . 5 𝑃 = (Base‘𝐺)
6 tkgeom.d . . . . 5 = (dist‘𝐺)
7 tkgeom.i . . . . 5 𝐼 = (Itv‘𝐺)
8 tkgeom.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
98adantr 474 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
10 tgcgrextend.a . . . . . 6 (𝜑𝐴𝑃)
1110adantr 474 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐴𝑃)
12 tgcgrextend.b . . . . . 6 (𝜑𝐵𝑃)
1312adantr 474 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐵𝑃)
14 tgcgrextend.d . . . . . 6 (𝜑𝐷𝑃)
1514adantr 474 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐷𝑃)
16 tgcgrextend.e . . . . . 6 (𝜑𝐸𝑃)
1716adantr 474 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐸𝑃)
18 tgcgrextend.3 . . . . . 6 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
1918adantr 474 . . . . 5 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐷 𝐸))
205, 6, 7, 9, 11, 13, 15, 17, 19, 3tgcgreq 25850 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐷 = 𝐸)
2120oveq1d 6939 . . 3 ((𝜑𝐴 = 𝐵) → (𝐷 𝐹) = (𝐸 𝐹))
222, 4, 213eqtr4d 2824 . 2 ((𝜑𝐴 = 𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
238adantr 474 . . 3 ((𝜑𝐴𝐵) → 𝐺 ∈ TarskiG)
24 tgcgrextend.c . . . 4 (𝜑𝐶𝑃)
2524adantr 474 . . 3 ((𝜑𝐴𝐵) → 𝐶𝑃)
2610adantr 474 . . 3 ((𝜑𝐴𝐵) → 𝐴𝑃)
27 tgcgrextend.f . . . 4 (𝜑𝐹𝑃)
2827adantr 474 . . 3 ((𝜑𝐴𝐵) → 𝐹𝑃)
2914adantr 474 . . 3 ((𝜑𝐴𝐵) → 𝐷𝑃)
3012adantr 474 . . . 4 ((𝜑𝐴𝐵) → 𝐵𝑃)
3116adantr 474 . . . 4 ((𝜑𝐴𝐵) → 𝐸𝑃)
32 simpr 479 . . . 4 ((𝜑𝐴𝐵) → 𝐴𝐵)
33 tgcgrextend.1 . . . . 5 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
3433adantr 474 . . . 4 ((𝜑𝐴𝐵) → 𝐵 ∈ (𝐴𝐼𝐶))
35 tgcgrextend.2 . . . . 5 (𝜑𝐸 ∈ (𝐷𝐼𝐹))
3635adantr 474 . . . 4 ((𝜑𝐴𝐵) → 𝐸 ∈ (𝐷𝐼𝐹))
3718adantr 474 . . . 4 ((𝜑𝐴𝐵) → (𝐴 𝐵) = (𝐷 𝐸))
381adantr 474 . . . 4 ((𝜑𝐴𝐵) → (𝐵 𝐶) = (𝐸 𝐹))
395, 6, 7, 23, 26, 29tgcgrtriv 25852 . . . 4 ((𝜑𝐴𝐵) → (𝐴 𝐴) = (𝐷 𝐷))
405, 6, 7, 23, 26, 30, 29, 31, 37tgcgrcomlr 25848 . . . 4 ((𝜑𝐴𝐵) → (𝐵 𝐴) = (𝐸 𝐷))
415, 6, 7, 23, 26, 30, 25, 29, 31, 28, 26, 29, 32, 34, 36, 37, 38, 39, 40axtg5seg 25833 . . 3 ((𝜑𝐴𝐵) → (𝐶 𝐴) = (𝐹 𝐷))
425, 6, 7, 23, 25, 26, 28, 29, 41tgcgrcomlr 25848 . 2 ((𝜑𝐴𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
4322, 42pm2.61dane 3057 1 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ‘cfv 6137  (class class class)co 6924  Basecbs 16266  distcds 16358  TarskiGcstrkg 25798  Itvcitv 25804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5027 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-iota 6101  df-fv 6145  df-ov 6927  df-trkgc 25816  df-trkgcb 25818  df-trkg 25821 This theorem is referenced by:  tgsegconeq  25854  tgcgrxfr  25886  lnext  25935  tgbtwnconn1lem1  25940  tgbtwnconn1lem2  25941  tgbtwnconn1lem3  25942  miriso  26038  mircgrextend  26050  midexlem  26060  opphllem  26100  flatcgra  26189  dfcgra2  26195
 Copyright terms: Public domain W3C validator