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Theorem tgcgrextend 26282
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 484 . . 3 ((𝜑𝐴 = 𝐵) → (𝐵 𝐶) = (𝐸 𝐹))
3 simpr 488 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
43oveq1d 7164 . . 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 484 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
10 tgcgrextend.a . . . . . 6 (𝜑𝐴𝑃)
1110adantr 484 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐴𝑃)
12 tgcgrextend.b . . . . . 6 (𝜑𝐵𝑃)
1312adantr 484 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐵𝑃)
14 tgcgrextend.d . . . . . 6 (𝜑𝐷𝑃)
1514adantr 484 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐷𝑃)
16 tgcgrextend.e . . . . . 6 (𝜑𝐸𝑃)
1716adantr 484 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐸𝑃)
18 tgcgrextend.3 . . . . . 6 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
1918adantr 484 . . . . 5 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐷 𝐸))
205, 6, 7, 9, 11, 13, 15, 17, 19, 3tgcgreq 26279 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐷 = 𝐸)
2120oveq1d 7164 . . 3 ((𝜑𝐴 = 𝐵) → (𝐷 𝐹) = (𝐸 𝐹))
222, 4, 213eqtr4d 2869 . 2 ((𝜑𝐴 = 𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
238adantr 484 . . 3 ((𝜑𝐴𝐵) → 𝐺 ∈ TarskiG)
24 tgcgrextend.c . . . 4 (𝜑𝐶𝑃)
2524adantr 484 . . 3 ((𝜑𝐴𝐵) → 𝐶𝑃)
2610adantr 484 . . 3 ((𝜑𝐴𝐵) → 𝐴𝑃)
27 tgcgrextend.f . . . 4 (𝜑𝐹𝑃)
2827adantr 484 . . 3 ((𝜑𝐴𝐵) → 𝐹𝑃)
2914adantr 484 . . 3 ((𝜑𝐴𝐵) → 𝐷𝑃)
3012adantr 484 . . . 4 ((𝜑𝐴𝐵) → 𝐵𝑃)
3116adantr 484 . . . 4 ((𝜑𝐴𝐵) → 𝐸𝑃)
32 simpr 488 . . . 4 ((𝜑𝐴𝐵) → 𝐴𝐵)
33 tgcgrextend.1 . . . . 5 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
3433adantr 484 . . . 4 ((𝜑𝐴𝐵) → 𝐵 ∈ (𝐴𝐼𝐶))
35 tgcgrextend.2 . . . . 5 (𝜑𝐸 ∈ (𝐷𝐼𝐹))
3635adantr 484 . . . 4 ((𝜑𝐴𝐵) → 𝐸 ∈ (𝐷𝐼𝐹))
3718adantr 484 . . . 4 ((𝜑𝐴𝐵) → (𝐴 𝐵) = (𝐷 𝐸))
381adantr 484 . . . 4 ((𝜑𝐴𝐵) → (𝐵 𝐶) = (𝐸 𝐹))
395, 6, 7, 23, 26, 29tgcgrtriv 26281 . . . 4 ((𝜑𝐴𝐵) → (𝐴 𝐴) = (𝐷 𝐷))
405, 6, 7, 23, 26, 30, 29, 31, 37tgcgrcomlr 26277 . . . 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 26262 . . 3 ((𝜑𝐴𝐵) → (𝐶 𝐴) = (𝐹 𝐷))
425, 6, 7, 23, 25, 26, 28, 29, 41tgcgrcomlr 26277 . 2 ((𝜑𝐴𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
4322, 42pm2.61dane 3101 1 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3014  cfv 6343  (class class class)co 7149  Basecbs 16483  distcds 16574  TarskiGcstrkg 26227  Itvcitv 26233
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-trkgc 26245  df-trkgcb 26247  df-trkg 26250
This theorem is referenced by:  tgsegconeq  26283  tgcgrxfr  26315  lnext  26364  tgbtwnconn1lem1  26369  tgbtwnconn1lem2  26370  tgbtwnconn1lem3  26371  miriso  26467  mircgrextend  26479  midexlem  26489  opphllem  26532  flatcgra  26621  dfcgra2  26627
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