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Theorem tgcgrextend 25601
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 466 . . 3 ((𝜑𝐴 = 𝐵) → (𝐵 𝐶) = (𝐸 𝐹))
3 simpr 471 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
43oveq1d 6808 . . 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 466 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
10 tgcgrextend.a . . . . . 6 (𝜑𝐴𝑃)
1110adantr 466 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐴𝑃)
12 tgcgrextend.b . . . . . 6 (𝜑𝐵𝑃)
1312adantr 466 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐵𝑃)
14 tgcgrextend.d . . . . . 6 (𝜑𝐷𝑃)
1514adantr 466 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐷𝑃)
16 tgcgrextend.e . . . . . 6 (𝜑𝐸𝑃)
1716adantr 466 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐸𝑃)
18 tgcgrextend.3 . . . . . 6 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
1918adantr 466 . . . . 5 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐷 𝐸))
205, 6, 7, 9, 11, 13, 15, 17, 19, 3tgcgreq 25598 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐷 = 𝐸)
2120oveq1d 6808 . . 3 ((𝜑𝐴 = 𝐵) → (𝐷 𝐹) = (𝐸 𝐹))
222, 4, 213eqtr4d 2815 . 2 ((𝜑𝐴 = 𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
238adantr 466 . . 3 ((𝜑𝐴𝐵) → 𝐺 ∈ TarskiG)
24 tgcgrextend.c . . . 4 (𝜑𝐶𝑃)
2524adantr 466 . . 3 ((𝜑𝐴𝐵) → 𝐶𝑃)
2610adantr 466 . . 3 ((𝜑𝐴𝐵) → 𝐴𝑃)
27 tgcgrextend.f . . . 4 (𝜑𝐹𝑃)
2827adantr 466 . . 3 ((𝜑𝐴𝐵) → 𝐹𝑃)
2914adantr 466 . . 3 ((𝜑𝐴𝐵) → 𝐷𝑃)
3012adantr 466 . . . 4 ((𝜑𝐴𝐵) → 𝐵𝑃)
3116adantr 466 . . . 4 ((𝜑𝐴𝐵) → 𝐸𝑃)
32 simpr 471 . . . 4 ((𝜑𝐴𝐵) → 𝐴𝐵)
33 tgcgrextend.1 . . . . 5 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
3433adantr 466 . . . 4 ((𝜑𝐴𝐵) → 𝐵 ∈ (𝐴𝐼𝐶))
35 tgcgrextend.2 . . . . 5 (𝜑𝐸 ∈ (𝐷𝐼𝐹))
3635adantr 466 . . . 4 ((𝜑𝐴𝐵) → 𝐸 ∈ (𝐷𝐼𝐹))
3718adantr 466 . . . 4 ((𝜑𝐴𝐵) → (𝐴 𝐵) = (𝐷 𝐸))
381adantr 466 . . . 4 ((𝜑𝐴𝐵) → (𝐵 𝐶) = (𝐸 𝐹))
395, 6, 7, 23, 26, 29tgcgrtriv 25600 . . . 4 ((𝜑𝐴𝐵) → (𝐴 𝐴) = (𝐷 𝐷))
405, 6, 7, 23, 26, 30, 29, 31, 37tgcgrcomlr 25596 . . . 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 25585 . . 3 ((𝜑𝐴𝐵) → (𝐶 𝐴) = (𝐹 𝐷))
425, 6, 7, 23, 25, 26, 28, 29, 41tgcgrcomlr 25596 . 2 ((𝜑𝐴𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
4322, 42pm2.61dane 3030 1 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  cfv 6031  (class class class)co 6793  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-trkgc 25568  df-trkgcb 25570  df-trkg 25573
This theorem is referenced by:  tgsegconeq  25602  tgcgrxfr  25634  lnext  25683  tgbtwnconn1lem1  25688  tgbtwnconn1lem2  25689  tgbtwnconn1lem3  25690  miriso  25786  mircgrextend  25798  midexlem  25808  opphllem  25848  dfcgra2  25942
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