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Theorem tgcgrextend 25280
 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 481 . . 3 ((𝜑𝐴 = 𝐵) → (𝐵 𝐶) = (𝐸 𝐹))
3 simpr 477 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
43oveq1d 6619 . . 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 481 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
10 tgcgrextend.a . . . . . 6 (𝜑𝐴𝑃)
1110adantr 481 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐴𝑃)
12 tgcgrextend.b . . . . . 6 (𝜑𝐵𝑃)
1312adantr 481 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐵𝑃)
14 tgcgrextend.d . . . . . 6 (𝜑𝐷𝑃)
1514adantr 481 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐷𝑃)
16 tgcgrextend.e . . . . . 6 (𝜑𝐸𝑃)
1716adantr 481 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐸𝑃)
18 tgcgrextend.3 . . . . . 6 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
1918adantr 481 . . . . 5 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐷 𝐸))
205, 6, 7, 9, 11, 13, 15, 17, 19, 3tgcgreq 25277 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐷 = 𝐸)
2120oveq1d 6619 . . 3 ((𝜑𝐴 = 𝐵) → (𝐷 𝐹) = (𝐸 𝐹))
222, 4, 213eqtr4d 2665 . 2 ((𝜑𝐴 = 𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
238adantr 481 . . 3 ((𝜑𝐴𝐵) → 𝐺 ∈ TarskiG)
24 tgcgrextend.c . . . 4 (𝜑𝐶𝑃)
2524adantr 481 . . 3 ((𝜑𝐴𝐵) → 𝐶𝑃)
2610adantr 481 . . 3 ((𝜑𝐴𝐵) → 𝐴𝑃)
27 tgcgrextend.f . . . 4 (𝜑𝐹𝑃)
2827adantr 481 . . 3 ((𝜑𝐴𝐵) → 𝐹𝑃)
2914adantr 481 . . 3 ((𝜑𝐴𝐵) → 𝐷𝑃)
3012adantr 481 . . . 4 ((𝜑𝐴𝐵) → 𝐵𝑃)
3116adantr 481 . . . 4 ((𝜑𝐴𝐵) → 𝐸𝑃)
32 simpr 477 . . . 4 ((𝜑𝐴𝐵) → 𝐴𝐵)
33 tgcgrextend.1 . . . . 5 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
3433adantr 481 . . . 4 ((𝜑𝐴𝐵) → 𝐵 ∈ (𝐴𝐼𝐶))
35 tgcgrextend.2 . . . . 5 (𝜑𝐸 ∈ (𝐷𝐼𝐹))
3635adantr 481 . . . 4 ((𝜑𝐴𝐵) → 𝐸 ∈ (𝐷𝐼𝐹))
3718adantr 481 . . . 4 ((𝜑𝐴𝐵) → (𝐴 𝐵) = (𝐷 𝐸))
381adantr 481 . . . 4 ((𝜑𝐴𝐵) → (𝐵 𝐶) = (𝐸 𝐹))
395, 6, 7, 23, 26, 29tgcgrtriv 25279 . . . 4 ((𝜑𝐴𝐵) → (𝐴 𝐴) = (𝐷 𝐷))
405, 6, 7, 23, 26, 30, 29, 31, 37tgcgrcomlr 25275 . . . 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 25264 . . 3 ((𝜑𝐴𝐵) → (𝐶 𝐴) = (𝐹 𝐷))
425, 6, 7, 23, 25, 26, 28, 29, 41tgcgrcomlr 25275 . 2 ((𝜑𝐴𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
4322, 42pm2.61dane 2877 1 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  distcds 15871  TarskiGcstrkg 25229  Itvcitv 25235 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-trkgc 25247  df-trkgcb 25249  df-trkg 25252 This theorem is referenced by:  tgsegconeq  25281  tgcgrxfr  25313  lnext  25362  tgbtwnconn1lem1  25367  tgbtwnconn1lem2  25368  tgbtwnconn1lem3  25369  miriso  25465  mircgrextend  25477  midexlem  25487  opphllem  25527  dfcgra2  25621
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