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Theorem tgcgrextend 28000
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 = (Baseβ€˜πΊ)
tkgeom.d βˆ’ = (distβ€˜πΊ)
tkgeom.i 𝐼 = (Itvβ€˜πΊ)
tkgeom.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tgcgrextend.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
tgcgrextend.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
tgcgrextend.c (πœ‘ β†’ 𝐢 ∈ 𝑃)
tgcgrextend.d (πœ‘ β†’ 𝐷 ∈ 𝑃)
tgcgrextend.e (πœ‘ β†’ 𝐸 ∈ 𝑃)
tgcgrextend.f (πœ‘ β†’ 𝐹 ∈ 𝑃)
tgcgrextend.1 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐢))
tgcgrextend.2 (πœ‘ β†’ 𝐸 ∈ (𝐷𝐼𝐹))
tgcgrextend.3 (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))
tgcgrextend.4 (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))
Assertion
Ref Expression
tgcgrextend (πœ‘ β†’ (𝐴 βˆ’ 𝐢) = (𝐷 βˆ’ 𝐹))

Proof of Theorem tgcgrextend
StepHypRef Expression
1 tgcgrextend.4 . . . 4 (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))
21adantr 480 . . 3 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))
3 simpr 484 . . . 4 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐴 = 𝐡)
43oveq1d 7427 . . 3 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ (𝐴 βˆ’ 𝐢) = (𝐡 βˆ’ 𝐢))
5 tkgeom.p . . . . 5 𝑃 = (Baseβ€˜πΊ)
6 tkgeom.d . . . . 5 βˆ’ = (distβ€˜πΊ)
7 tkgeom.i . . . . 5 𝐼 = (Itvβ€˜πΊ)
8 tkgeom.g . . . . . 6 (πœ‘ β†’ 𝐺 ∈ TarskiG)
98adantr 480 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐺 ∈ TarskiG)
10 tgcgrextend.a . . . . . 6 (πœ‘ β†’ 𝐴 ∈ 𝑃)
1110adantr 480 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐴 ∈ 𝑃)
12 tgcgrextend.b . . . . . 6 (πœ‘ β†’ 𝐡 ∈ 𝑃)
1312adantr 480 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐡 ∈ 𝑃)
14 tgcgrextend.d . . . . . 6 (πœ‘ β†’ 𝐷 ∈ 𝑃)
1514adantr 480 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐷 ∈ 𝑃)
16 tgcgrextend.e . . . . . 6 (πœ‘ β†’ 𝐸 ∈ 𝑃)
1716adantr 480 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐸 ∈ 𝑃)
18 tgcgrextend.3 . . . . . 6 (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))
1918adantr 480 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))
205, 6, 7, 9, 11, 13, 15, 17, 19, 3tgcgreq 27997 . . . 4 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐷 = 𝐸)
2120oveq1d 7427 . . 3 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ (𝐷 βˆ’ 𝐹) = (𝐸 βˆ’ 𝐹))
222, 4, 213eqtr4d 2781 . 2 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ (𝐴 βˆ’ 𝐢) = (𝐷 βˆ’ 𝐹))
238adantr 480 . . 3 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐺 ∈ TarskiG)
24 tgcgrextend.c . . . 4 (πœ‘ β†’ 𝐢 ∈ 𝑃)
2524adantr 480 . . 3 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐢 ∈ 𝑃)
2610adantr 480 . . 3 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐴 ∈ 𝑃)
27 tgcgrextend.f . . . 4 (πœ‘ β†’ 𝐹 ∈ 𝑃)
2827adantr 480 . . 3 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐹 ∈ 𝑃)
2914adantr 480 . . 3 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐷 ∈ 𝑃)
3012adantr 480 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐡 ∈ 𝑃)
3116adantr 480 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐸 ∈ 𝑃)
32 simpr 484 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐴 β‰  𝐡)
33 tgcgrextend.1 . . . . 5 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐢))
3433adantr 480 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐡 ∈ (𝐴𝐼𝐢))
35 tgcgrextend.2 . . . . 5 (πœ‘ β†’ 𝐸 ∈ (𝐷𝐼𝐹))
3635adantr 480 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐸 ∈ (𝐷𝐼𝐹))
3718adantr 480 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))
381adantr 480 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))
395, 6, 7, 23, 26, 29tgcgrtriv 27999 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (𝐴 βˆ’ 𝐴) = (𝐷 βˆ’ 𝐷))
405, 6, 7, 23, 26, 30, 29, 31, 37tgcgrcomlr 27995 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (𝐡 βˆ’ 𝐴) = (𝐸 βˆ’ 𝐷))
415, 6, 7, 23, 26, 30, 25, 29, 31, 28, 26, 29, 32, 34, 36, 37, 38, 39, 40axtg5seg 27980 . . 3 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (𝐢 βˆ’ 𝐴) = (𝐹 βˆ’ 𝐷))
425, 6, 7, 23, 25, 26, 28, 29, 41tgcgrcomlr 27995 . 2 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (𝐴 βˆ’ 𝐢) = (𝐷 βˆ’ 𝐹))
4322, 42pm2.61dane 3028 1 (πœ‘ β†’ (𝐴 βˆ’ 𝐢) = (𝐷 βˆ’ 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  β€˜cfv 6544  (class class class)co 7412  Basecbs 17149  distcds 17211  TarskiGcstrkg 27942  Itvcitv 27948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7415  df-trkgc 27963  df-trkgcb 27965  df-trkg 27968
This theorem is referenced by:  tgsegconeq  28001  tgcgrxfr  28033  lnext  28082  tgbtwnconn1lem1  28087  tgbtwnconn1lem2  28088  tgbtwnconn1lem3  28089  miriso  28185  mircgrextend  28197  midexlem  28207  opphllem  28250  flatcgra  28339  dfcgra2  28345
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