MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgcgrextend Structured version   Visualization version   GIF version
Theorem tgcgrextend 27774
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 481 . . 3 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))
3 simpr 485 . . . 4 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐴 = 𝐡)
43oveq1d 7426 . . 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 481 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐺 ∈ TarskiG)
10 tgcgrextend.a . . . . . 6 (πœ‘ β†’ 𝐴 ∈ 𝑃)
1110adantr 481 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐴 ∈ 𝑃)
12 tgcgrextend.b . . . . . 6 (πœ‘ β†’ 𝐡 ∈ 𝑃)
1312adantr 481 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐡 ∈ 𝑃)
14 tgcgrextend.d . . . . . 6 (πœ‘ β†’ 𝐷 ∈ 𝑃)
1514adantr 481 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐷 ∈ 𝑃)
16 tgcgrextend.e . . . . . 6 (πœ‘ β†’ 𝐸 ∈ 𝑃)
1716adantr 481 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐸 ∈ 𝑃)
18 tgcgrextend.3 . . . . . 6 (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))
1918adantr 481 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))
205, 6, 7, 9, 11, 13, 15, 17, 19, 3tgcgreq 27771 . . . 4 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐷 = 𝐸)
2120oveq1d 7426 . . 3 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ (𝐷 βˆ’ 𝐹) = (𝐸 βˆ’ 𝐹))
222, 4, 213eqtr4d 2782 . 2 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ (𝐴 βˆ’ 𝐢) = (𝐷 βˆ’ 𝐹))
238adantr 481 . . 3 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐺 ∈ TarskiG)
24 tgcgrextend.c . . . 4 (πœ‘ β†’ 𝐢 ∈ 𝑃)
2524adantr 481 . . 3 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐢 ∈ 𝑃)
2610adantr 481 . . 3 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐴 ∈ 𝑃)
27 tgcgrextend.f . . . 4 (πœ‘ β†’ 𝐹 ∈ 𝑃)
2827adantr 481 . . 3 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐹 ∈ 𝑃)
2914adantr 481 . . 3 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐷 ∈ 𝑃)
3012adantr 481 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐡 ∈ 𝑃)
3116adantr 481 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐸 ∈ 𝑃)
32 simpr 485 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐴 β‰  𝐡)
33 tgcgrextend.1 . . . . 5 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐢))
3433adantr 481 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐡 ∈ (𝐴𝐼𝐢))
35 tgcgrextend.2 . . . . 5 (πœ‘ β†’ 𝐸 ∈ (𝐷𝐼𝐹))
3635adantr 481 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐸 ∈ (𝐷𝐼𝐹))
3718adantr 481 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))
381adantr 481 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))
395, 6, 7, 23, 26, 29tgcgrtriv 27773 . . . 4 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (𝐴 βˆ’ 𝐴) = (𝐷 βˆ’ 𝐷))
405, 6, 7, 23, 26, 30, 29, 31, 37tgcgrcomlr 27769 . . . 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 27754 . . 3 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (𝐢 βˆ’ 𝐴) = (𝐹 βˆ’ 𝐷))
425, 6, 7, 23, 25, 26, 28, 29, 41tgcgrcomlr 27769 . 2 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (𝐴 βˆ’ 𝐢) = (𝐷 βˆ’ 𝐹))
4322, 42pm2.61dane 3029 1 (πœ‘ β†’ (𝐴 βˆ’ 𝐢) = (𝐷 βˆ’ 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  distcds 17208  TarskiGcstrkg 27716  Itvcitv 27722
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7414  df-trkgc 27737  df-trkgcb 27739  df-trkg 27742
This theorem is referenced by:  tgsegconeq  27775  tgcgrxfr  27807  lnext  27856  tgbtwnconn1lem1  27861  tgbtwnconn1lem2  27862  tgbtwnconn1lem3  27863  miriso  27959  mircgrextend  27971  midexlem  27981  opphllem  28024  flatcgra  28113  dfcgra2  28119
  Copyright terms: Public domain W3C validator