MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mirconn Structured version   Visualization version   GIF version

Theorem mirconn 28686
Description: Point inversion of connectedness. (Contributed by Thierry Arnoux, 2-Mar-2020.)
Hypotheses
Ref Expression
mirval.p 𝑃 = (Base‘𝐺)
mirval.d = (dist‘𝐺)
mirval.i 𝐼 = (Itv‘𝐺)
mirval.l 𝐿 = (LineG‘𝐺)
mirval.s 𝑆 = (pInvG‘𝐺)
mirval.g (𝜑𝐺 ∈ TarskiG)
mirconn.m 𝑀 = (𝑆𝐴)
mirconn.a (𝜑𝐴𝑃)
mirconn.x (𝜑𝑋𝑃)
mirconn.y (𝜑𝑌𝑃)
mirconn.1 (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋)))
Assertion
Ref Expression
mirconn (𝜑𝐴 ∈ (𝑋𝐼(𝑀𝑌)))

Proof of Theorem mirconn
StepHypRef Expression
1 mirval.p . . 3 𝑃 = (Base‘𝐺)
2 mirval.d . . 3 = (dist‘𝐺)
3 mirval.i . . 3 𝐼 = (Itv‘𝐺)
4 mirval.g . . . 4 (𝜑𝐺 ∈ TarskiG)
54adantr 480 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝐺 ∈ TarskiG)
6 mirconn.x . . . 4 (𝜑𝑋𝑃)
76adantr 480 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝑋𝑃)
8 mirconn.a . . . 4 (𝜑𝐴𝑃)
98adantr 480 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴𝑃)
10 mirval.l . . . . 5 𝐿 = (LineG‘𝐺)
11 mirval.s . . . . 5 𝑆 = (pInvG‘𝐺)
12 mirconn.m . . . . 5 𝑀 = (𝑆𝐴)
13 mirconn.y . . . . 5 (𝜑𝑌𝑃)
141, 2, 3, 10, 11, 4, 8, 12, 13mircl 28669 . . . 4 (𝜑 → (𝑀𝑌) ∈ 𝑃)
1514adantr 480 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → (𝑀𝑌) ∈ 𝑃)
1613adantr 480 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝑌𝑃)
17 simpr 484 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝑋 ∈ (𝐴𝐼𝑌))
181, 2, 3, 10, 11, 4, 8, 12, 13mirbtwn 28666 . . . 4 (𝜑𝐴 ∈ ((𝑀𝑌)𝐼𝑌))
1918adantr 480 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ ((𝑀𝑌)𝐼𝑌))
201, 2, 3, 5, 7, 9, 15, 16, 17, 19tgbtwnintr 28501 . 2 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
211, 2, 3, 4, 6, 8tgbtwntriv2 28495 . . . . . 6 (𝜑𝐴 ∈ (𝑋𝐼𝐴))
2221adantr 480 . . . . 5 ((𝜑𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼𝐴))
23 simpr 484 . . . . . . . 8 ((𝜑𝑌 = 𝐴) → 𝑌 = 𝐴)
2423fveq2d 6910 . . . . . . 7 ((𝜑𝑌 = 𝐴) → (𝑀𝑌) = (𝑀𝐴))
251, 2, 3, 10, 11, 4, 8, 12mircinv 28676 . . . . . . . 8 (𝜑 → (𝑀𝐴) = 𝐴)
2625adantr 480 . . . . . . 7 ((𝜑𝑌 = 𝐴) → (𝑀𝐴) = 𝐴)
2724, 26eqtrd 2777 . . . . . 6 ((𝜑𝑌 = 𝐴) → (𝑀𝑌) = 𝐴)
2827oveq2d 7447 . . . . 5 ((𝜑𝑌 = 𝐴) → (𝑋𝐼(𝑀𝑌)) = (𝑋𝐼𝐴))
2922, 28eleqtrrd 2844 . . . 4 ((𝜑𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
3029adantlr 715 . . 3 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
314ad2antrr 726 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐺 ∈ TarskiG)
326ad2antrr 726 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑋𝑃)
3313ad2antrr 726 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑌𝑃)
348ad2antrr 726 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐴𝑃)
3514ad2antrr 726 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → (𝑀𝑌) ∈ 𝑃)
36 simpr 484 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑌𝐴)
37 simplr 769 . . . . 5 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑌 ∈ (𝐴𝐼𝑋))
381, 2, 3, 31, 34, 33, 32, 37tgbtwncom 28496 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑌 ∈ (𝑋𝐼𝐴))
391, 2, 3, 4, 14, 8, 13, 18tgbtwncom 28496 . . . . 5 (𝜑𝐴 ∈ (𝑌𝐼(𝑀𝑌)))
4039ad2antrr 726 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐴 ∈ (𝑌𝐼(𝑀𝑌)))
411, 2, 3, 31, 32, 33, 34, 35, 36, 38, 40tgbtwnouttr2 28503 . . 3 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
4230, 41pm2.61dane 3029 . 2 ((𝜑𝑌 ∈ (𝐴𝐼𝑋)) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
43 mirconn.1 . 2 (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋)))
4420, 42, 43mpjaodan 961 1 (𝜑𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  cfv 6561  (class class class)co 7431  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442  pInvGcmir 28660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461  df-mir 28661
This theorem is referenced by:  mirbtwnhl  28688
  Copyright terms: Public domain W3C validator