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Theorem mirconn 28700
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 28683 . . . 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 28680 . . . 4 (𝜑𝐴 ∈ ((𝑀𝑌)𝐼𝑌))
1918adantr 480 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ ((𝑀𝑌)𝐼𝑌))
201, 2, 3, 5, 7, 9, 15, 16, 17, 19tgbtwnintr 28515 . 2 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
211, 2, 3, 4, 6, 8tgbtwntriv2 28509 . . . . . 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 28690 . . . . . . . 8 (𝜑 → (𝑀𝐴) = 𝐴)
2625adantr 480 . . . . . . 7 ((𝜑𝑌 = 𝐴) → (𝑀𝐴) = 𝐴)
2724, 26eqtrd 2774 . . . . . 6 ((𝜑𝑌 = 𝐴) → (𝑀𝑌) = 𝐴)
2827oveq2d 7446 . . . . 5 ((𝜑𝑌 = 𝐴) → (𝑋𝐼(𝑀𝑌)) = (𝑋𝐼𝐴))
2922, 28eleqtrrd 2841 . . . 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 28510 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑌 ∈ (𝑋𝐼𝐴))
391, 2, 3, 4, 14, 8, 13, 18tgbtwncom 28510 . . . . 5 (𝜑𝐴 ∈ (𝑌𝐼(𝑀𝑌)))
4039ad2antrr 726 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐴 ∈ (𝑌𝐼(𝑀𝑌)))
411, 2, 3, 31, 32, 33, 34, 35, 36, 38, 40tgbtwnouttr2 28517 . . 3 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
4230, 41pm2.61dane 3026 . 2 ((𝜑𝑌 ∈ (𝐴𝐼𝑋)) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
43 mirconn.1 . 2 (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋)))
4420, 42, 43mpjaodan 960 1 (𝜑𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1536  wcel 2105  wne 2937  cfv 6562  (class class class)co 7430  Basecbs 17244  distcds 17306  TarskiGcstrkg 28449  Itvcitv 28455  LineGclng 28456  pInvGcmir 28674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-trkgc 28470  df-trkgb 28471  df-trkgcb 28472  df-trkg 28475  df-mir 28675
This theorem is referenced by:  mirbtwnhl  28702
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