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Theorem mirconn 26470
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 484 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝐺 ∈ TarskiG)
6 mirconn.x . . . 4 (𝜑𝑋𝑃)
76adantr 484 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝑋𝑃)
8 mirconn.a . . . 4 (𝜑𝐴𝑃)
98adantr 484 . . 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 26453 . . . 4 (𝜑 → (𝑀𝑌) ∈ 𝑃)
1514adantr 484 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → (𝑀𝑌) ∈ 𝑃)
1613adantr 484 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝑌𝑃)
17 simpr 488 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝑋 ∈ (𝐴𝐼𝑌))
181, 2, 3, 10, 11, 4, 8, 12, 13mirbtwn 26450 . . . 4 (𝜑𝐴 ∈ ((𝑀𝑌)𝐼𝑌))
1918adantr 484 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ ((𝑀𝑌)𝐼𝑌))
201, 2, 3, 5, 7, 9, 15, 16, 17, 19tgbtwnintr 26285 . 2 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
211, 2, 3, 4, 6, 8tgbtwntriv2 26279 . . . . . 6 (𝜑𝐴 ∈ (𝑋𝐼𝐴))
2221adantr 484 . . . . 5 ((𝜑𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼𝐴))
23 simpr 488 . . . . . . . 8 ((𝜑𝑌 = 𝐴) → 𝑌 = 𝐴)
2423fveq2d 6656 . . . . . . 7 ((𝜑𝑌 = 𝐴) → (𝑀𝑌) = (𝑀𝐴))
251, 2, 3, 10, 11, 4, 8, 12mircinv 26460 . . . . . . . 8 (𝜑 → (𝑀𝐴) = 𝐴)
2625adantr 484 . . . . . . 7 ((𝜑𝑌 = 𝐴) → (𝑀𝐴) = 𝐴)
2724, 26eqtrd 2857 . . . . . 6 ((𝜑𝑌 = 𝐴) → (𝑀𝑌) = 𝐴)
2827oveq2d 7156 . . . . 5 ((𝜑𝑌 = 𝐴) → (𝑋𝐼(𝑀𝑌)) = (𝑋𝐼𝐴))
2922, 28eleqtrrd 2917 . . . 4 ((𝜑𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
3029adantlr 714 . . 3 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
314ad2antrr 725 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐺 ∈ TarskiG)
326ad2antrr 725 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑋𝑃)
3313ad2antrr 725 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑌𝑃)
348ad2antrr 725 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐴𝑃)
3514ad2antrr 725 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → (𝑀𝑌) ∈ 𝑃)
36 simpr 488 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑌𝐴)
37 simplr 768 . . . . 5 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑌 ∈ (𝐴𝐼𝑋))
381, 2, 3, 31, 34, 33, 32, 37tgbtwncom 26280 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑌 ∈ (𝑋𝐼𝐴))
391, 2, 3, 4, 14, 8, 13, 18tgbtwncom 26280 . . . . 5 (𝜑𝐴 ∈ (𝑌𝐼(𝑀𝑌)))
4039ad2antrr 725 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐴 ∈ (𝑌𝐼(𝑀𝑌)))
411, 2, 3, 31, 32, 33, 34, 35, 36, 38, 40tgbtwnouttr2 26287 . . 3 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
4230, 41pm2.61dane 3098 . 2 ((𝜑𝑌 ∈ (𝐴𝐼𝑋)) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
43 mirconn.1 . 2 (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋)))
4420, 42, 43mpjaodan 956 1 (𝜑𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844   = wceq 1538  wcel 2114  wne 3011  cfv 6334  (class class class)co 7140  Basecbs 16474  distcds 16565  TarskiGcstrkg 26222  Itvcitv 26228  LineGclng 26229  pInvGcmir 26444
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-trkgc 26240  df-trkgb 26241  df-trkgcb 26242  df-trkg 26245  df-mir 26445
This theorem is referenced by:  mirbtwnhl  26472
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