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Theorem mirconn 25794
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 466 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝐺 ∈ TarskiG)
6 mirconn.x . . . 4 (𝜑𝑋𝑃)
76adantr 466 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝑋𝑃)
8 mirconn.a . . . 4 (𝜑𝐴𝑃)
98adantr 466 . . 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 25777 . . . 4 (𝜑 → (𝑀𝑌) ∈ 𝑃)
1514adantr 466 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → (𝑀𝑌) ∈ 𝑃)
1613adantr 466 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝑌𝑃)
17 simpr 471 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝑋 ∈ (𝐴𝐼𝑌))
181, 2, 3, 10, 11, 4, 8, 12, 13mirbtwn 25774 . . . 4 (𝜑𝐴 ∈ ((𝑀𝑌)𝐼𝑌))
1918adantr 466 . . 3 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ ((𝑀𝑌)𝐼𝑌))
201, 2, 3, 5, 7, 9, 15, 16, 17, 19tgbtwnintr 25609 . 2 ((𝜑𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
211, 2, 3, 4, 6, 8tgbtwntriv2 25603 . . . . . 6 (𝜑𝐴 ∈ (𝑋𝐼𝐴))
2221adantr 466 . . . . 5 ((𝜑𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼𝐴))
23 simpr 471 . . . . . . . 8 ((𝜑𝑌 = 𝐴) → 𝑌 = 𝐴)
2423fveq2d 6337 . . . . . . 7 ((𝜑𝑌 = 𝐴) → (𝑀𝑌) = (𝑀𝐴))
251, 2, 3, 10, 11, 4, 8, 12mircinv 25784 . . . . . . . 8 (𝜑 → (𝑀𝐴) = 𝐴)
2625adantr 466 . . . . . . 7 ((𝜑𝑌 = 𝐴) → (𝑀𝐴) = 𝐴)
2724, 26eqtrd 2805 . . . . . 6 ((𝜑𝑌 = 𝐴) → (𝑀𝑌) = 𝐴)
2827oveq2d 6812 . . . . 5 ((𝜑𝑌 = 𝐴) → (𝑋𝐼(𝑀𝑌)) = (𝑋𝐼𝐴))
2922, 28eleqtrrd 2853 . . . 4 ((𝜑𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
3029adantlr 694 . . 3 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
314ad2antrr 705 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐺 ∈ TarskiG)
326ad2antrr 705 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑋𝑃)
3313ad2antrr 705 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑌𝑃)
348ad2antrr 705 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐴𝑃)
3514ad2antrr 705 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → (𝑀𝑌) ∈ 𝑃)
36 simpr 471 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑌𝐴)
37 simplr 752 . . . . 5 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑌 ∈ (𝐴𝐼𝑋))
381, 2, 3, 31, 34, 33, 32, 37tgbtwncom 25604 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝑌 ∈ (𝑋𝐼𝐴))
391, 2, 3, 4, 14, 8, 13, 18tgbtwncom 25604 . . . . 5 (𝜑𝐴 ∈ (𝑌𝐼(𝑀𝑌)))
4039ad2antrr 705 . . . 4 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐴 ∈ (𝑌𝐼(𝑀𝑌)))
411, 2, 3, 31, 32, 33, 34, 35, 36, 38, 40tgbtwnouttr2 25611 . . 3 (((𝜑𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
4230, 41pm2.61dane 3030 . 2 ((𝜑𝑌 ∈ (𝐴𝐼𝑋)) → 𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
43 mirconn.1 . 2 (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋)))
4420, 42, 43mpjaodan 943 1 (𝜑𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 836   = wceq 1631  wcel 2145  wne 2943  cfv 6030  (class class class)co 6796  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557  pInvGcmir 25768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573  df-mir 25769
This theorem is referenced by:  mirbtwnhl  25796
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