Theorem mirconn 28913

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mirconn.m	⊢ 𝑀 = (𝑆‘𝐴)
mirconn.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mirconn.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
mirconn.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
mirconn.1	⊢ (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋)))

Assertion

Ref	Expression
mirconn	⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))

Proof of Theorem mirconn

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝐺 ∈ TarskiG)
6		mirconn.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7	6	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝑋 ∈ 𝑃)
8		mirconn.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9	8	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ 𝑃)
10		mirval.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
11		mirval.s	. . . . 5 ⊢ 𝑆 = (pInvG‘𝐺)
12		mirconn.m	. . . . 5 ⊢ 𝑀 = (𝑆‘𝐴)
13		mirconn.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
14	1, 2, 3, 10, 11, 4, 8, 12, 13	mircl 28896	. . . 4 ⊢ (𝜑 → (𝑀‘𝑌) ∈ 𝑃)
15	14	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → (𝑀‘𝑌) ∈ 𝑃)
16	13	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝑌 ∈ 𝑃)
17		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝑋 ∈ (𝐴𝐼𝑌))
18	1, 2, 3, 10, 11, 4, 8, 12, 13	mirbtwn 28893	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝑌)𝐼𝑌))
19	18	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ ((𝑀‘𝑌)𝐼𝑌))
20	1, 2, 3, 5, 7, 9, 15, 16, 17, 19	tgbtwnintr 28724	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
21	1, 2, 3, 4, 6, 8	tgbtwntriv2 28718	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝐴))
22	21	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼𝐴))
23		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → 𝑌 = 𝐴)
24	23	fveq2d 6883	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → (𝑀‘𝑌) = (𝑀‘𝐴))
25	1, 2, 3, 10, 11, 4, 8, 12	mircinv 28903	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴)
26	25	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → (𝑀‘𝐴) = 𝐴)
27	24, 26	eqtrd 2804	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → (𝑀‘𝑌) = 𝐴)
28	27	oveq2d 7424	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → (𝑋𝐼(𝑀‘𝑌)) = (𝑋𝐼𝐴))
29	22, 28	eleqtrrd 2872	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
30	29	adantlr 727	. . 3 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
31	4	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝐺 ∈ TarskiG)
32	6	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑋 ∈ 𝑃)
33	13	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑌 ∈ 𝑃)
34	8	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝐴 ∈ 𝑃)
35	14	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → (𝑀‘𝑌) ∈ 𝑃)
36		simpr 489	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑌 ≠ 𝐴)
37		simplr 780	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑌 ∈ (𝐴𝐼𝑋))
38	1, 2, 3, 31, 34, 33, 32, 37	tgbtwncom 28719	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑌 ∈ (𝑋𝐼𝐴))
39	1, 2, 3, 4, 14, 8, 13, 18	tgbtwncom 28719	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑌𝐼(𝑀‘𝑌)))
40	39	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝐴 ∈ (𝑌𝐼(𝑀‘𝑌)))
41	1, 2, 3, 31, 32, 33, 34, 35, 36, 38, 40	tgbtwnouttr2 28726	. . 3 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
42	30, 41	pm2.61dane 3051	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
43		mirconn.1	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋)))
44	20, 42, 43	mpjaodan 973	1 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ‘cfv 6534  (class class class)co 7408  Basecbs 17265  distcds 17315  TarskiGcstrkg 28658  Itvcitv 28664  LineGclng 28665  pInvGcmir 28887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-trkgc 28679  df-trkgb 28680  df-trkgcb 28681  df-trkg 28684  df-mir 28888

This theorem is referenced by:  mirbtwnhl  28915