Theorem mirconn 28623

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 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝐺 ∈ TarskiG)
6		mirconn.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝑋 ∈ 𝑃)
8		mirconn.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9	8	adantr 480	. . 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 28606	. . . 4 ⊢ (𝜑 → (𝑀‘𝑌) ∈ 𝑃)
15	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → (𝑀‘𝑌) ∈ 𝑃)
16	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝑌 ∈ 𝑃)
17		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝑋 ∈ (𝐴𝐼𝑌))
18	1, 2, 3, 10, 11, 4, 8, 12, 13	mirbtwn 28603	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝑌)𝐼𝑌))
19	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ ((𝑀‘𝑌)𝐼𝑌))
20	1, 2, 3, 5, 7, 9, 15, 16, 17, 19	tgbtwnintr 28438	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
21	1, 2, 3, 4, 6, 8	tgbtwntriv2 28432	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝐴))
22	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼𝐴))
23		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → 𝑌 = 𝐴)
24	23	fveq2d 6826	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → (𝑀‘𝑌) = (𝑀‘𝐴))
25	1, 2, 3, 10, 11, 4, 8, 12	mircinv 28613	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴)
26	25	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → (𝑀‘𝐴) = 𝐴)
27	24, 26	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → (𝑀‘𝑌) = 𝐴)
28	27	oveq2d 7365	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → (𝑋𝐼(𝑀‘𝑌)) = (𝑋𝐼𝐴))
29	22, 28	eleqtrrd 2831	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
30	29	adantlr 715	. . 3 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
31	4	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝐺 ∈ TarskiG)
32	6	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑋 ∈ 𝑃)
33	13	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑌 ∈ 𝑃)
34	8	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝐴 ∈ 𝑃)
35	14	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → (𝑀‘𝑌) ∈ 𝑃)
36		simpr 484	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑌 ≠ 𝐴)
37		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑌 ∈ (𝐴𝐼𝑋))
38	1, 2, 3, 31, 34, 33, 32, 37	tgbtwncom 28433	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑌 ∈ (𝑋𝐼𝐴))
39	1, 2, 3, 4, 14, 8, 13, 18	tgbtwncom 28433	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑌𝐼(𝑀‘𝑌)))
40	39	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝐴 ∈ (𝑌𝐼(𝑀‘𝑌)))
41	1, 2, 3, 31, 32, 33, 34, 35, 36, 38, 40	tgbtwnouttr2 28440	. . 3 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
42	30, 41	pm2.61dane 3012	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
43		mirconn.1	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋)))
44	20, 42, 43	mpjaodan 960	1 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  distcds 17170  TarskiGcstrkg 28372  Itvcitv 28378  LineGclng 28379  pInvGcmir 28597

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-trkgc 28393  df-trkgb 28394  df-trkgcb 28395  df-trkg 28398  df-mir 28598

This theorem is referenced by:  mirbtwnhl  28625