Theorem mirconn 26046

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 474	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝐺 ∈ TarskiG)
6		mirconn.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7	6	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝑋 ∈ 𝑃)
8		mirconn.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9	8	adantr 474	. . 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 26029	. . . 4 ⊢ (𝜑 → (𝑀‘𝑌) ∈ 𝑃)
15	14	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → (𝑀‘𝑌) ∈ 𝑃)
16	13	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝑌 ∈ 𝑃)
17		simpr 479	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝑋 ∈ (𝐴𝐼𝑌))
18	1, 2, 3, 10, 11, 4, 8, 12, 13	mirbtwn 26026	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝑌)𝐼𝑌))
19	18	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ ((𝑀‘𝑌)𝐼𝑌))
20	1, 2, 3, 5, 7, 9, 15, 16, 17, 19	tgbtwnintr 25861	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐼𝑌)) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
21	1, 2, 3, 4, 6, 8	tgbtwntriv2 25855	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝐴))
22	21	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼𝐴))
23		simpr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → 𝑌 = 𝐴)
24	23	fveq2d 6452	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → (𝑀‘𝑌) = (𝑀‘𝐴))
25	1, 2, 3, 10, 11, 4, 8, 12	mircinv 26036	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴)
26	25	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → (𝑀‘𝐴) = 𝐴)
27	24, 26	eqtrd 2814	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → (𝑀‘𝑌) = 𝐴)
28	27	oveq2d 6940	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → (𝑋𝐼(𝑀‘𝑌)) = (𝑋𝐼𝐴))
29	22, 28	eleqtrrd 2862	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
30	29	adantlr 705	. . 3 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 = 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
31	4	ad2antrr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝐺 ∈ TarskiG)
32	6	ad2antrr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑋 ∈ 𝑃)
33	13	ad2antrr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑌 ∈ 𝑃)
34	8	ad2antrr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝐴 ∈ 𝑃)
35	14	ad2antrr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → (𝑀‘𝑌) ∈ 𝑃)
36		simpr 479	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑌 ≠ 𝐴)
37		simplr 759	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑌 ∈ (𝐴𝐼𝑋))
38	1, 2, 3, 31, 34, 33, 32, 37	tgbtwncom 25856	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝑌 ∈ (𝑋𝐼𝐴))
39	1, 2, 3, 4, 14, 8, 13, 18	tgbtwncom 25856	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑌𝐼(𝑀‘𝑌)))
40	39	ad2antrr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝐴 ∈ (𝑌𝐼(𝑀‘𝑌)))
41	1, 2, 3, 31, 32, 33, 34, 35, 36, 38, 40	tgbtwnouttr2 25863	. . 3 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) ∧ 𝑌 ≠ 𝐴) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
42	30, 41	pm2.61dane 3057	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴𝐼𝑋)) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
43		mirconn.1	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋)))
44	20, 42, 43	mpjaodan 944	1 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 386   ∨ wo 836   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ‘cfv 6137  (class class class)co 6924  Basecbs 16266  distcds 16358  TarskiGcstrkg 25798  Itvcitv 25804  LineGclng 25805  pInvGcmir 26020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-trkgc 25816  df-trkgb 25817  df-trkgcb 25818  df-trkg 25821  df-mir 26021

This theorem is referenced by:  mirbtwnhl  26048