Theorem footexlem2 29222

Description: Lemma for footex 29223. (Contributed by Thierry Arnoux, 19-Oct-2019.) (Revised by Thierry Arnoux, 1-Jul-2023.)

Hypotheses

Ref	Expression
isperp.p	⊢ 𝑃 = (Base‘𝐺)
isperp.d	⊢ − = (dist‘𝐺)
isperp.i	⊢ 𝐼 = (Itv‘𝐺)
isperp.l	⊢ 𝐿 = (LineG‘𝐺)
isperp.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
isperp.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
foot.x	⊢ (𝜑 → 𝐶 ∈ 𝑃)
foot.y	⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
footexlem.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
footexlem.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
footexlem.r	⊢ (𝜑 → 𝑅 ∈ 𝑃)
footexlem.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
footexlem.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
footexlem.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
footexlem.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
footexlem.1	⊢ (𝜑 → 𝐴 = (𝐸𝐿𝐹))
footexlem.2	⊢ (𝜑 → 𝐸 ≠ 𝐹)
footexlem.3	⊢ (𝜑 → 𝐸 ∈ (𝐹𝐼𝑌))
footexlem.4	⊢ (𝜑 → (𝐸 − 𝑌) = (𝐸 − 𝐶))
footexlem.5	⊢ (𝜑 → 𝐶 = (((pInvG‘𝐺)‘𝑅)‘𝑌))
footexlem.6	⊢ (𝜑 → 𝑌 ∈ (𝐸𝐼𝑍))
footexlem.7	⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝑅))
footexlem.q	⊢ (𝜑 → 𝑄 ∈ 𝑃)
footexlem.8	⊢ (𝜑 → 𝑌 ∈ (𝑅𝐼𝑄))
footexlem.9	⊢ (𝜑 → (𝑌 − 𝑄) = (𝑌 − 𝐸))
footexlem.10	⊢ (𝜑 → 𝑌 ∈ ((((pInvG‘𝐺)‘𝑍)‘𝑄)𝐼𝐷))
footexlem.11	⊢ (𝜑 → (𝑌 − 𝐷) = (𝑌 − 𝐶))
footexlem.12	⊢ (𝜑 → 𝐷 = (((pInvG‘𝐺)‘𝑋)‘𝐶))

Assertion

Ref	Expression
footexlem2	⊢ (𝜑 → (𝐶𝐿𝑋)(⟂G‘𝐺)𝐴)

Proof of Theorem footexlem2

Step	Hyp	Ref	Expression
1		isperp.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		isperp.d	. 2 ⊢ − = (dist‘𝐺)
3		isperp.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
4		isperp.l	. 2 ⊢ 𝐿 = (LineG‘𝐺)
5		isperp.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		foot.x	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
7		footexlem.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
8		isperp.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
9		foot.y	. . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
10		footexlem.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
11		footexlem.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
12		footexlem.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
13		footexlem.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
14		footexlem.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
15		footexlem.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
16		footexlem.1	. . . . . 6 ⊢ (𝜑 → 𝐴 = (𝐸𝐿𝐹))
17		footexlem.2	. . . . . 6 ⊢ (𝜑 → 𝐸 ≠ 𝐹)
18		footexlem.3	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝐹𝐼𝑌))
19		footexlem.4	. . . . . 6 ⊢ (𝜑 → (𝐸 − 𝑌) = (𝐸 − 𝐶))
20		footexlem.5	. . . . . 6 ⊢ (𝜑 → 𝐶 = (((pInvG‘𝐺)‘𝑅)‘𝑌))
21		footexlem.6	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐸𝐼𝑍))
22		footexlem.7	. . . . . 6 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝑅))
23		footexlem.q	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝑃)
24		footexlem.8	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑅𝐼𝑄))
25		footexlem.9	. . . . . 6 ⊢ (𝜑 → (𝑌 − 𝑄) = (𝑌 − 𝐸))
26		footexlem.10	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ((((pInvG‘𝐺)‘𝑍)‘𝑄)𝐼𝐷))
27		footexlem.11	. . . . . 6 ⊢ (𝜑 → (𝑌 − 𝐷) = (𝑌 − 𝐶))
28		footexlem.12	. . . . . 6 ⊢ (𝜑 → 𝐷 = (((pInvG‘𝐺)‘𝑋)‘𝐶))
29	1, 2, 3, 4, 5, 8, 6, 9, 10, 11, 12, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28	footexlem1 29221	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
30		nelne2 3057	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑋 ≠ 𝐶)
31	29, 9, 30	syl2anc 596	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝐶)
32	31	necomd 3014	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝑋)
33	1, 3, 4, 5, 6, 7, 32	tgelrnln 29125	. 2 ⊢ (𝜑 → (𝐶𝐿𝑋) ∈ ran 𝐿)
34	1, 3, 4, 5, 6, 7, 32	tglinerflx2 29129	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐶𝐿𝑋))
35	34, 29	elind 4100	. 2 ⊢ (𝜑 → 𝑋 ∈ ((𝐶𝐿𝑋) ∩ 𝐴))
36	1, 3, 4, 5, 6, 7, 32	tglinerflx1 29128	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐶𝐿𝑋))
37	17	necomd 3014	. . . . 5 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
38	1, 3, 4, 5, 11, 10, 13, 37, 18	btwnlng3 29116	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐹𝐿𝐸))
39	1, 3, 4, 5, 10, 11, 13, 17, 38	lncom 29117	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐸𝐿𝐹))
40	39, 16	eleqtrrd 2867	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
41		eqid 2763	. . . . 5 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
42	5	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐺 ∈ TarskiG)
43	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 ∈ 𝑃)
44	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 ∈ 𝑃)
45	12	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑅 ∈ 𝑃)
46	6	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐶 ∈ 𝑃)
47		eqidd 2764	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐶 = 𝐶)
48		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑋)
49		eqidd 2764	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 = 𝐸)
50	47, 48, 49	s3eqd 15040	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐶𝑌𝐸”⟩ = ⟨“𝐶𝑋𝐸”⟩)
51	7	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑋 ∈ 𝑃)
52	14	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ∈ 𝑃)
53		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ ((pInvG‘𝐺)‘𝑍) = ((pInvG‘𝐺)‘𝑍)
54	1, 2, 3, 4, 41, 5, 14, 53, 23	mircl 29160	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑍)‘𝑄) ∈ 𝑃)
55	1, 2, 3, 5, 10, 13, 10, 6, 19	tgcgrcomlr 28969	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑌 − 𝐸) = (𝐶 − 𝐸))
56	25, 55	eqtr2d 2799	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐶 − 𝐸) = (𝑌 − 𝑄))
57	1, 3, 4, 5, 10, 11, 17	tglinerflx1 29128	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸 ∈ (𝐸𝐿𝐹))
58	57, 16	eleqtrrd 2867	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
59		nelne2 3057	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐸 ≠ 𝐶)
60	58, 9, 59	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ≠ 𝐶)
61	60	necomd 3014	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ≠ 𝐸)
62	1, 2, 3, 5, 6, 10, 13, 23, 56, 61	tgcgrneq 28972	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ≠ 𝑄)
63	62	necomd 3014	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ≠ 𝑌)
64		nelne2 3057	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑌 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑌 ≠ 𝐶)
65	40, 9, 64	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑌 ≠ 𝐶)
66	65	necomd 3014	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶 ≠ 𝑌)
67	20, 66	eqnetrrd 3027	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌)
68		eqid 2763	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((pInvG‘𝐺)‘𝑅) = ((pInvG‘𝐺)‘𝑅)
69	1, 2, 3, 4, 41, 5, 12, 68, 13	mirinv 29165	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) = 𝑌 ↔ 𝑅 = 𝑌))
70	69	necon3bid 3003	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌 ↔ 𝑅 ≠ 𝑌))
71	67, 70	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ≠ 𝑌)
72	1, 2, 3, 4, 41, 5, 12, 68, 13	mirbtwn 29157	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
73	20	oveq1d 7455	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐶𝐼𝑌) = ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
74	72, 73	eleqtrrd 2867	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ (𝐶𝐼𝑌))
75	1, 2, 3, 5, 6, 12, 13, 23, 71, 74, 24	tgbtwnouttr2 28985	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐼𝑄))
76	1, 2, 3, 5, 6, 13, 23, 75	tgbtwncom 28978	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (𝑄𝐼𝐶))
77		eqid 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
78	20	oveq2d 7456	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 − 𝐶) = (𝐸 − (((pInvG‘𝐺)‘𝑅)‘𝑌)))
79	19, 78	eqtrd 2798	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 − 𝑌) = (𝐸 − (((pInvG‘𝐺)‘𝑅)‘𝑌)))
80	1, 2, 3, 4, 41, 5, 10, 12, 13	israg 29199	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺) ↔ (𝐸 − 𝑌) = (𝐸 − (((pInvG‘𝐺)‘𝑅)‘𝑌))))
81	79, 80	mpbird 260	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺))
82	1, 2, 3, 5, 12, 13, 23, 24	tgbtwncom 28978	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑌 ∈ (𝑄𝐼𝑅))
83	1, 2, 3, 5, 13, 23, 13, 10, 25	tgcgrcomlr 28969	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑄 − 𝑌) = (𝐸 − 𝑌))
84	22	eqcomd 2769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑌 − 𝑅) = (𝑌 − 𝑍))
85	1, 2, 3, 5, 23, 10	axtgcgrrflx 28951	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑄 − 𝐸) = (𝐸 − 𝑄))
86	25	eqcomd 2769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑌 − 𝐸) = (𝑌 − 𝑄))
87	1, 2, 3, 5, 23, 13, 12, 10, 13, 14, 10, 23, 63, 82, 21, 83, 84, 85, 86	axtg5seg 28954	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑅 − 𝐸) = (𝑍 − 𝑄))
88	1, 2, 3, 5, 12, 10, 14, 23, 87	tgcgrcomlr 28969	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 − 𝑅) = (𝑄 − 𝑍))
89	1, 2, 3, 5, 13, 12, 13, 14, 84	tgcgrcomlr 28969	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑅 − 𝑌) = (𝑍 − 𝑌))
90	1, 2, 77, 5, 10, 12, 13, 23, 14, 13, 88, 89, 86	trgcgr 29006	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨“𝐸𝑅𝑌”⟩(cgrG‘𝐺)⟨“𝑄𝑍𝑌”⟩)
91	1, 2, 3, 4, 41, 5, 10, 12, 13, 77, 23, 14, 13, 81, 90	ragcgr 29209	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ⟨“𝑄𝑍𝑌”⟩ ∈ (∟G‘𝐺))
92	1, 2, 3, 4, 41, 5, 23, 14, 13, 91	ragcom 29200	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺))
93	1, 2, 3, 4, 41, 5, 13, 14, 23	israg 29199	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝑌 − 𝑄) = (𝑌 − (((pInvG‘𝐺)‘𝑍)‘𝑄))))
94	92, 93	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑌 − 𝑄) = (𝑌 − (((pInvG‘𝐺)‘𝑍)‘𝑄)))
95	1, 2, 3, 5, 13, 23, 13, 54, 94	tgcgrcomlr 28969	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 − 𝑌) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) − 𝑌))
96	27	eqcomd 2769	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 − 𝐶) = (𝑌 − 𝐷))
97	1, 2, 3, 4, 41, 5, 14, 53, 23	mircgr 29156	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑍 − (((pInvG‘𝐺)‘𝑍)‘𝑄)) = (𝑍 − 𝑄))
98	97	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑍 − 𝑄) = (𝑍 − (((pInvG‘𝐺)‘𝑍)‘𝑄)))
99	1, 2, 3, 5, 14, 23, 14, 54, 98	tgcgrcomlr 28969	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 − 𝑍) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) − 𝑍))
100		eqidd 2764	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝑍))
101	1, 2, 3, 5, 23, 13, 6, 54, 13, 15, 14, 14, 63, 76, 26, 95, 96, 99, 100	axtg5seg 28954	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 − 𝑍) = (𝐷 − 𝑍))
102	1, 2, 3, 5, 6, 14, 15, 14, 101	tgcgrcomlr 28969	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 − 𝐶) = (𝑍 − 𝐷))
103	28	oveq2d 7456	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 − 𝐷) = (𝑍 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
104	102, 103	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑍 − 𝐶) = (𝑍 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
105	1, 2, 3, 4, 41, 5, 14, 7, 6	israg 29199	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨“𝑍𝑋𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝑍 − 𝐶) = (𝑍 − (((pInvG‘𝐺)‘𝑋)‘𝐶))))
106	104, 105	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨“𝑍𝑋𝐶”⟩ ∈ (∟G‘𝐺))
107	106	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝑍𝑋𝐶”⟩ ∈ (∟G‘𝐺))
108	71	necomd 3014	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ≠ 𝑅)
109	1, 2, 3, 5, 13, 12, 13, 14, 84, 108	tgcgrneq 28972	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
110	109	necomd 3014	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
111	110	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ≠ 𝑌)
112	111, 48	neeqtrd 3028	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ≠ 𝑋)
113	19	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 − 𝐶) = (𝐸 − 𝑌))
114	113	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → (𝐸 − 𝐶) = (𝐸 − 𝑌))
115	60	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 ≠ 𝐶)
116	1, 2, 3, 42, 43, 46, 43, 44, 114, 115	tgcgrneq 28972	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 ≠ 𝑌)
117	116	necomd 3014	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 ≠ 𝐸)
118	1, 2, 3, 5, 10, 6, 10, 13, 113, 60	tgcgrneq 28972	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ≠ 𝑌)
119	1, 3, 4, 5, 10, 13, 14, 118, 21	btwnlng3 29116	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ (𝐸𝐿𝑌))
120	119	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ∈ (𝐸𝐿𝑌))
121	1, 3, 4, 42, 44, 43, 52, 117, 120	lncom 29117	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ∈ (𝑌𝐿𝐸))
122	48	oveq1d 7455	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → (𝑌𝐿𝐸) = (𝑋𝐿𝐸))
123	121, 122	eleqtrd 2866	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ∈ (𝑋𝐿𝐸))
124	123	orcd 887	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → (𝑍 ∈ (𝑋𝐿𝐸) ∨ 𝑋 = 𝐸))
125	1, 2, 3, 4, 41, 42, 52, 51, 46, 43, 107, 112, 124	ragcol 29201	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐸𝑋𝐶”⟩ ∈ (∟G‘𝐺))
126	1, 2, 3, 4, 41, 42, 43, 51, 46, 125	ragcom 29200	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐶𝑋𝐸”⟩ ∈ (∟G‘𝐺))
127	50, 126	eqeltrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐶𝑌𝐸”⟩ ∈ (∟G‘𝐺))
128	66	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐶 ≠ 𝑌)
129	1, 2, 3, 5, 6, 12, 13, 74	tgbtwncom 28978	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (𝑌𝐼𝐶))
130	1, 4, 3, 5, 13, 12, 6, 129	btwncolg3 29047	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ (𝑌𝐿𝑅) ∨ 𝑌 = 𝑅))
131	130	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → (𝐶 ∈ (𝑌𝐿𝑅) ∨ 𝑌 = 𝑅))
132	1, 2, 3, 4, 41, 42, 46, 44, 43, 45, 127, 128, 131	ragcol 29201	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝑅𝑌𝐸”⟩ ∈ (∟G‘𝐺))
133	1, 2, 3, 4, 41, 42, 45, 44, 43, 132	ragcom 29200	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐸𝑌𝑅”⟩ ∈ (∟G‘𝐺))
134	81	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺))
135	1, 2, 3, 4, 41, 42, 43, 44, 45, 133, 134	ragflat 29206	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑅)
136	108	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 ≠ 𝑅)
137	136	neneqd 2964	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ¬ 𝑌 = 𝑅)
138	135, 137	pm2.65da 829	. . 3 ⊢ (𝜑 → ¬ 𝑌 = 𝑋)
139	138	neqned 2966	. 2 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
140	28	oveq2d 7456	. . . . 5 ⊢ (𝜑 → (𝑌 − 𝐷) = (𝑌 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
141	96, 140	eqtrd 2798	. . . 4 ⊢ (𝜑 → (𝑌 − 𝐶) = (𝑌 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
142	1, 2, 3, 4, 41, 5, 13, 7, 6	israg 29199	. . . 4 ⊢ (𝜑 → (⟨“𝑌𝑋𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝑌 − 𝐶) = (𝑌 − (((pInvG‘𝐺)‘𝑋)‘𝐶))))
143	141, 142	mpbird 260	. . 3 ⊢ (𝜑 → ⟨“𝑌𝑋𝐶”⟩ ∈ (∟G‘𝐺))
144	1, 2, 3, 4, 41, 5, 13, 7, 6, 143	ragcom 29200	. 2 ⊢ (𝜑 → ⟨“𝐶𝑋𝑌”⟩ ∈ (∟G‘𝐺))
145	1, 2, 3, 4, 5, 33, 8, 35, 36, 40, 32, 139, 144	ragperp 29219	1 ⊢ (𝜑 → (𝐶𝐿𝑋)(⟂G‘𝐺)𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   class class class wbr 5106  ran crn 5656  ‘cfv 6555  (class class class)co 7440  ⟨“cs3 15018  Basecbs 17413  distcds 17463  TarskiGcstrkg 28916  Itvcitv 28922  LineGclng 28923  cgrGccgrg 29000  pInvGcmir 29151  ∟Gcrag 29195  ⟂Gcperpg 29197

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-oadd 8505  df-er 8742  df-map 8874  df-pm 8875  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-dju 10006  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-n0 12631  df-xnn0 12704  df-z 12718  df-uz 12990  df-fz 13664  df-fzo 13813  df-hash 14499  df-word 14683  df-concat 14740  df-s1 14767  df-s2 15024  df-s3 15025  df-trkgc 28937  df-trkgb 28938  df-trkgcb 28939  df-trkg 28942  df-cgrg 29001  df-leg 29073  df-mir 29152  df-rag 29196  df-perpg 29198

This theorem is used by:  footex  29223