Theorem footexlem2 26985

Description: Lemma for footex 26986. (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 26984	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
30		nelne2 3041	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑋 ≠ 𝐶)
31	29, 9, 30	syl2anc 583	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝐶)
32	31	necomd 2998	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝑋)
33	1, 3, 4, 5, 6, 7, 32	tgelrnln 26895	. 2 ⊢ (𝜑 → (𝐶𝐿𝑋) ∈ ran 𝐿)
34	1, 3, 4, 5, 6, 7, 32	tglinerflx2 26899	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐶𝐿𝑋))
35	34, 29	elind 4124	. 2 ⊢ (𝜑 → 𝑋 ∈ ((𝐶𝐿𝑋) ∩ 𝐴))
36	1, 3, 4, 5, 6, 7, 32	tglinerflx1 26898	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐶𝐿𝑋))
37	17	necomd 2998	. . . . 5 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
38	1, 3, 4, 5, 11, 10, 13, 37, 18	btwnlng3 26886	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐹𝐿𝐸))
39	1, 3, 4, 5, 10, 11, 13, 17, 38	lncom 26887	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐸𝐿𝐹))
40	39, 16	eleqtrrd 2842	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
41		eqid 2738	. . . . 5 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
42	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐺 ∈ TarskiG)
43	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 ∈ 𝑃)
44	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 ∈ 𝑃)
45	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑅 ∈ 𝑃)
46	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐶 ∈ 𝑃)
47		eqidd 2739	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐶 = 𝐶)
48		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑋)
49		eqidd 2739	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 = 𝐸)
50	47, 48, 49	s3eqd 14505	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐶𝑌𝐸”⟩ = ⟨“𝐶𝑋𝐸”⟩)
51	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑋 ∈ 𝑃)
52	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ∈ 𝑃)
53		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ ((pInvG‘𝐺)‘𝑍) = ((pInvG‘𝐺)‘𝑍)
54	1, 2, 3, 4, 41, 5, 14, 53, 23	mircl 26926	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑍)‘𝑄) ∈ 𝑃)
55	1, 2, 3, 5, 10, 13, 10, 6, 19	tgcgrcomlr 26745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑌 − 𝐸) = (𝐶 − 𝐸))
56	25, 55	eqtr2d 2779	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐶 − 𝐸) = (𝑌 − 𝑄))
57	1, 3, 4, 5, 10, 11, 17	tglinerflx1 26898	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸 ∈ (𝐸𝐿𝐹))
58	57, 16	eleqtrrd 2842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
59		nelne2 3041	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐸 ≠ 𝐶)
60	58, 9, 59	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ≠ 𝐶)
61	60	necomd 2998	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ≠ 𝐸)
62	1, 2, 3, 5, 6, 10, 13, 23, 56, 61	tgcgrneq 26748	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ≠ 𝑄)
63	62	necomd 2998	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ≠ 𝑌)
64		nelne2 3041	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑌 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑌 ≠ 𝐶)
65	40, 9, 64	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑌 ≠ 𝐶)
66	65	necomd 2998	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶 ≠ 𝑌)
67	20, 66	eqnetrrd 3011	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌)
68		eqid 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((pInvG‘𝐺)‘𝑅) = ((pInvG‘𝐺)‘𝑅)
69	1, 2, 3, 4, 41, 5, 12, 68, 13	mirinv 26931	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) = 𝑌 ↔ 𝑅 = 𝑌))
70	69	necon3bid 2987	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌 ↔ 𝑅 ≠ 𝑌))
71	67, 70	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ≠ 𝑌)
72	1, 2, 3, 4, 41, 5, 12, 68, 13	mirbtwn 26923	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
73	20	oveq1d 7270	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐶𝐼𝑌) = ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
74	72, 73	eleqtrrd 2842	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ (𝐶𝐼𝑌))
75	1, 2, 3, 5, 6, 12, 13, 23, 71, 74, 24	tgbtwnouttr2 26760	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐼𝑄))
76	1, 2, 3, 5, 6, 13, 23, 75	tgbtwncom 26753	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (𝑄𝐼𝐶))
77		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
78	20	oveq2d 7271	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 − 𝐶) = (𝐸 − (((pInvG‘𝐺)‘𝑅)‘𝑌)))
79	19, 78	eqtrd 2778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 − 𝑌) = (𝐸 − (((pInvG‘𝐺)‘𝑅)‘𝑌)))
80	1, 2, 3, 4, 41, 5, 10, 12, 13	israg 26962	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺) ↔ (𝐸 − 𝑌) = (𝐸 − (((pInvG‘𝐺)‘𝑅)‘𝑌))))
81	79, 80	mpbird 256	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺))
82	1, 2, 3, 5, 12, 13, 23, 24	tgbtwncom 26753	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑌 ∈ (𝑄𝐼𝑅))
83	1, 2, 3, 5, 13, 23, 13, 10, 25	tgcgrcomlr 26745	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑄 − 𝑌) = (𝐸 − 𝑌))
84	22	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑌 − 𝑅) = (𝑌 − 𝑍))
85	1, 2, 3, 5, 23, 10	axtgcgrrflx 26727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑄 − 𝐸) = (𝐸 − 𝑄))
86	25	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . 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 26730	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑅 − 𝐸) = (𝑍 − 𝑄))
88	1, 2, 3, 5, 12, 10, 14, 23, 87	tgcgrcomlr 26745	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 − 𝑅) = (𝑄 − 𝑍))
89	1, 2, 3, 5, 13, 12, 13, 14, 84	tgcgrcomlr 26745	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑅 − 𝑌) = (𝑍 − 𝑌))
90	1, 2, 77, 5, 10, 12, 13, 23, 14, 13, 88, 89, 86	trgcgr 26781	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨“𝐸𝑅𝑌”⟩(cgrG‘𝐺)⟨“𝑄𝑍𝑌”⟩)
91	1, 2, 3, 4, 41, 5, 10, 12, 13, 77, 23, 14, 13, 81, 90	ragcgr 26972	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ⟨“𝑄𝑍𝑌”⟩ ∈ (∟G‘𝐺))
92	1, 2, 3, 4, 41, 5, 23, 14, 13, 91	ragcom 26963	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺))
93	1, 2, 3, 4, 41, 5, 13, 14, 23	israg 26962	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝑌 − 𝑄) = (𝑌 − (((pInvG‘𝐺)‘𝑍)‘𝑄))))
94	92, 93	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑌 − 𝑄) = (𝑌 − (((pInvG‘𝐺)‘𝑍)‘𝑄)))
95	1, 2, 3, 5, 13, 23, 13, 54, 94	tgcgrcomlr 26745	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 − 𝑌) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) − 𝑌))
96	27	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 − 𝐶) = (𝑌 − 𝐷))
97	1, 2, 3, 4, 41, 5, 14, 53, 23	mircgr 26922	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑍 − (((pInvG‘𝐺)‘𝑍)‘𝑄)) = (𝑍 − 𝑄))
98	97	eqcomd 2744	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑍 − 𝑄) = (𝑍 − (((pInvG‘𝐺)‘𝑍)‘𝑄)))
99	1, 2, 3, 5, 14, 23, 14, 54, 98	tgcgrcomlr 26745	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 − 𝑍) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) − 𝑍))
100		eqidd 2739	. . . . . . . . . . . . . . 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 26730	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 − 𝑍) = (𝐷 − 𝑍))
102	1, 2, 3, 5, 6, 14, 15, 14, 101	tgcgrcomlr 26745	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 − 𝐶) = (𝑍 − 𝐷))
103	28	oveq2d 7271	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 − 𝐷) = (𝑍 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
104	102, 103	eqtrd 2778	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑍 − 𝐶) = (𝑍 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
105	1, 2, 3, 4, 41, 5, 14, 7, 6	israg 26962	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨“𝑍𝑋𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝑍 − 𝐶) = (𝑍 − (((pInvG‘𝐺)‘𝑋)‘𝐶))))
106	104, 105	mpbird 256	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨“𝑍𝑋𝐶”⟩ ∈ (∟G‘𝐺))
107	106	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝑍𝑋𝐶”⟩ ∈ (∟G‘𝐺))
108	71	necomd 2998	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ≠ 𝑅)
109	1, 2, 3, 5, 13, 12, 13, 14, 84, 108	tgcgrneq 26748	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
110	109	necomd 2998	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
111	110	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ≠ 𝑌)
112	111, 48	neeqtrd 3012	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ≠ 𝑋)
113	19	eqcomd 2744	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 − 𝐶) = (𝐸 − 𝑌))
114	113	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → (𝐸 − 𝐶) = (𝐸 − 𝑌))
115	60	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 ≠ 𝐶)
116	1, 2, 3, 42, 43, 46, 43, 44, 114, 115	tgcgrneq 26748	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 ≠ 𝑌)
117	116	necomd 2998	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 ≠ 𝐸)
118	1, 2, 3, 5, 10, 6, 10, 13, 113, 60	tgcgrneq 26748	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ≠ 𝑌)
119	1, 3, 4, 5, 10, 13, 14, 118, 21	btwnlng3 26886	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ (𝐸𝐿𝑌))
120	119	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ∈ (𝐸𝐿𝑌))
121	1, 3, 4, 42, 44, 43, 52, 117, 120	lncom 26887	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ∈ (𝑌𝐿𝐸))
122	48	oveq1d 7270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → (𝑌𝐿𝐸) = (𝑋𝐿𝐸))
123	121, 122	eleqtrd 2841	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ∈ (𝑋𝐿𝐸))
124	123	orcd 869	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → (𝑍 ∈ (𝑋𝐿𝐸) ∨ 𝑋 = 𝐸))
125	1, 2, 3, 4, 41, 42, 52, 51, 46, 43, 107, 112, 124	ragcol 26964	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐸𝑋𝐶”⟩ ∈ (∟G‘𝐺))
126	1, 2, 3, 4, 41, 42, 43, 51, 46, 125	ragcom 26963	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐶𝑋𝐸”⟩ ∈ (∟G‘𝐺))
127	50, 126	eqeltrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐶𝑌𝐸”⟩ ∈ (∟G‘𝐺))
128	66	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐶 ≠ 𝑌)
129	1, 2, 3, 5, 6, 12, 13, 74	tgbtwncom 26753	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (𝑌𝐼𝐶))
130	1, 4, 3, 5, 13, 12, 6, 129	btwncolg3 26822	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ (𝑌𝐿𝑅) ∨ 𝑌 = 𝑅))
131	130	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → (𝐶 ∈ (𝑌𝐿𝑅) ∨ 𝑌 = 𝑅))
132	1, 2, 3, 4, 41, 42, 46, 44, 43, 45, 127, 128, 131	ragcol 26964	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝑅𝑌𝐸”⟩ ∈ (∟G‘𝐺))
133	1, 2, 3, 4, 41, 42, 45, 44, 43, 132	ragcom 26963	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐸𝑌𝑅”⟩ ∈ (∟G‘𝐺))
134	81	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺))
135	1, 2, 3, 4, 41, 42, 43, 44, 45, 133, 134	ragflat 26969	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑅)
136	108	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 ≠ 𝑅)
137	136	neneqd 2947	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ¬ 𝑌 = 𝑅)
138	135, 137	pm2.65da 813	. . 3 ⊢ (𝜑 → ¬ 𝑌 = 𝑋)
139	138	neqned 2949	. 2 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
140	28	oveq2d 7271	. . . . 5 ⊢ (𝜑 → (𝑌 − 𝐷) = (𝑌 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
141	96, 140	eqtrd 2778	. . . 4 ⊢ (𝜑 → (𝑌 − 𝐶) = (𝑌 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
142	1, 2, 3, 4, 41, 5, 13, 7, 6	israg 26962	. . . 4 ⊢ (𝜑 → (⟨“𝑌𝑋𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝑌 − 𝐶) = (𝑌 − (((pInvG‘𝐺)‘𝑋)‘𝐶))))
143	141, 142	mpbird 256	. . 3 ⊢ (𝜑 → ⟨“𝑌𝑋𝐶”⟩ ∈ (∟G‘𝐺))
144	1, 2, 3, 4, 41, 5, 13, 7, 6, 143	ragcom 26963	. 2 ⊢ (𝜑 → ⟨“𝐶𝑋𝑌”⟩ ∈ (∟G‘𝐺))
145	1, 2, 3, 4, 5, 33, 8, 35, 36, 40, 32, 139, 144	ragperp 26982	1 ⊢ (𝜑 → (𝐶𝐿𝑋)(⟂G‘𝐺)𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   class class class wbr 5070  ran crn 5581  ‘cfv 6418  (class class class)co 7255  ⟨“cs3 14483  Basecbs 16840  distcds 16897  TarskiGcstrkg 26693  Itvcitv 26699  LineGclng 26700  cgrGccgrg 26775  pInvGcmir 26917  ∟Gcrag 26958  ⟂Gcperpg 26960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-concat 14202  df-s1 14229  df-s2 14489  df-s3 14490  df-trkgc 26713  df-trkgb 26714  df-trkgcb 26715  df-trkg 26718  df-cgrg 26776  df-leg 26848  df-mir 26918  df-rag 26959  df-perpg 26961

This theorem is referenced by:  footex  26986