Theorem footexlem2 28698

Description: Lemma for footex 28699. (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 28697	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
30		nelne2 3026	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑋 ≠ 𝐶)
31	29, 9, 30	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝐶)
32	31	necomd 2983	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝑋)
33	1, 3, 4, 5, 6, 7, 32	tgelrnln 28608	. 2 ⊢ (𝜑 → (𝐶𝐿𝑋) ∈ ran 𝐿)
34	1, 3, 4, 5, 6, 7, 32	tglinerflx2 28612	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐶𝐿𝑋))
35	34, 29	elind 4147	. 2 ⊢ (𝜑 → 𝑋 ∈ ((𝐶𝐿𝑋) ∩ 𝐴))
36	1, 3, 4, 5, 6, 7, 32	tglinerflx1 28611	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐶𝐿𝑋))
37	17	necomd 2983	. . . . 5 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
38	1, 3, 4, 5, 11, 10, 13, 37, 18	btwnlng3 28599	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐹𝐿𝐸))
39	1, 3, 4, 5, 10, 11, 13, 17, 38	lncom 28600	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐸𝐿𝐹))
40	39, 16	eleqtrrd 2834	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
41		eqid 2731	. . . . 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 2732	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐶 = 𝐶)
48		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑋)
49		eqidd 2732	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 = 𝐸)
50	47, 48, 49	s3eqd 14771	. . . . . . . 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 2731	. . . . . . . . . . . . . . . 16 ⊢ ((pInvG‘𝐺)‘𝑍) = ((pInvG‘𝐺)‘𝑍)
54	1, 2, 3, 4, 41, 5, 14, 53, 23	mircl 28639	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑍)‘𝑄) ∈ 𝑃)
55	1, 2, 3, 5, 10, 13, 10, 6, 19	tgcgrcomlr 28458	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑌 − 𝐸) = (𝐶 − 𝐸))
56	25, 55	eqtr2d 2767	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐶 − 𝐸) = (𝑌 − 𝑄))
57	1, 3, 4, 5, 10, 11, 17	tglinerflx1 28611	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸 ∈ (𝐸𝐿𝐹))
58	57, 16	eleqtrrd 2834	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
59		nelne2 3026	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐸 ≠ 𝐶)
60	58, 9, 59	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ≠ 𝐶)
61	60	necomd 2983	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ≠ 𝐸)
62	1, 2, 3, 5, 6, 10, 13, 23, 56, 61	tgcgrneq 28461	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ≠ 𝑄)
63	62	necomd 2983	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ≠ 𝑌)
64		nelne2 3026	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑌 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑌 ≠ 𝐶)
65	40, 9, 64	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑌 ≠ 𝐶)
66	65	necomd 2983	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶 ≠ 𝑌)
67	20, 66	eqnetrrd 2996	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌)
68		eqid 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((pInvG‘𝐺)‘𝑅) = ((pInvG‘𝐺)‘𝑅)
69	1, 2, 3, 4, 41, 5, 12, 68, 13	mirinv 28644	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) = 𝑌 ↔ 𝑅 = 𝑌))
70	69	necon3bid 2972	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌 ↔ 𝑅 ≠ 𝑌))
71	67, 70	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ≠ 𝑌)
72	1, 2, 3, 4, 41, 5, 12, 68, 13	mirbtwn 28636	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
73	20	oveq1d 7361	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐶𝐼𝑌) = ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
74	72, 73	eleqtrrd 2834	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ (𝐶𝐼𝑌))
75	1, 2, 3, 5, 6, 12, 13, 23, 71, 74, 24	tgbtwnouttr2 28473	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐼𝑄))
76	1, 2, 3, 5, 6, 13, 23, 75	tgbtwncom 28466	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (𝑄𝐼𝐶))
77		eqid 2731	. . . . . . . . . . . . . . . . . . 19 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
78	20	oveq2d 7362	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 − 𝐶) = (𝐸 − (((pInvG‘𝐺)‘𝑅)‘𝑌)))
79	19, 78	eqtrd 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 − 𝑌) = (𝐸 − (((pInvG‘𝐺)‘𝑅)‘𝑌)))
80	1, 2, 3, 4, 41, 5, 10, 12, 13	israg 28675	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺) ↔ (𝐸 − 𝑌) = (𝐸 − (((pInvG‘𝐺)‘𝑅)‘𝑌))))
81	79, 80	mpbird 257	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺))
82	1, 2, 3, 5, 12, 13, 23, 24	tgbtwncom 28466	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑌 ∈ (𝑄𝐼𝑅))
83	1, 2, 3, 5, 13, 23, 13, 10, 25	tgcgrcomlr 28458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑄 − 𝑌) = (𝐸 − 𝑌))
84	22	eqcomd 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑌 − 𝑅) = (𝑌 − 𝑍))
85	1, 2, 3, 5, 23, 10	axtgcgrrflx 28440	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑄 − 𝐸) = (𝐸 − 𝑄))
86	25	eqcomd 2737	. . . . . . . . . . . . . . . . . . . . . 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 28443	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑅 − 𝐸) = (𝑍 − 𝑄))
88	1, 2, 3, 5, 12, 10, 14, 23, 87	tgcgrcomlr 28458	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 − 𝑅) = (𝑄 − 𝑍))
89	1, 2, 3, 5, 13, 12, 13, 14, 84	tgcgrcomlr 28458	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑅 − 𝑌) = (𝑍 − 𝑌))
90	1, 2, 77, 5, 10, 12, 13, 23, 14, 13, 88, 89, 86	trgcgr 28494	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨“𝐸𝑅𝑌”⟩(cgrG‘𝐺)⟨“𝑄𝑍𝑌”⟩)
91	1, 2, 3, 4, 41, 5, 10, 12, 13, 77, 23, 14, 13, 81, 90	ragcgr 28685	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ⟨“𝑄𝑍𝑌”⟩ ∈ (∟G‘𝐺))
92	1, 2, 3, 4, 41, 5, 23, 14, 13, 91	ragcom 28676	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺))
93	1, 2, 3, 4, 41, 5, 13, 14, 23	israg 28675	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝑌 − 𝑄) = (𝑌 − (((pInvG‘𝐺)‘𝑍)‘𝑄))))
94	92, 93	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑌 − 𝑄) = (𝑌 − (((pInvG‘𝐺)‘𝑍)‘𝑄)))
95	1, 2, 3, 5, 13, 23, 13, 54, 94	tgcgrcomlr 28458	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 − 𝑌) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) − 𝑌))
96	27	eqcomd 2737	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 − 𝐶) = (𝑌 − 𝐷))
97	1, 2, 3, 4, 41, 5, 14, 53, 23	mircgr 28635	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑍 − (((pInvG‘𝐺)‘𝑍)‘𝑄)) = (𝑍 − 𝑄))
98	97	eqcomd 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑍 − 𝑄) = (𝑍 − (((pInvG‘𝐺)‘𝑍)‘𝑄)))
99	1, 2, 3, 5, 14, 23, 14, 54, 98	tgcgrcomlr 28458	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 − 𝑍) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) − 𝑍))
100		eqidd 2732	. . . . . . . . . . . . . . 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 28443	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 − 𝑍) = (𝐷 − 𝑍))
102	1, 2, 3, 5, 6, 14, 15, 14, 101	tgcgrcomlr 28458	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 − 𝐶) = (𝑍 − 𝐷))
103	28	oveq2d 7362	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 − 𝐷) = (𝑍 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
104	102, 103	eqtrd 2766	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑍 − 𝐶) = (𝑍 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
105	1, 2, 3, 4, 41, 5, 14, 7, 6	israg 28675	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨“𝑍𝑋𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝑍 − 𝐶) = (𝑍 − (((pInvG‘𝐺)‘𝑋)‘𝐶))))
106	104, 105	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨“𝑍𝑋𝐶”⟩ ∈ (∟G‘𝐺))
107	106	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝑍𝑋𝐶”⟩ ∈ (∟G‘𝐺))
108	71	necomd 2983	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ≠ 𝑅)
109	1, 2, 3, 5, 13, 12, 13, 14, 84, 108	tgcgrneq 28461	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
110	109	necomd 2983	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
111	110	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ≠ 𝑌)
112	111, 48	neeqtrd 2997	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ≠ 𝑋)
113	19	eqcomd 2737	. . . . . . . . . . . . . . . 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 28461	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 ≠ 𝑌)
117	116	necomd 2983	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 ≠ 𝐸)
118	1, 2, 3, 5, 10, 6, 10, 13, 113, 60	tgcgrneq 28461	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ≠ 𝑌)
119	1, 3, 4, 5, 10, 13, 14, 118, 21	btwnlng3 28599	. . . . . . . . . . . . . 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 28600	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ∈ (𝑌𝐿𝐸))
122	48	oveq1d 7361	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → (𝑌𝐿𝐸) = (𝑋𝐿𝐸))
123	121, 122	eleqtrd 2833	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ∈ (𝑋𝐿𝐸))
124	123	orcd 873	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → (𝑍 ∈ (𝑋𝐿𝐸) ∨ 𝑋 = 𝐸))
125	1, 2, 3, 4, 41, 42, 52, 51, 46, 43, 107, 112, 124	ragcol 28677	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐸𝑋𝐶”⟩ ∈ (∟G‘𝐺))
126	1, 2, 3, 4, 41, 42, 43, 51, 46, 125	ragcom 28676	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐶𝑋𝐸”⟩ ∈ (∟G‘𝐺))
127	50, 126	eqeltrd 2831	. . . . . . 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 28466	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (𝑌𝐼𝐶))
130	1, 4, 3, 5, 13, 12, 6, 129	btwncolg3 28535	. . . . . . . 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 28677	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝑅𝑌𝐸”⟩ ∈ (∟G‘𝐺))
133	1, 2, 3, 4, 41, 42, 45, 44, 43, 132	ragcom 28676	. . . . 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 28682	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑅)
136	108	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 ≠ 𝑅)
137	136	neneqd 2933	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ¬ 𝑌 = 𝑅)
138	135, 137	pm2.65da 816	. . 3 ⊢ (𝜑 → ¬ 𝑌 = 𝑋)
139	138	neqned 2935	. 2 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
140	28	oveq2d 7362	. . . . 5 ⊢ (𝜑 → (𝑌 − 𝐷) = (𝑌 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
141	96, 140	eqtrd 2766	. . . 4 ⊢ (𝜑 → (𝑌 − 𝐶) = (𝑌 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
142	1, 2, 3, 4, 41, 5, 13, 7, 6	israg 28675	. . . 4 ⊢ (𝜑 → (⟨“𝑌𝑋𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝑌 − 𝐶) = (𝑌 − (((pInvG‘𝐺)‘𝑋)‘𝐶))))
143	141, 142	mpbird 257	. . 3 ⊢ (𝜑 → ⟨“𝑌𝑋𝐶”⟩ ∈ (∟G‘𝐺))
144	1, 2, 3, 4, 41, 5, 13, 7, 6, 143	ragcom 28676	. 2 ⊢ (𝜑 → ⟨“𝐶𝑋𝑌”⟩ ∈ (∟G‘𝐺))
145	1, 2, 3, 4, 5, 33, 8, 35, 36, 40, 32, 139, 144	ragperp 28695	1 ⊢ (𝜑 → (𝐶𝐿𝑋)(⟂G‘𝐺)𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111   ≠ wne 2928   class class class wbr 5089  ran crn 5615  ‘cfv 6481  (class class class)co 7346  ⟨“cs3 14749  Basecbs 17120  distcds 17170  TarskiGcstrkg 28405  Itvcitv 28411  LineGclng 28412  cgrGccgrg 28488  pInvGcmir 28630  ∟Gcrag 28671  ⟂Gcperpg 28673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-er 8622  df-map 8752  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-dju 9794  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-xnn0 12455  df-z 12469  df-uz 12733  df-fz 13408  df-fzo 13555  df-hash 14238  df-word 14421  df-concat 14478  df-s1 14504  df-s2 14755  df-s3 14756  df-trkgc 28426  df-trkgb 28427  df-trkgcb 28428  df-trkg 28431  df-cgrg 28489  df-leg 28561  df-mir 28631  df-rag 28672  df-perpg 28674

This theorem is referenced by:  footex  28699