Theorem footexlem2 28647

Description: Lemma for footex 28648. (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 28646	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
30		nelne2 3023	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑋 ≠ 𝐶)
31	29, 9, 30	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝐶)
32	31	necomd 2980	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝑋)
33	1, 3, 4, 5, 6, 7, 32	tgelrnln 28557	. 2 ⊢ (𝜑 → (𝐶𝐿𝑋) ∈ ran 𝐿)
34	1, 3, 4, 5, 6, 7, 32	tglinerflx2 28561	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐶𝐿𝑋))
35	34, 29	elind 4163	. 2 ⊢ (𝜑 → 𝑋 ∈ ((𝐶𝐿𝑋) ∩ 𝐴))
36	1, 3, 4, 5, 6, 7, 32	tglinerflx1 28560	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐶𝐿𝑋))
37	17	necomd 2980	. . . . 5 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
38	1, 3, 4, 5, 11, 10, 13, 37, 18	btwnlng3 28548	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐹𝐿𝐸))
39	1, 3, 4, 5, 10, 11, 13, 17, 38	lncom 28549	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐸𝐿𝐹))
40	39, 16	eleqtrrd 2831	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
41		eqid 2729	. . . . 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 2730	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐶 = 𝐶)
48		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑋)
49		eqidd 2730	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 = 𝐸)
50	47, 48, 49	s3eqd 14830	. . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ ((pInvG‘𝐺)‘𝑍) = ((pInvG‘𝐺)‘𝑍)
54	1, 2, 3, 4, 41, 5, 14, 53, 23	mircl 28588	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑍)‘𝑄) ∈ 𝑃)
55	1, 2, 3, 5, 10, 13, 10, 6, 19	tgcgrcomlr 28407	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑌 − 𝐸) = (𝐶 − 𝐸))
56	25, 55	eqtr2d 2765	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐶 − 𝐸) = (𝑌 − 𝑄))
57	1, 3, 4, 5, 10, 11, 17	tglinerflx1 28560	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸 ∈ (𝐸𝐿𝐹))
58	57, 16	eleqtrrd 2831	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
59		nelne2 3023	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐸 ≠ 𝐶)
60	58, 9, 59	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ≠ 𝐶)
61	60	necomd 2980	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ≠ 𝐸)
62	1, 2, 3, 5, 6, 10, 13, 23, 56, 61	tgcgrneq 28410	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ≠ 𝑄)
63	62	necomd 2980	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ≠ 𝑌)
64		nelne2 3023	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑌 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑌 ≠ 𝐶)
65	40, 9, 64	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑌 ≠ 𝐶)
66	65	necomd 2980	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶 ≠ 𝑌)
67	20, 66	eqnetrrd 2993	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌)
68		eqid 2729	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((pInvG‘𝐺)‘𝑅) = ((pInvG‘𝐺)‘𝑅)
69	1, 2, 3, 4, 41, 5, 12, 68, 13	mirinv 28593	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) = 𝑌 ↔ 𝑅 = 𝑌))
70	69	necon3bid 2969	. . . . . . . . . . . . . . . . . 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 28585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
73	20	oveq1d 7402	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐶𝐼𝑌) = ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
74	72, 73	eleqtrrd 2831	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ (𝐶𝐼𝑌))
75	1, 2, 3, 5, 6, 12, 13, 23, 71, 74, 24	tgbtwnouttr2 28422	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐼𝑄))
76	1, 2, 3, 5, 6, 13, 23, 75	tgbtwncom 28415	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (𝑄𝐼𝐶))
77		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
78	20	oveq2d 7403	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 − 𝐶) = (𝐸 − (((pInvG‘𝐺)‘𝑅)‘𝑌)))
79	19, 78	eqtrd 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 − 𝑌) = (𝐸 − (((pInvG‘𝐺)‘𝑅)‘𝑌)))
80	1, 2, 3, 4, 41, 5, 10, 12, 13	israg 28624	. . . . . . . . . . . . . . . . . . . 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 28415	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑌 ∈ (𝑄𝐼𝑅))
83	1, 2, 3, 5, 13, 23, 13, 10, 25	tgcgrcomlr 28407	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑄 − 𝑌) = (𝐸 − 𝑌))
84	22	eqcomd 2735	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑌 − 𝑅) = (𝑌 − 𝑍))
85	1, 2, 3, 5, 23, 10	axtgcgrrflx 28389	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑄 − 𝐸) = (𝐸 − 𝑄))
86	25	eqcomd 2735	. . . . . . . . . . . . . . . . . . . . . 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 28392	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑅 − 𝐸) = (𝑍 − 𝑄))
88	1, 2, 3, 5, 12, 10, 14, 23, 87	tgcgrcomlr 28407	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 − 𝑅) = (𝑄 − 𝑍))
89	1, 2, 3, 5, 13, 12, 13, 14, 84	tgcgrcomlr 28407	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑅 − 𝑌) = (𝑍 − 𝑌))
90	1, 2, 77, 5, 10, 12, 13, 23, 14, 13, 88, 89, 86	trgcgr 28443	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨“𝐸𝑅𝑌”⟩(cgrG‘𝐺)⟨“𝑄𝑍𝑌”⟩)
91	1, 2, 3, 4, 41, 5, 10, 12, 13, 77, 23, 14, 13, 81, 90	ragcgr 28634	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ⟨“𝑄𝑍𝑌”⟩ ∈ (∟G‘𝐺))
92	1, 2, 3, 4, 41, 5, 23, 14, 13, 91	ragcom 28625	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺))
93	1, 2, 3, 4, 41, 5, 13, 14, 23	israg 28624	. . . . . . . . . . . . . . . . 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 28407	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 − 𝑌) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) − 𝑌))
96	27	eqcomd 2735	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 − 𝐶) = (𝑌 − 𝐷))
97	1, 2, 3, 4, 41, 5, 14, 53, 23	mircgr 28584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑍 − (((pInvG‘𝐺)‘𝑍)‘𝑄)) = (𝑍 − 𝑄))
98	97	eqcomd 2735	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑍 − 𝑄) = (𝑍 − (((pInvG‘𝐺)‘𝑍)‘𝑄)))
99	1, 2, 3, 5, 14, 23, 14, 54, 98	tgcgrcomlr 28407	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 − 𝑍) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) − 𝑍))
100		eqidd 2730	. . . . . . . . . . . . . . 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 28392	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 − 𝑍) = (𝐷 − 𝑍))
102	1, 2, 3, 5, 6, 14, 15, 14, 101	tgcgrcomlr 28407	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 − 𝐶) = (𝑍 − 𝐷))
103	28	oveq2d 7403	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 − 𝐷) = (𝑍 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
104	102, 103	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑍 − 𝐶) = (𝑍 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
105	1, 2, 3, 4, 41, 5, 14, 7, 6	israg 28624	. . . . . . . . . . . 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 2980	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ≠ 𝑅)
109	1, 2, 3, 5, 13, 12, 13, 14, 84, 108	tgcgrneq 28410	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
110	109	necomd 2980	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
111	110	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ≠ 𝑌)
112	111, 48	neeqtrd 2994	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ≠ 𝑋)
113	19	eqcomd 2735	. . . . . . . . . . . . . . . 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 28410	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 ≠ 𝑌)
117	116	necomd 2980	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 ≠ 𝐸)
118	1, 2, 3, 5, 10, 6, 10, 13, 113, 60	tgcgrneq 28410	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ≠ 𝑌)
119	1, 3, 4, 5, 10, 13, 14, 118, 21	btwnlng3 28548	. . . . . . . . . . . . . 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 28549	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ∈ (𝑌𝐿𝐸))
122	48	oveq1d 7402	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → (𝑌𝐿𝐸) = (𝑋𝐿𝐸))
123	121, 122	eleqtrd 2830	. . . . . . . . . . 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 28626	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐸𝑋𝐶”⟩ ∈ (∟G‘𝐺))
126	1, 2, 3, 4, 41, 42, 43, 51, 46, 125	ragcom 28625	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐶𝑋𝐸”⟩ ∈ (∟G‘𝐺))
127	50, 126	eqeltrd 2828	. . . . . . 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 28415	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (𝑌𝐼𝐶))
130	1, 4, 3, 5, 13, 12, 6, 129	btwncolg3 28484	. . . . . . . 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 28626	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝑅𝑌𝐸”⟩ ∈ (∟G‘𝐺))
133	1, 2, 3, 4, 41, 42, 45, 44, 43, 132	ragcom 28625	. . . . 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 28631	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑅)
136	108	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 ≠ 𝑅)
137	136	neneqd 2930	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ¬ 𝑌 = 𝑅)
138	135, 137	pm2.65da 816	. . 3 ⊢ (𝜑 → ¬ 𝑌 = 𝑋)
139	138	neqned 2932	. 2 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
140	28	oveq2d 7403	. . . . 5 ⊢ (𝜑 → (𝑌 − 𝐷) = (𝑌 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
141	96, 140	eqtrd 2764	. . . 4 ⊢ (𝜑 → (𝑌 − 𝐶) = (𝑌 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
142	1, 2, 3, 4, 41, 5, 13, 7, 6	israg 28624	. . . 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 28625	. 2 ⊢ (𝜑 → ⟨“𝐶𝑋𝑌”⟩ ∈ (∟G‘𝐺))
145	1, 2, 3, 4, 5, 33, 8, 35, 36, 40, 32, 139, 144	ragperp 28644	1 ⊢ (𝜑 → (𝐶𝐿𝑋)(⟂G‘𝐺)𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5107  ran crn 5639  ‘cfv 6511  (class class class)co 7387  ⟨“cs3 14808  Basecbs 17179  distcds 17229  TarskiGcstrkg 28354  Itvcitv 28360  LineGclng 28361  cgrGccgrg 28437  pInvGcmir 28579  ∟Gcrag 28620  ⟂Gcperpg 28622

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oadd 8438  df-er 8671  df-map 8801  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-dju 9854  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-xnn0 12516  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-hash 14296  df-word 14479  df-concat 14536  df-s1 14561  df-s2 14814  df-s3 14815  df-trkgc 28375  df-trkgb 28376  df-trkgcb 28377  df-trkg 28380  df-cgrg 28438  df-leg 28510  df-mir 28580  df-rag 28621  df-perpg 28623

This theorem is referenced by:  footex  28648