Theorem footexlem2 28773

Description: Lemma for footex 28774. (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 28772	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
30		nelne2 3029	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑋 ≠ 𝐶)
31	29, 9, 30	syl2anc 585	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝐶)
32	31	necomd 2986	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝑋)
33	1, 3, 4, 5, 6, 7, 32	tgelrnln 28683	. 2 ⊢ (𝜑 → (𝐶𝐿𝑋) ∈ ran 𝐿)
34	1, 3, 4, 5, 6, 7, 32	tglinerflx2 28687	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐶𝐿𝑋))
35	34, 29	elind 4151	. 2 ⊢ (𝜑 → 𝑋 ∈ ((𝐶𝐿𝑋) ∩ 𝐴))
36	1, 3, 4, 5, 6, 7, 32	tglinerflx1 28686	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐶𝐿𝑋))
37	17	necomd 2986	. . . . 5 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
38	1, 3, 4, 5, 11, 10, 13, 37, 18	btwnlng3 28674	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐹𝐿𝐸))
39	1, 3, 4, 5, 10, 11, 13, 17, 38	lncom 28675	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐸𝐿𝐹))
40	39, 16	eleqtrrd 2838	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
41		eqid 2735	. . . . 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 2736	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐶 = 𝐶)
48		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑋)
49		eqidd 2736	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 = 𝐸)
50	47, 48, 49	s3eqd 14789	. . . . . . . 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 2735	. . . . . . . . . . . . . . . 16 ⊢ ((pInvG‘𝐺)‘𝑍) = ((pInvG‘𝐺)‘𝑍)
54	1, 2, 3, 4, 41, 5, 14, 53, 23	mircl 28714	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑍)‘𝑄) ∈ 𝑃)
55	1, 2, 3, 5, 10, 13, 10, 6, 19	tgcgrcomlr 28533	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑌 − 𝐸) = (𝐶 − 𝐸))
56	25, 55	eqtr2d 2771	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐶 − 𝐸) = (𝑌 − 𝑄))
57	1, 3, 4, 5, 10, 11, 17	tglinerflx1 28686	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸 ∈ (𝐸𝐿𝐹))
58	57, 16	eleqtrrd 2838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
59		nelne2 3029	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐸 ≠ 𝐶)
60	58, 9, 59	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ≠ 𝐶)
61	60	necomd 2986	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ≠ 𝐸)
62	1, 2, 3, 5, 6, 10, 13, 23, 56, 61	tgcgrneq 28536	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ≠ 𝑄)
63	62	necomd 2986	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ≠ 𝑌)
64		nelne2 3029	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑌 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑌 ≠ 𝐶)
65	40, 9, 64	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑌 ≠ 𝐶)
66	65	necomd 2986	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶 ≠ 𝑌)
67	20, 66	eqnetrrd 2999	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌)
68		eqid 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((pInvG‘𝐺)‘𝑅) = ((pInvG‘𝐺)‘𝑅)
69	1, 2, 3, 4, 41, 5, 12, 68, 13	mirinv 28719	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) = 𝑌 ↔ 𝑅 = 𝑌))
70	69	necon3bid 2975	. . . . . . . . . . . . . . . . . 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 28711	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
73	20	oveq1d 7373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐶𝐼𝑌) = ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
74	72, 73	eleqtrrd 2838	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ (𝐶𝐼𝑌))
75	1, 2, 3, 5, 6, 12, 13, 23, 71, 74, 24	tgbtwnouttr2 28548	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐼𝑄))
76	1, 2, 3, 5, 6, 13, 23, 75	tgbtwncom 28541	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (𝑄𝐼𝐶))
77		eqid 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
78	20	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 − 𝐶) = (𝐸 − (((pInvG‘𝐺)‘𝑅)‘𝑌)))
79	19, 78	eqtrd 2770	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 − 𝑌) = (𝐸 − (((pInvG‘𝐺)‘𝑅)‘𝑌)))
80	1, 2, 3, 4, 41, 5, 10, 12, 13	israg 28750	. . . . . . . . . . . . . . . . . . . 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 28541	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑌 ∈ (𝑄𝐼𝑅))
83	1, 2, 3, 5, 13, 23, 13, 10, 25	tgcgrcomlr 28533	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑄 − 𝑌) = (𝐸 − 𝑌))
84	22	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑌 − 𝑅) = (𝑌 − 𝑍))
85	1, 2, 3, 5, 23, 10	axtgcgrrflx 28515	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑄 − 𝐸) = (𝐸 − 𝑄))
86	25	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . 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 28518	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑅 − 𝐸) = (𝑍 − 𝑄))
88	1, 2, 3, 5, 12, 10, 14, 23, 87	tgcgrcomlr 28533	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 − 𝑅) = (𝑄 − 𝑍))
89	1, 2, 3, 5, 13, 12, 13, 14, 84	tgcgrcomlr 28533	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑅 − 𝑌) = (𝑍 − 𝑌))
90	1, 2, 77, 5, 10, 12, 13, 23, 14, 13, 88, 89, 86	trgcgr 28569	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨“𝐸𝑅𝑌”⟩(cgrG‘𝐺)⟨“𝑄𝑍𝑌”⟩)
91	1, 2, 3, 4, 41, 5, 10, 12, 13, 77, 23, 14, 13, 81, 90	ragcgr 28760	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ⟨“𝑄𝑍𝑌”⟩ ∈ (∟G‘𝐺))
92	1, 2, 3, 4, 41, 5, 23, 14, 13, 91	ragcom 28751	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺))
93	1, 2, 3, 4, 41, 5, 13, 14, 23	israg 28750	. . . . . . . . . . . . . . . . 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 28533	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 − 𝑌) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) − 𝑌))
96	27	eqcomd 2741	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 − 𝐶) = (𝑌 − 𝐷))
97	1, 2, 3, 4, 41, 5, 14, 53, 23	mircgr 28710	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑍 − (((pInvG‘𝐺)‘𝑍)‘𝑄)) = (𝑍 − 𝑄))
98	97	eqcomd 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑍 − 𝑄) = (𝑍 − (((pInvG‘𝐺)‘𝑍)‘𝑄)))
99	1, 2, 3, 5, 14, 23, 14, 54, 98	tgcgrcomlr 28533	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 − 𝑍) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) − 𝑍))
100		eqidd 2736	. . . . . . . . . . . . . . 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 28518	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 − 𝑍) = (𝐷 − 𝑍))
102	1, 2, 3, 5, 6, 14, 15, 14, 101	tgcgrcomlr 28533	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 − 𝐶) = (𝑍 − 𝐷))
103	28	oveq2d 7374	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 − 𝐷) = (𝑍 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
104	102, 103	eqtrd 2770	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑍 − 𝐶) = (𝑍 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
105	1, 2, 3, 4, 41, 5, 14, 7, 6	israg 28750	. . . . . . . . . . . 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 2986	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ≠ 𝑅)
109	1, 2, 3, 5, 13, 12, 13, 14, 84, 108	tgcgrneq 28536	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
110	109	necomd 2986	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
111	110	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ≠ 𝑌)
112	111, 48	neeqtrd 3000	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ≠ 𝑋)
113	19	eqcomd 2741	. . . . . . . . . . . . . . . 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 28536	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝐸 ≠ 𝑌)
117	116	necomd 2986	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 ≠ 𝐸)
118	1, 2, 3, 5, 10, 6, 10, 13, 113, 60	tgcgrneq 28536	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ≠ 𝑌)
119	1, 3, 4, 5, 10, 13, 14, 118, 21	btwnlng3 28674	. . . . . . . . . . . . . 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 28675	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ∈ (𝑌𝐿𝐸))
122	48	oveq1d 7373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → (𝑌𝐿𝐸) = (𝑋𝐿𝐸))
123	121, 122	eleqtrd 2837	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑍 ∈ (𝑋𝐿𝐸))
124	123	orcd 874	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → (𝑍 ∈ (𝑋𝐿𝐸) ∨ 𝑋 = 𝐸))
125	1, 2, 3, 4, 41, 42, 52, 51, 46, 43, 107, 112, 124	ragcol 28752	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐸𝑋𝐶”⟩ ∈ (∟G‘𝐺))
126	1, 2, 3, 4, 41, 42, 43, 51, 46, 125	ragcom 28751	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝐶𝑋𝐸”⟩ ∈ (∟G‘𝐺))
127	50, 126	eqeltrd 2835	. . . . . . 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 28541	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (𝑌𝐼𝐶))
130	1, 4, 3, 5, 13, 12, 6, 129	btwncolg3 28610	. . . . . . . 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 28752	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ⟨“𝑅𝑌𝐸”⟩ ∈ (∟G‘𝐺))
133	1, 2, 3, 4, 41, 42, 45, 44, 43, 132	ragcom 28751	. . . . 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 28757	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑅)
136	108	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → 𝑌 ≠ 𝑅)
137	136	neneqd 2936	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑋) → ¬ 𝑌 = 𝑅)
138	135, 137	pm2.65da 817	. . 3 ⊢ (𝜑 → ¬ 𝑌 = 𝑋)
139	138	neqned 2938	. 2 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
140	28	oveq2d 7374	. . . . 5 ⊢ (𝜑 → (𝑌 − 𝐷) = (𝑌 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
141	96, 140	eqtrd 2770	. . . 4 ⊢ (𝜑 → (𝑌 − 𝐶) = (𝑌 − (((pInvG‘𝐺)‘𝑋)‘𝐶)))
142	1, 2, 3, 4, 41, 5, 13, 7, 6	israg 28750	. . . 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 28751	. 2 ⊢ (𝜑 → ⟨“𝐶𝑋𝑌”⟩ ∈ (∟G‘𝐺))
145	1, 2, 3, 4, 5, 33, 8, 35, 36, 40, 32, 139, 144	ragperp 28770	1 ⊢ (𝜑 → (𝐶𝐿𝑋)(⟂G‘𝐺)𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2931   class class class wbr 5097  ran crn 5624  ‘cfv 6491  (class class class)co 7358  ⟨“cs3 14767  Basecbs 17138  distcds 17188  TarskiGcstrkg 28480  Itvcitv 28486  LineGclng 28487  cgrGccgrg 28563  pInvGcmir 28705  ∟Gcrag 28746  ⟂Gcperpg 28748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-map 8767  df-pm 8768  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-dju 9815  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-n0 12404  df-xnn0 12477  df-z 12491  df-uz 12754  df-fz 13426  df-fzo 13573  df-hash 14256  df-word 14439  df-concat 14496  df-s1 14522  df-s2 14773  df-s3 14774  df-trkgc 28501  df-trkgb 28502  df-trkgcb 28503  df-trkg 28506  df-cgrg 28564  df-leg 28636  df-mir 28706  df-rag 28747  df-perpg 28749

This theorem is referenced by:  footex  28774