Theorem sacgrOLD 26180

Description: Obsolete version of sacgr 26179 as of 16-Feb-2023. (Contributed by Thierry Arnoux, 30-Sep-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dfcgra2.p	⊢ 𝑃 = (Base‘𝐺)
dfcgra2.i	⊢ 𝐼 = (Itv‘𝐺)
dfcgra2.m	⊢ − = (dist‘𝐺)
dfcgra2.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
dfcgra2.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
dfcgra2.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
dfcgra2.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
dfcgra2.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
dfcgra2.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
dfcgra2.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
sacgr.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
sacgr.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
sacgr.1	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
sacgr.2	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝑋))
sacgr.3	⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝑌))
sacgr.4	⊢ (𝜑 → 𝐵 ≠ 𝑋)
sacgr.5	⊢ (𝜑 → 𝐸 ≠ 𝑌)

Assertion

Ref	Expression
sacgrOLD	⊢ (𝜑 → ⟨“𝑋𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑌𝐸𝐹”⟩)

Proof of Theorem sacgrOLD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcgra2.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		dfcgra2.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
3		eqid 2777	. . 3 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		dfcgra2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 720	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐺 ∈ TarskiG)
6		sacgr.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7	6	ad3antrrr 720	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑋 ∈ 𝑃)
8		dfcgra2.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad3antrrr 720	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐵 ∈ 𝑃)
10		dfcgra2.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	ad3antrrr 720	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐶 ∈ 𝑃)
12		sacgr.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
13	12	ad3antrrr 720	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑌 ∈ 𝑃)
14		dfcgra2.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
15	14	ad3antrrr 720	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸 ∈ 𝑃)
16		dfcgra2.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
17	16	ad3antrrr 720	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐹 ∈ 𝑃)
18		dfcgra2.m	. . . 4 ⊢ − = (dist‘𝐺)
19		eqid 2777	. . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
20		eqid 2777	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
21		eqid 2777	. . . 4 ⊢ ((pInvG‘𝐺)‘𝐸) = ((pInvG‘𝐺)‘𝐸)
22		simpllr 766	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑥 ∈ 𝑃)
23	1, 18, 2, 19, 20, 5, 15, 21, 22	mircl 26012	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (((pInvG‘𝐺)‘𝐸)‘𝑥) ∈ 𝑃)
24		simplr 759	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑦 ∈ 𝑃)
25		eqid 2777	. . . 4 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
26	1, 18, 2, 19, 20, 5, 15, 21, 22	mircgr 26008	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (𝐸 − (((pInvG‘𝐺)‘𝐸)‘𝑥)) = (𝐸 − 𝑥))
27	1, 18, 2, 5, 15, 23, 15, 22, 26	tgcgrcomlr 25831	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ((((pInvG‘𝐺)‘𝐸)‘𝑥) − 𝐸) = (𝑥 − 𝐸))
28		eqid 2777	. . . . . . . 8 ⊢ ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
29	1, 18, 2, 19, 20, 4, 8, 28, 6	mircl 26012	. . . . . . 7 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝐵)‘𝑋) ∈ 𝑃)
30	29	ad3antrrr 720	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (((pInvG‘𝐺)‘𝐵)‘𝑋) ∈ 𝑃)
31		simpr1 1205	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩)
32	1, 18, 2, 25, 5, 30, 9, 11, 22, 15, 24, 31	cgr3simp1 25871	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ((((pInvG‘𝐺)‘𝐵)‘𝑋) − 𝐵) = (𝑥 − 𝐸))
33	1, 18, 2, 19, 20, 4, 8, 28, 6	mircgr 26008	. . . . . . 7 ⊢ (𝜑 → (𝐵 − (((pInvG‘𝐺)‘𝐵)‘𝑋)) = (𝐵 − 𝑋))
34	1, 18, 2, 4, 8, 29, 8, 6, 33	tgcgrcomlr 25831	. . . . . 6 ⊢ (𝜑 → ((((pInvG‘𝐺)‘𝐵)‘𝑋) − 𝐵) = (𝑋 − 𝐵))
35	34	ad3antrrr 720	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ((((pInvG‘𝐺)‘𝐵)‘𝑋) − 𝐵) = (𝑋 − 𝐵))
36	27, 32, 35	3eqtr2rd 2820	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (𝑋 − 𝐵) = ((((pInvG‘𝐺)‘𝐸)‘𝑥) − 𝐸))
37	1, 18, 2, 25, 5, 30, 9, 11, 22, 15, 24, 31	cgr3simp2 25872	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (𝐵 − 𝐶) = (𝐸 − 𝑦))
38	1, 18, 2, 19, 20, 4, 8, 28, 6	mirmir 26013	. . . . . . . . . 10 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋)) = 𝑋)
39		eqidd 2778	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = 𝐵)
40		eqidd 2778	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 = 𝐶)
41	38, 39, 40	s3eqd 14015	. . . . . . . . 9 ⊢ (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))𝐵𝐶”⟩ = ⟨“𝑋𝐵𝐶”⟩)
42	41	ad3antrrr 720	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))𝐵𝐶”⟩ = ⟨“𝑋𝐵𝐶”⟩)
43		sacgr.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ 𝑋)
44	43	necomd 3023	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ≠ 𝐵)
45	1, 18, 2, 19, 20, 4, 8, 28, 6, 44	mirne 26018	. . . . . . . . . 10 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝐵)‘𝑋) ≠ 𝐵)
46	45	ad3antrrr 720	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (((pInvG‘𝐺)‘𝐵)‘𝑋) ≠ 𝐵)
47	1, 18, 2, 19, 20, 5, 25, 28, 21, 30, 9, 22, 15, 11, 24, 46, 31	mirtrcgr 26034	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))𝐵𝐶”⟩(cgrG‘𝐺)⟨“(((pInvG‘𝐺)‘𝐸)‘𝑥)𝐸𝑦”⟩)
48	42, 47	eqbrtrrd 4910	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“𝑋𝐵𝐶”⟩(cgrG‘𝐺)⟨“(((pInvG‘𝐺)‘𝐸)‘𝑥)𝐸𝑦”⟩)
49	1, 18, 2, 25, 5, 7, 9, 11, 23, 15, 24, 48	cgr3swap13 25876	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“𝐶𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝑦𝐸(((pInvG‘𝐺)‘𝐸)‘𝑥)”⟩)
50	1, 18, 2, 25, 5, 11, 9, 7, 24, 15, 23, 49	cgr3simp3 25873	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (𝑋 − 𝐶) = ((((pInvG‘𝐺)‘𝐸)‘𝑥) − 𝑦))
51	1, 18, 2, 5, 7, 11, 23, 24, 50	tgcgrcomlr 25831	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (𝐶 − 𝑋) = (𝑦 − (((pInvG‘𝐺)‘𝐸)‘𝑥)))
52	1, 18, 25, 5, 7, 9, 11, 23, 15, 24, 36, 37, 51	trgcgr 25867	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“𝑋𝐵𝐶”⟩(cgrG‘𝐺)⟨“(((pInvG‘𝐺)‘𝐸)‘𝑥)𝐸𝑦”⟩)
53		sacgr.5	. . . . . . 7 ⊢ (𝜑 → 𝐸 ≠ 𝑌)
54	53	ad3antrrr 720	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸 ≠ 𝑌)
55	54	necomd 3023	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑌 ≠ 𝐸)
56		dfcgra2.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
57	56	ad3antrrr 720	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐷 ∈ 𝑃)
58		simpr2 1207	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑥((hlG‘𝐺)‘𝐸)𝐷)
59	1, 2, 3, 22, 57, 15, 5, 58	hlne1 25956	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑥 ≠ 𝐸)
60	1, 18, 2, 19, 20, 5, 15, 21, 22, 59	mirne 26018	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (((pInvG‘𝐺)‘𝐸)‘𝑥) ≠ 𝐸)
61	1, 2, 3, 22, 57, 15, 5, 58	hlcomd 25955	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐷((hlG‘𝐺)‘𝐸)𝑥)
62		sacgr.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝑌))
63	62	ad3antrrr 720	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸 ∈ (𝐷𝐼𝑌))
64	1, 2, 3, 57, 22, 13, 5, 15, 61, 63	btwnhl 25965	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸 ∈ (𝑥𝐼𝑌))
65	1, 18, 2, 5, 22, 15, 13, 64	tgbtwncom 25839	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸 ∈ (𝑌𝐼𝑥))
66	1, 18, 2, 19, 20, 5, 15, 21, 22	mirmir 26013	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (((pInvG‘𝐺)‘𝐸)‘(((pInvG‘𝐺)‘𝐸)‘𝑥)) = 𝑥)
67	66	oveq2d 6938	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (𝑌𝐼(((pInvG‘𝐺)‘𝐸)‘(((pInvG‘𝐺)‘𝐸)‘𝑥))) = (𝑌𝐼𝑥))
68	65, 67	eleqtrrd 2861	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸 ∈ (𝑌𝐼(((pInvG‘𝐺)‘𝐸)‘(((pInvG‘𝐺)‘𝐸)‘𝑥))))
69	1, 18, 2, 19, 20, 5, 21, 3, 15, 13, 23, 15, 55, 60, 68	mirhl2 26032	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑌((hlG‘𝐺)‘𝐸)(((pInvG‘𝐺)‘𝐸)‘𝑥))
70	1, 2, 3, 13, 23, 15, 5, 69	hlcomd 25955	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (((pInvG‘𝐺)‘𝐸)‘𝑥)((hlG‘𝐺)‘𝐸)𝑌)
71		simpr3 1209	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑦((hlG‘𝐺)‘𝐸)𝐹)
72	1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 23, 24, 52, 70, 71	iscgrad 26159	. 2 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“𝑋𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑌𝐸𝐹”⟩)
73		dfcgra2.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
74		sacgr.1	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
75	1, 2, 3, 4, 73, 8, 10, 56, 14, 16, 74	cgrane2 26161	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
76	1, 2, 4, 3, 29, 8, 10, 45, 75	cgraid 26167	. . . . 5 ⊢ (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrA‘𝐺)⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩)
77	1, 2, 3, 4, 73, 8, 10, 56, 14, 16, 74	cgrane1 26160	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
78		sacgr.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝑋))
79	38	oveq2d 6938	. . . . . . 7 ⊢ (𝜑 → (𝐴𝐼(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))) = (𝐴𝐼𝑋))
80	78, 79	eleqtrrd 2861	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))))
81	1, 18, 2, 19, 20, 4, 28, 3, 8, 73, 29, 73, 77, 45, 80	mirhl2 26032	. . . . 5 ⊢ (𝜑 → 𝐴((hlG‘𝐺)‘𝐵)(((pInvG‘𝐺)‘𝐵)‘𝑋))
82	1, 2, 3, 4, 29, 8, 10, 29, 8, 10, 76, 73, 81	cgrahl1 26164	. . . 4 ⊢ (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
83	1, 2, 4, 3, 29, 8, 10, 73, 8, 10, 82, 56, 14, 16, 74	cgratr 26171	. . 3 ⊢ (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
84	1, 2, 3, 4, 29, 8, 10, 56, 14, 16	iscgra 26157	. . 3 ⊢ (𝜑 → (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹)))
85	83, 84	mpbid 224	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹))
86	72, 85	r19.29vva 3266	1 ⊢ (𝜑 → ⟨“𝑋𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑌𝐸𝐹”⟩)

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106   ≠ wne 2968  ∃wrex 3090   class class class wbr 4886  ‘cfv 6135  (class class class)co 6922  ⟨“cs3 13993  Basecbs 16255  distcds 16347  TarskiGcstrkg 25781  Itvcitv 25787  LineGclng 25788  cgrGccgrg 25861  hlGchlg 25951  pInvGcmir 26003  cgrAccgra 26155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-hash 13436  df-word 13600  df-concat 13661  df-s1 13686  df-s2 13999  df-s3 14000  df-trkgc 25799  df-trkgb 25800  df-trkgcb 25801  df-trkg 25804  df-cgrg 25862  df-leg 25934  df-hlg 25952  df-mir 26004  df-cgra 26156

This theorem is referenced by: (None)