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Theorem sacgr 28349

Description: Supplementary angles of congruent angles are themselves congruent. Theorem 11.13 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 30-Sep-2020.) (Proof shortened by Igor Ieskov, 16-Feb-2023.)

**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
sacgr	⊢ (𝜑 → ⟨“𝑋𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑌𝐸𝐹”⟩)

Proof of Theorem sacgr 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 2730	. . 3 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		dfcgra2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 726	. . 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 726	. . 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 726	. . 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 726	. . 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 726	. . 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 726	. . 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 726	. . 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 2730	. . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
20		eqid 2730	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
21		eqid 2730	. . . 4 ⊢ ((pInvG‘𝐺)‘𝐸) = ((pInvG‘𝐺)‘𝐸)
22		simpllr 772	. . . 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 28179	. . 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 765	. . 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 2730	. . . . . . 7 ⊢ ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
26	1, 18, 2, 19, 20, 4, 8, 25, 6	mirmir 28180	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋)) = 𝑋)
27		eqidd 2731	. . . . . 6 ⊢ (𝜑 → 𝐵 = 𝐵)
28		eqidd 2731	. . . . . 6 ⊢ (𝜑 → 𝐶 = 𝐶)
29	26, 27, 28	s3eqd 14819	. . . . 5 ⊢ (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))𝐵𝐶”⟩ = ⟨“𝑋𝐵𝐶”⟩)
30	29	ad3antrrr 726	. . . 4 ⊢ ((((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”⟩)
31		eqid 2730	. . . . 5 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
32	1, 18, 2, 19, 20, 4, 8, 25, 6	mircl 28179	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝐵)‘𝑋) ∈ 𝑃)
33	32	ad3antrrr 726	. . . . 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) ∈ 𝑃)
34		sacgr.4	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ 𝑋)
35	34	necomd 2994	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ 𝐵)
36	1, 18, 2, 19, 20, 4, 8, 25, 6, 35	mirne 28185	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝐵)‘𝑋) ≠ 𝐵)
37	36	ad3antrrr 726	. . . . 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)
38		simpr1 1192	. . . . 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)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩)
39	1, 18, 2, 19, 20, 5, 31, 25, 21, 33, 9, 22, 15, 11, 24, 37, 38	mirtrcgr 28201	. . . 4 ⊢ ((((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”⟩)
40	30, 39	eqbrtrrd 5171	. . 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”⟩)
41		sacgr.5	. . . . . . 7 ⊢ (𝜑 → 𝐸 ≠ 𝑌)
42	41	ad3antrrr 726	. . . . . 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)
43	42	necomd 2994	. . . . 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)
44		dfcgra2.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
45	44	ad3antrrr 726	. . . . . . 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)
46		simpr2 1193	. . . . . . 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)
47	1, 2, 3, 22, 45, 15, 5, 46	hlne1 28123	. . . . . 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)
48	1, 18, 2, 19, 20, 5, 15, 21, 22, 47	mirne 28185	. . . . 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)
49	1, 2, 3, 22, 45, 15, 5, 46	hlcomd 28122	. . . . . . . 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)
50		sacgr.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝑌))
51	50	ad3antrrr 726	. . . . . . . 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))
52	1, 2, 3, 45, 22, 13, 5, 15, 49, 51	btwnhl 28132	. . . . . . 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))
53	1, 18, 2, 5, 22, 15, 13, 52	tgbtwncom 28006	. . . . . 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))
54	1, 18, 2, 19, 20, 5, 15, 21, 22	mirmir 28180	. . . . . . 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)
55	54	oveq2d 7427	. . . . . 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))
56	53, 55	eleqtrrd 2834	. . . . 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))))
57	1, 18, 2, 19, 20, 5, 21, 3, 15, 13, 23, 15, 43, 48, 56	mirhl2 28199	. . . 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))
58	1, 2, 3, 13, 23, 15, 5, 57	hlcomd 28122	. . 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)
59		simpr3 1194	. . 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)
60	1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 23, 24, 40, 58, 59	iscgrad 28329	. 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”⟩)
61		dfcgra2.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
62		sacgr.1	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
63	1, 2, 3, 4, 61, 8, 10, 44, 14, 16, 62	cgrane2 28331	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
64	1, 2, 4, 3, 32, 8, 10, 36, 63	cgraid 28337	. . . . 5 ⊢ (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrA‘𝐺)⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩)
65	1, 2, 3, 4, 61, 8, 10, 44, 14, 16, 62	cgrane1 28330	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
66		sacgr.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝑋))
67	26	oveq2d 7427	. . . . . . 7 ⊢ (𝜑 → (𝐴𝐼(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))) = (𝐴𝐼𝑋))
68	66, 67	eleqtrrd 2834	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))))
69	1, 18, 2, 19, 20, 4, 25, 3, 8, 61, 32, 61, 65, 36, 68	mirhl2 28199	. . . . 5 ⊢ (𝜑 → 𝐴((hlG‘𝐺)‘𝐵)(((pInvG‘𝐺)‘𝐵)‘𝑋))
70	1, 2, 3, 4, 32, 8, 10, 32, 8, 10, 64, 61, 69	cgrahl1 28334	. . . 4 ⊢ (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
71	1, 2, 4, 3, 32, 8, 10, 61, 8, 10, 70, 44, 14, 16, 62	cgratr 28341	. . 3 ⊢ (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
72	1, 2, 3, 4, 32, 8, 10, 44, 14, 16	iscgra 28327	. . 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)))
73	71, 72	mpbid 231	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷 ∧ 𝑦((hlG‘𝐺)‘𝐸)𝐹))
74	60, 73	r19.29vva 3211	1 ⊢ (𝜑 → ⟨“𝑋𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑌𝐸𝐹”⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∃wrex 3068 class class class wbr 5147 ‘cfv 6542 (class class class)co 7411 ⟨“cs3 14797 Basecbs 17148 distcds 17210 TarskiGcstrkg 27945 Itvcitv 27951 LineGclng 27952 cgrGccgrg 28028 hlGchlg 28118 pInvGcmir 28170 cgrAccgra 28325

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-hash 14295 df-word 14469 df-concat 14525 df-s1 14550 df-s2 14803 df-s3 14804 df-trkgc 27966 df-trkgb 27967 df-trkgcb 27968 df-trkg 27971 df-cgrg 28029 df-leg 28101 df-hlg 28119 df-mir 28171 df-cgra 28326

This theorem is referenced by: oacgr 28350

W3C validator