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Theorem sacgr 25943
Description: Supplementary angles of congruent angles are themselves congruent. Theorem 11.13 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 30-Sep-2020.)
Hypotheses
Ref Expression
dfcgra2.p 𝑃 = (Base‘𝐺)
dfcgra2.i 𝐼 = (Itv‘𝐺)
dfcgra2.m = (dist‘𝐺)
dfcgra2.g (𝜑𝐺 ∈ TarskiG)
dfcgra2.a (𝜑𝐴𝑃)
dfcgra2.b (𝜑𝐵𝑃)
dfcgra2.c (𝜑𝐶𝑃)
dfcgra2.d (𝜑𝐷𝑃)
dfcgra2.e (𝜑𝐸𝑃)
dfcgra2.f (𝜑𝐹𝑃)
sacgr.x (𝜑𝑋𝑃)
sacgr.y (𝜑𝑌𝑃)
sacgr.1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
sacgr.2 (𝜑𝐵 ∈ (𝐴𝐼𝑋))
sacgr.3 (𝜑𝐸 ∈ (𝐷𝐼𝑌))
sacgr.4 (𝜑𝐵𝑋)
sacgr.5 (𝜑𝐸𝑌)
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.
StepHypRef Expression
1 dfcgra2.p . . 3 𝑃 = (Base‘𝐺)
2 dfcgra2.i . . 3 𝐼 = (Itv‘𝐺)
3 eqid 2771 . . 3 (hlG‘𝐺) = (hlG‘𝐺)
4 dfcgra2.g . . . 4 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 709 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐺 ∈ TarskiG)
6 sacgr.x . . . 4 (𝜑𝑋𝑃)
76ad3antrrr 709 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑋𝑃)
8 dfcgra2.b . . . 4 (𝜑𝐵𝑃)
98ad3antrrr 709 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐵𝑃)
10 dfcgra2.c . . . 4 (𝜑𝐶𝑃)
1110ad3antrrr 709 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐶𝑃)
12 sacgr.y . . . 4 (𝜑𝑌𝑃)
1312ad3antrrr 709 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑌𝑃)
14 dfcgra2.e . . . 4 (𝜑𝐸𝑃)
1514ad3antrrr 709 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸𝑃)
16 dfcgra2.f . . . 4 (𝜑𝐹𝑃)
1716ad3antrrr 709 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐹𝑃)
18 dfcgra2.m . . . 4 = (dist‘𝐺)
19 eqid 2771 . . . 4 (LineG‘𝐺) = (LineG‘𝐺)
20 eqid 2771 . . . 4 (pInvG‘𝐺) = (pInvG‘𝐺)
21 eqid 2771 . . . 4 ((pInvG‘𝐺)‘𝐸) = ((pInvG‘𝐺)‘𝐸)
22 simpllr 760 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑥𝑃)
231, 18, 2, 19, 20, 5, 15, 21, 22mircl 25777 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (((pInvG‘𝐺)‘𝐸)‘𝑥) ∈ 𝑃)
24 simplr 752 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑦𝑃)
25 eqid 2771 . . . 4 (cgrG‘𝐺) = (cgrG‘𝐺)
261, 18, 2, 19, 20, 5, 15, 21, 22mircgr 25773 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (𝐸 (((pInvG‘𝐺)‘𝐸)‘𝑥)) = (𝐸 𝑥))
271, 18, 2, 5, 15, 23, 15, 22, 26tgcgrcomlr 25596 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ((((pInvG‘𝐺)‘𝐸)‘𝑥) 𝐸) = (𝑥 𝐸))
28 eqid 2771 . . . . . . . 8 ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
291, 18, 2, 19, 20, 4, 8, 28, 6mircl 25777 . . . . . . 7 (𝜑 → (((pInvG‘𝐺)‘𝐵)‘𝑋) ∈ 𝑃)
3029ad3antrrr 709 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (((pInvG‘𝐺)‘𝐵)‘𝑋) ∈ 𝑃)
31 simpr1 1233 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩)
321, 18, 2, 25, 5, 30, 9, 11, 22, 15, 24, 31cgr3simp1 25636 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ((((pInvG‘𝐺)‘𝐵)‘𝑋) 𝐵) = (𝑥 𝐸))
331, 18, 2, 19, 20, 4, 8, 28, 6mircgr 25773 . . . . . . 7 (𝜑 → (𝐵 (((pInvG‘𝐺)‘𝐵)‘𝑋)) = (𝐵 𝑋))
341, 18, 2, 4, 8, 29, 8, 6, 33tgcgrcomlr 25596 . . . . . 6 (𝜑 → ((((pInvG‘𝐺)‘𝐵)‘𝑋) 𝐵) = (𝑋 𝐵))
3534ad3antrrr 709 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ((((pInvG‘𝐺)‘𝐵)‘𝑋) 𝐵) = (𝑋 𝐵))
3627, 32, 353eqtr2rd 2812 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (𝑋 𝐵) = ((((pInvG‘𝐺)‘𝐸)‘𝑥) 𝐸))
371, 18, 2, 25, 5, 30, 9, 11, 22, 15, 24, 31cgr3simp2 25637 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (𝐵 𝐶) = (𝐸 𝑦))
381, 18, 2, 19, 20, 4, 8, 28, 6mirmir 25778 . . . . . . . . . 10 (𝜑 → (((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋)) = 𝑋)
39 eqidd 2772 . . . . . . . . . 10 (𝜑𝐵 = 𝐵)
40 eqidd 2772 . . . . . . . . . 10 (𝜑𝐶 = 𝐶)
4138, 39, 40s3eqd 13818 . . . . . . . . 9 (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))𝐵𝐶”⟩ = ⟨“𝑋𝐵𝐶”⟩)
4241ad3antrrr 709 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))𝐵𝐶”⟩ = ⟨“𝑋𝐵𝐶”⟩)
43 sacgr.4 . . . . . . . . . . . 12 (𝜑𝐵𝑋)
4443necomd 2998 . . . . . . . . . . 11 (𝜑𝑋𝐵)
451, 18, 2, 19, 20, 4, 8, 28, 6, 44mirne 25783 . . . . . . . . . 10 (𝜑 → (((pInvG‘𝐺)‘𝐵)‘𝑋) ≠ 𝐵)
4645ad3antrrr 709 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (((pInvG‘𝐺)‘𝐵)‘𝑋) ≠ 𝐵)
471, 18, 2, 19, 20, 5, 25, 28, 21, 30, 9, 22, 15, 11, 24, 46, 31mirtrcgr 25799 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))𝐵𝐶”⟩(cgrG‘𝐺)⟨“(((pInvG‘𝐺)‘𝐸)‘𝑥)𝐸𝑦”⟩)
4842, 47eqbrtrrd 4810 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“𝑋𝐵𝐶”⟩(cgrG‘𝐺)⟨“(((pInvG‘𝐺)‘𝐸)‘𝑥)𝐸𝑦”⟩)
491, 18, 2, 25, 5, 7, 9, 11, 23, 15, 24, 48cgr3swap13 25641 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“𝐶𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝑦𝐸(((pInvG‘𝐺)‘𝐸)‘𝑥)”⟩)
501, 18, 2, 25, 5, 11, 9, 7, 24, 15, 23, 49cgr3simp3 25638 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (𝑋 𝐶) = ((((pInvG‘𝐺)‘𝐸)‘𝑥) 𝑦))
511, 18, 2, 5, 7, 11, 23, 24, 50tgcgrcomlr 25596 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (𝐶 𝑋) = (𝑦 (((pInvG‘𝐺)‘𝐸)‘𝑥)))
521, 18, 25, 5, 7, 9, 11, 23, 15, 24, 36, 37, 51trgcgr 25632 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“𝑋𝐵𝐶”⟩(cgrG‘𝐺)⟨“(((pInvG‘𝐺)‘𝐸)‘𝑥)𝐸𝑦”⟩)
53 sacgr.5 . . . . . . 7 (𝜑𝐸𝑌)
5453ad3antrrr 709 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸𝑌)
5554necomd 2998 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑌𝐸)
56 dfcgra2.d . . . . . . . 8 (𝜑𝐷𝑃)
5756ad3antrrr 709 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐷𝑃)
58 simpr2 1235 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑥((hlG‘𝐺)‘𝐸)𝐷)
591, 2, 3, 22, 57, 15, 5, 58hlne1 25721 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑥𝐸)
601, 18, 2, 19, 20, 5, 15, 21, 22, 59mirne 25783 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (((pInvG‘𝐺)‘𝐸)‘𝑥) ≠ 𝐸)
611, 2, 3, 22, 57, 15, 5, 58hlcomd 25720 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐷((hlG‘𝐺)‘𝐸)𝑥)
62 sacgr.3 . . . . . . . . 9 (𝜑𝐸 ∈ (𝐷𝐼𝑌))
6362ad3antrrr 709 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸 ∈ (𝐷𝐼𝑌))
641, 2, 3, 57, 22, 13, 5, 15, 61, 63btwnhl 25730 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸 ∈ (𝑥𝐼𝑌))
651, 18, 2, 5, 22, 15, 13, 64tgbtwncom 25604 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸 ∈ (𝑌𝐼𝑥))
661, 18, 2, 19, 20, 5, 15, 21, 22mirmir 25778 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (((pInvG‘𝐺)‘𝐸)‘(((pInvG‘𝐺)‘𝐸)‘𝑥)) = 𝑥)
6766oveq2d 6809 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (𝑌𝐼(((pInvG‘𝐺)‘𝐸)‘(((pInvG‘𝐺)‘𝐸)‘𝑥))) = (𝑌𝐼𝑥))
6865, 67eleqtrrd 2853 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸 ∈ (𝑌𝐼(((pInvG‘𝐺)‘𝐸)‘(((pInvG‘𝐺)‘𝐸)‘𝑥))))
691, 18, 2, 19, 20, 5, 21, 3, 15, 13, 23, 15, 55, 60, 68mirhl2 25797 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑌((hlG‘𝐺)‘𝐸)(((pInvG‘𝐺)‘𝐸)‘𝑥))
701, 2, 3, 13, 23, 15, 5, 69hlcomd 25720 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → (((pInvG‘𝐺)‘𝐸)‘𝑥)((hlG‘𝐺)‘𝐸)𝑌)
71 simpr3 1237 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑦((hlG‘𝐺)‘𝐸)𝐹)
721, 2, 3, 5, 7, 9, 11, 13, 15, 17, 23, 24, 52, 70, 71iscgrad 25924 . 2 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → ⟨“𝑋𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑌𝐸𝐹”⟩)
73 dfcgra2.a . . . 4 (𝜑𝐴𝑃)
74 sacgr.1 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
751, 2, 3, 4, 73, 8, 10, 56, 14, 16, 74cgrane2 25926 . . . . . 6 (𝜑𝐵𝐶)
761, 2, 4, 3, 29, 8, 10, 45, 75cgraid 25932 . . . . 5 (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrA‘𝐺)⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩)
771, 2, 3, 4, 73, 8, 10, 56, 14, 16, 74cgrane1 25925 . . . . . 6 (𝜑𝐴𝐵)
78 sacgr.2 . . . . . . 7 (𝜑𝐵 ∈ (𝐴𝐼𝑋))
7938oveq2d 6809 . . . . . . 7 (𝜑 → (𝐴𝐼(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))) = (𝐴𝐼𝑋))
8078, 79eleqtrrd 2853 . . . . . 6 (𝜑𝐵 ∈ (𝐴𝐼(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))))
811, 18, 2, 19, 20, 4, 28, 3, 8, 73, 29, 73, 77, 45, 80mirhl2 25797 . . . . 5 (𝜑𝐴((hlG‘𝐺)‘𝐵)(((pInvG‘𝐺)‘𝐵)‘𝑋))
821, 2, 3, 4, 29, 8, 10, 29, 8, 10, 76, 73, 81cgrahl1 25929 . . . 4 (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
831, 2, 4, 3, 29, 8, 10, 73, 8, 10, 82, 56, 14, 16, 74cgratr 25936 . . 3 (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
841, 2, 3, 4, 29, 8, 10, 56, 14, 16iscgra 25922 . . 3 (𝜑 → (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥𝑃𝑦𝑃 (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)))
8583, 84mpbid 222 . 2 (𝜑 → ∃𝑥𝑃𝑦𝑃 (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹))
8672, 85r19.29vva 3229 1 (𝜑 → ⟨“𝑋𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑌𝐸𝐹”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wrex 3062   class class class wbr 4786  cfv 6031  (class class class)co 6793  ⟨“cs3 13796  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557  cgrGccgrg 25626  hlGchlg 25716  pInvGcmir 25768  cgrAccgra 25920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-concat 13497  df-s1 13498  df-s2 13802  df-s3 13803  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573  df-cgrg 25627  df-leg 25699  df-hlg 25717  df-mir 25769  df-cgra 25921
This theorem is referenced by:  oacgr  25944
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