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Theorem sacgr 25716
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 2621 . . 3 (hlG‘𝐺) = (hlG‘𝐺)
4 dfcgra2.g . . . 4 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 766 . . 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 766 . . 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 766 . . 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 766 . . 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 766 . . 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 766 . . 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 766 . . 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 2621 . . . 4 (LineG‘𝐺) = (LineG‘𝐺)
20 eqid 2621 . . . 4 (pInvG‘𝐺) = (pInvG‘𝐺)
21 eqid 2621 . . . 4 ((pInvG‘𝐺)‘𝐸) = ((pInvG‘𝐺)‘𝐸)
22 simpllr 799 . . . 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 25550 . . 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 792 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑦𝑃)
25 eqid 2621 . . . 4 (cgrG‘𝐺) = (cgrG‘𝐺)
261, 18, 2, 19, 20, 5, 15, 21, 22mircgr 25546 . . . . . 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 25369 . . . . 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 2621 . . . . . . . 8 ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
291, 18, 2, 19, 20, 4, 8, 28, 6mircl 25550 . . . . . . 7 (𝜑 → (((pInvG‘𝐺)‘𝐵)‘𝑋) ∈ 𝑃)
3029ad3antrrr 766 . . . . . 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 1066 . . . . . 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 25409 . . . . 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 25546 . . . . . . 7 (𝜑 → (𝐵 (((pInvG‘𝐺)‘𝐵)‘𝑋)) = (𝐵 𝑋))
341, 18, 2, 4, 8, 29, 8, 6, 33tgcgrcomlr 25369 . . . . . 6 (𝜑 → ((((pInvG‘𝐺)‘𝐵)‘𝑋) 𝐵) = (𝑋 𝐵))
3534ad3antrrr 766 . . . . 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 2662 . . . 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 25410 . . . 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 25551 . . . . . . . . . 10 (𝜑 → (((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋)) = 𝑋)
39 eqidd 2622 . . . . . . . . . 10 (𝜑𝐵 = 𝐵)
40 eqidd 2622 . . . . . . . . . 10 (𝜑𝐶 = 𝐶)
4138, 39, 40s3eqd 13603 . . . . . . . . 9 (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))𝐵𝐶”⟩ = ⟨“𝑋𝐵𝐶”⟩)
4241ad3antrrr 766 . . . . . . . 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 2848 . . . . . . . . . . 11 (𝜑𝑋𝐵)
451, 18, 2, 19, 20, 4, 8, 28, 6, 44mirne 25556 . . . . . . . . . 10 (𝜑 → (((pInvG‘𝐺)‘𝐵)‘𝑋) ≠ 𝐵)
4645ad3antrrr 766 . . . . . . . . 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 25572 . . . . . . . 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 4675 . . . . . . 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 25414 . . . . . 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 25411 . . . . 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 25369 . . . 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 25405 . . 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 766 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸𝑌)
5554necomd 2848 . . . . 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 766 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐷𝑃)
58 simpr2 1067 . . . . . . 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 25494 . . . . . 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 25556 . . . . 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 25493 . . . . . . . 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 766 . . . . . . . 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 25503 . . . . . . 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 25377 . . . . . 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 25551 . . . . . . 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 6663 . . . . . 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 2703 . . . . 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 25570 . . . 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 25493 . . 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 1068 . . 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 25697 . 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 25699 . . . . . 6 (𝜑𝐵𝐶)
761, 2, 4, 3, 29, 8, 10, 45, 75cgraid 25705 . . . . 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 25698 . . . . . 6 (𝜑𝐴𝐵)
78 sacgr.2 . . . . . . 7 (𝜑𝐵 ∈ (𝐴𝐼𝑋))
7938oveq2d 6663 . . . . . . 7 (𝜑 → (𝐴𝐼(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))) = (𝐴𝐼𝑋))
8078, 79eleqtrrd 2703 . . . . . 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 25570 . . . . 5 (𝜑𝐴((hlG‘𝐺)‘𝐵)(((pInvG‘𝐺)‘𝐵)‘𝑋))
821, 2, 3, 4, 29, 8, 10, 29, 8, 10, 76, 73, 81cgrahl1 25702 . . . 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 25709 . . 3 (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
841, 2, 3, 4, 29, 8, 10, 56, 14, 16iscgra 25695 . . 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 3079 1 (𝜑 → ⟨“𝑋𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑌𝐸𝐹”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989  wne 2793  wrex 2912   class class class wbr 4651  cfv 5886  (class class class)co 6647  ⟨“cs3 13581  Basecbs 15851  distcds 15944  TarskiGcstrkg 25323  Itvcitv 25329  LineGclng 25330  cgrGccgrg 25399  hlGchlg 25489  pInvGcmir 25541  cgrAccgra 25693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-er 7739  df-map 7856  df-pm 7857  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-card 8762  df-cda 8987  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-3 11077  df-n0 11290  df-xnn0 11361  df-z 11375  df-uz 11685  df-fz 12324  df-fzo 12462  df-hash 13113  df-word 13294  df-concat 13296  df-s1 13297  df-s2 13587  df-s3 13588  df-trkgc 25341  df-trkgb 25342  df-trkgcb 25343  df-trkg 25346  df-cgrg 25400  df-leg 25472  df-hlg 25490  df-mir 25542  df-cgra 25694
This theorem is referenced by:  oacgr  25717
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