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Theorem sacgr 25436
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 2605 . . 3 (hlG‘𝐺) = (hlG‘𝐺)
4 dfcgra2.g . . . 4 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 761 . . 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 761 . . 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 761 . . 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 761 . . 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 761 . . 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 761 . . 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 761 . . 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 2605 . . . 4 (LineG‘𝐺) = (LineG‘𝐺)
20 eqid 2605 . . . 4 (pInvG‘𝐺) = (pInvG‘𝐺)
21 eqid 2605 . . . 4 ((pInvG‘𝐺)‘𝐸) = ((pInvG‘𝐺)‘𝐸)
22 simpllr 794 . . . 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 25270 . . 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 787 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝑦𝑃)
25 eqid 2605 . . . 4 (cgrG‘𝐺) = (cgrG‘𝐺)
261, 18, 2, 19, 20, 5, 15, 21, 22mircgr 25266 . . . . . 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 25088 . . . . 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 2605 . . . . . . . 8 ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
291, 18, 2, 19, 20, 4, 8, 28, 6mircl 25270 . . . . . . 7 (𝜑 → (((pInvG‘𝐺)‘𝐵)‘𝑋) ∈ 𝑃)
3029ad3antrrr 761 . . . . . 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 1059 . . . . . 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 25129 . . . . 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 25266 . . . . . . 7 (𝜑 → (𝐵 (((pInvG‘𝐺)‘𝐵)‘𝑋)) = (𝐵 𝑋))
341, 18, 2, 4, 8, 29, 8, 6, 33tgcgrcomlr 25088 . . . . . 6 (𝜑 → ((((pInvG‘𝐺)‘𝐵)‘𝑋) 𝐵) = (𝑋 𝐵))
3534ad3antrrr 761 . . . . 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 2646 . . . 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 25130 . . . 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 25271 . . . . . . . . . 10 (𝜑 → (((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋)) = 𝑋)
39 eqidd 2606 . . . . . . . . . 10 (𝜑𝐵 = 𝐵)
40 eqidd 2606 . . . . . . . . . 10 (𝜑𝐶 = 𝐶)
4138, 39, 40s3eqd 13402 . . . . . . . . 9 (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))𝐵𝐶”⟩ = ⟨“𝑋𝐵𝐶”⟩)
4241ad3antrrr 761 . . . . . . . 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 2832 . . . . . . . . . . 11 (𝜑𝑋𝐵)
451, 18, 2, 19, 20, 4, 8, 28, 6, 44mirne 25276 . . . . . . . . . 10 (𝜑 → (((pInvG‘𝐺)‘𝐵)‘𝑋) ≠ 𝐵)
4645ad3antrrr 761 . . . . . . . . 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 25292 . . . . . . . 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 4597 . . . . . . 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 25134 . . . . . 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 25131 . . . . 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 25088 . . . 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 25125 . . 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 761 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐸𝑌)
5554necomd 2832 . . . . 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 761 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)) → 𝐷𝑃)
58 simpr2 1060 . . . . . . 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 25214 . . . . . 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 25276 . . . . 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 25213 . . . . . . . 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 761 . . . . . . . 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 25223 . . . . . . 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 25096 . . . . . 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 25271 . . . . . . 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 6539 . . . . . 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 2686 . . . . 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 25290 . . . 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 25213 . . 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 1061 . . 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 25417 . 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 25419 . . . . . 6 (𝜑𝐵𝐶)
761, 2, 4, 3, 29, 8, 10, 45, 75cgraid 25425 . . . . 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 25418 . . . . . 6 (𝜑𝐴𝐵)
78 sacgr.2 . . . . . . 7 (𝜑𝐵 ∈ (𝐴𝐼𝑋))
7938oveq2d 6539 . . . . . . 7 (𝜑 → (𝐴𝐼(((pInvG‘𝐺)‘𝐵)‘(((pInvG‘𝐺)‘𝐵)‘𝑋))) = (𝐴𝐼𝑋))
8078, 79eleqtrrd 2686 . . . . . 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 25290 . . . . 5 (𝜑𝐴((hlG‘𝐺)‘𝐵)(((pInvG‘𝐺)‘𝐵)‘𝑋))
821, 2, 3, 4, 29, 8, 10, 29, 8, 10, 76, 73, 81cgrahl1 25422 . . . 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 25429 . . 3 (𝜑 → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
841, 2, 3, 4, 29, 8, 10, 56, 14, 16iscgra 25415 . . 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 220 . 2 (𝜑 → ∃𝑥𝑃𝑦𝑃 (⟨“(((pInvG‘𝐺)‘𝐵)‘𝑋)𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹))
8672, 85r19.29vva 3057 1 (𝜑 → ⟨“𝑋𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑌𝐸𝐹”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775  wrex 2892   class class class wbr 4573  cfv 5786  (class class class)co 6523  ⟨“cs3 13380  Basecbs 15637  distcds 15719  TarskiGcstrkg 25042  Itvcitv 25048  LineGclng 25049  cgrGccgrg 25119  hlGchlg 25209  pInvGcmir 25261  cgrAccgra 25413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-card 8621  df-cda 8846  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-3 10923  df-n0 11136  df-z 11207  df-uz 11516  df-fz 12149  df-fzo 12286  df-hash 12931  df-word 13096  df-concat 13098  df-s1 13099  df-s2 13386  df-s3 13387  df-trkgc 25060  df-trkgb 25061  df-trkgcb 25062  df-trkg 25065  df-cgrg 25120  df-leg 25192  df-hlg 25210  df-mir 25262  df-cgra 25414
This theorem is referenced by:  oacgr  25437
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