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Theorem trgcopy 26517
Description: Triangle construction: a copy of a given triangle can always be constructed in such a way that one side is lying on a half-line, and the third vertex is on a given half-plane: existence part. First part of Theorem 10.16 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 4-Aug-2020.)
Hypotheses
Ref Expression
trgcopy.p 𝑃 = (Base‘𝐺)
trgcopy.m = (dist‘𝐺)
trgcopy.i 𝐼 = (Itv‘𝐺)
trgcopy.l 𝐿 = (LineG‘𝐺)
trgcopy.k 𝐾 = (hlG‘𝐺)
trgcopy.g (𝜑𝐺 ∈ TarskiG)
trgcopy.a (𝜑𝐴𝑃)
trgcopy.b (𝜑𝐵𝑃)
trgcopy.c (𝜑𝐶𝑃)
trgcopy.d (𝜑𝐷𝑃)
trgcopy.e (𝜑𝐸𝑃)
trgcopy.f (𝜑𝐹𝑃)
trgcopy.1 (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
trgcopy.2 (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
trgcopy.3 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
Assertion
Ref Expression
trgcopy (𝜑 → ∃𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
Distinct variable groups:   ,𝑓   𝐴,𝑓   𝐵,𝑓   𝐶,𝑓   𝐷,𝑓   𝑓,𝐸   𝑓,𝐹   𝑓,𝐺   𝑓,𝐼   𝑓,𝐿   𝑃,𝑓   𝜑,𝑓   𝑓,𝐾

Proof of Theorem trgcopy
Dummy variables 𝑗 𝑘 𝑙 𝑞 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trgcopy.p . . . . . . 7 𝑃 = (Base‘𝐺)
2 trgcopy.m . . . . . . 7 = (dist‘𝐺)
3 eqid 2818 . . . . . . 7 (cgrG‘𝐺) = (cgrG‘𝐺)
4 trgcopy.g . . . . . . . . . . 11 (𝜑𝐺 ∈ TarskiG)
54ad2antrr 722 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → 𝐺 ∈ TarskiG)
65ad2antrr 722 . . . . . . . . 9 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐺 ∈ TarskiG)
76ad2antrr 722 . . . . . . . 8 (((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝐺 ∈ TarskiG)
87adantr 481 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝐺 ∈ TarskiG)
9 trgcopy.a . . . . . . . . . 10 (𝜑𝐴𝑃)
109ad2antrr 722 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → 𝐴𝑃)
1110ad2antrr 722 . . . . . . . 8 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴𝑃)
1211ad3antrrr 726 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝐴𝑃)
13 trgcopy.b . . . . . . . . . 10 (𝜑𝐵𝑃)
1413ad2antrr 722 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → 𝐵𝑃)
1514ad2antrr 722 . . . . . . . 8 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐵𝑃)
1615ad3antrrr 726 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝐵𝑃)
17 trgcopy.c . . . . . . . . 9 (𝜑𝐶𝑃)
1817ad6antr 732 . . . . . . . 8 (((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝐶𝑃)
1918adantr 481 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝐶𝑃)
20 trgcopy.d . . . . . . . . . 10 (𝜑𝐷𝑃)
2120ad2antrr 722 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → 𝐷𝑃)
2221ad2antrr 722 . . . . . . . 8 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷𝑃)
2322ad3antrrr 726 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝐷𝑃)
24 trgcopy.e . . . . . . . . . 10 (𝜑𝐸𝑃)
2524ad2antrr 722 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → 𝐸𝑃)
2625ad2antrr 722 . . . . . . . 8 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐸𝑃)
2726ad3antrrr 726 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝐸𝑃)
28 simprl 767 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑓𝑃)
29 trgcopy.3 . . . . . . . . 9 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
3029ad2antrr 722 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → (𝐴 𝐵) = (𝐷 𝐸))
3130ad5antr 730 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐴 𝐵) = (𝐷 𝐸))
32 trgcopy.i . . . . . . . 8 𝐼 = (Itv‘𝐺)
33 trgcopy.l . . . . . . . . . . 11 𝐿 = (LineG‘𝐺)
34 trgcopy.1 . . . . . . . . . . 11 (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
351, 33, 32, 4, 13, 17, 9, 34ncoltgdim2 26278 . . . . . . . . . 10 (𝜑𝐺DimTarskiG≥2)
3635ad4antr 728 . . . . . . . . 9 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐺DimTarskiG≥2)
3736ad3antrrr 726 . . . . . . . 8 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝐺DimTarskiG≥2)
381, 32, 33, 4, 9, 13, 17, 34ncolne1 26338 . . . . . . . . . . . . . 14 (𝜑𝐴𝐵)
391, 32, 33, 4, 9, 13, 38tgelrnln 26343 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝐿𝐵) ∈ ran 𝐿)
4039ad2antrr 722 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → (𝐴𝐿𝐵) ∈ ran 𝐿)
41 simplr 765 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → 𝑥 ∈ (𝐴𝐿𝐵))
421, 33, 32, 5, 40, 41tglnpt 26262 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → 𝑥𝑃)
4342ad2antrr 722 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑥𝑃)
4443ad2antrr 722 . . . . . . . . 9 (((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑥𝑃)
4544adantr 481 . . . . . . . 8 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑥𝑃)
46 simplr 765 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑦𝑃)
4746ad2antrr 722 . . . . . . . . 9 (((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑦𝑃)
4847adantr 481 . . . . . . . 8 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑦𝑃)
4941ad5antr 730 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑥 ∈ (𝐴𝐿𝐵))
5038ad7antr 734 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝐴𝐵)
511, 32, 33, 8, 12, 16, 50tglinecom 26348 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
5249, 51eleqtrd 2912 . . . . . . . . 9 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑥 ∈ (𝐵𝐿𝐴))
53 simp-6r 784 . . . . . . . . . . . 12 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵))
5433, 8, 53perpln1 26423 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐶𝐿𝑥) ∈ ran 𝐿)
5540ad5antr 730 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐴𝐿𝐵) ∈ ran 𝐿)
561, 2, 32, 33, 8, 54, 55, 53perpcom 26426 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐶𝐿𝑥))
571, 33, 32, 4, 13, 17, 9, 34ncolrot2 26276 . . . . . . . . . . . . . . . . . 18 (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
58 ioran 977 . . . . . . . . . . . . . . . . . 18 (¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ↔ (¬ 𝐶 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵))
5957, 58sylib 219 . . . . . . . . . . . . . . . . 17 (𝜑 → (¬ 𝐶 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵))
6059simpld 495 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ 𝐶 ∈ (𝐴𝐿𝐵))
6160ad2antrr 722 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → ¬ 𝐶 ∈ (𝐴𝐿𝐵))
62 nelne2 3112 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐶 ∈ (𝐴𝐿𝐵)) → 𝑥𝐶)
6341, 61, 62syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → 𝑥𝐶)
6463ad4antr 728 . . . . . . . . . . . . 13 (((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑥𝐶)
6564adantr 481 . . . . . . . . . . . 12 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑥𝐶)
6665necomd 3068 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝐶𝑥)
671, 32, 33, 8, 19, 45, 66tglinecom 26348 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐶𝐿𝑥) = (𝑥𝐿𝐶))
6856, 51, 673brtr3d 5088 . . . . . . . . 9 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐵𝐿𝐴)(⟂G‘𝐺)(𝑥𝐿𝐶))
691, 2, 32, 33, 8, 16, 12, 52, 19, 68perprag 26439 . . . . . . . 8 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → ⟨“𝐵𝑥𝐶”⟩ ∈ (∟G‘𝐺))
70 trgcopy.f . . . . . . . . . . . . 13 (𝜑𝐹𝑃)
71 trgcopy.2 . . . . . . . . . . . . 13 (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
721, 32, 33, 4, 20, 24, 70, 71ncolne1 26338 . . . . . . . . . . . 12 (𝜑𝐷𝐸)
7372necomd 3068 . . . . . . . . . . 11 (𝜑𝐸𝐷)
7473ad7antr 734 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝐸𝐷)
7572ad4antr 728 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷𝐸)
7675neneqd 3018 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ¬ 𝐷 = 𝐸)
7741orcd 869 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → (𝑥 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
781, 33, 32, 5, 10, 14, 42, 77colrot2 26273 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → (𝐵 ∈ (𝑥𝐿𝐴) ∨ 𝑥 = 𝐴))
791, 33, 32, 5, 42, 10, 14, 78colcom 26271 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → (𝐵 ∈ (𝐴𝐿𝑥) ∨ 𝐴 = 𝑥))
8079ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐵 ∈ (𝐴𝐿𝑥) ∨ 𝐴 = 𝑥))
81 simpr 485 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩)
821, 33, 32, 6, 11, 15, 43, 3, 22, 26, 46, 80, 81lnxfr 26279 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐸 ∈ (𝐷𝐿𝑦) ∨ 𝐷 = 𝑦))
831, 33, 32, 6, 22, 46, 26, 82colrot2 26273 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑦 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
841, 33, 32, 6, 26, 22, 46, 83colcom 26271 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑦 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
8584orcomd 865 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐷 = 𝐸𝑦 ∈ (𝐷𝐿𝐸)))
8685ord 858 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (¬ 𝐷 = 𝐸𝑦 ∈ (𝐷𝐿𝐸)))
8776, 86mpd 15 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑦 ∈ (𝐷𝐿𝐸))
8887ad3antrrr 726 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑦 ∈ (𝐷𝐿𝐸))
891, 32, 33, 8, 27, 23, 48, 74, 88lncom 26335 . . . . . . . . 9 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑦 ∈ (𝐸𝐿𝐷))
90 simprrr 778 . . . . . . . . . . . . 13 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝑦 𝑓) = (𝑥 𝐶))
9190eqcomd 2824 . . . . . . . . . . . 12 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝑥 𝐶) = (𝑦 𝑓))
921, 2, 32, 8, 45, 19, 48, 28, 91, 65tgcgrneq 26196 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑦𝑓)
931, 32, 33, 8, 48, 28, 92tgelrnln 26343 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝑦𝐿𝑓) ∈ ran 𝐿)
941, 32, 33, 8, 27, 23, 74tgelrnln 26343 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐸𝐿𝐷) ∈ ran 𝐿)
95 simpllr 772 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑞𝑃)
96 simplr 765 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑞𝑃)
97 simprl 767 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → (𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦))
9833, 7, 97perpln2 26424 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → (𝑞𝐿𝑦) ∈ ran 𝐿)
991, 32, 33, 7, 96, 47, 98tglnne 26341 . . . . . . . . . . . . . . 15 (((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑞𝑦)
10099adantr 481 . . . . . . . . . . . . . 14 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑞𝑦)
101100necomd 3068 . . . . . . . . . . . . 13 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑦𝑞)
1021, 32, 33, 8, 48, 95, 101tgelrnln 26343 . . . . . . . . . . . 12 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝑦𝐿𝑞) ∈ ran 𝐿)
10397adantr 481 . . . . . . . . . . . . 13 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦))
1041, 32, 33, 8, 27, 23, 74tglinecom 26348 . . . . . . . . . . . . 13 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐸𝐿𝐷) = (𝐷𝐿𝐸))
1051, 32, 33, 8, 48, 95, 101tglinecom 26348 . . . . . . . . . . . . 13 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝑦𝐿𝑞) = (𝑞𝐿𝑦))
106103, 104, 1053brtr4d 5089 . . . . . . . . . . . 12 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐸𝐿𝐷)(⟂G‘𝐺)(𝑦𝐿𝑞))
1071, 2, 32, 33, 8, 94, 102, 106perpcom 26426 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝑦𝐿𝑞)(⟂G‘𝐺)(𝐸𝐿𝐷))
108 trgcopy.k . . . . . . . . . . . . . 14 𝐾 = (hlG‘𝐺)
109 simprrl 777 . . . . . . . . . . . . . 14 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑓(𝐾𝑦)𝑞)
1101, 32, 108, 28, 95, 48, 8, 33, 109hlln 26320 . . . . . . . . . . . . 13 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑓 ∈ (𝑞𝐿𝑦))
1111, 32, 33, 8, 48, 95, 28, 101, 110lncom 26335 . . . . . . . . . . . 12 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑓 ∈ (𝑦𝐿𝑞))
112111orcd 869 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝑓 ∈ (𝑦𝐿𝑞) ∨ 𝑦 = 𝑞))
1131, 2, 32, 33, 8, 48, 95, 28, 107, 112, 92colperp 26442 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝑦𝐿𝑓)(⟂G‘𝐺)(𝐸𝐿𝐷))
1141, 2, 32, 33, 8, 93, 94, 113perpcom 26426 . . . . . . . . 9 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐸𝐿𝐷)(⟂G‘𝐺)(𝑦𝐿𝑓))
1151, 2, 32, 33, 8, 27, 23, 89, 28, 114perprag 26439 . . . . . . . 8 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → ⟨“𝐸𝑦𝑓”⟩ ∈ (∟G‘𝐺))
11681ad3antrrr 726 . . . . . . . . 9 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩)
1171, 2, 32, 3, 8, 12, 16, 45, 23, 27, 48, 116cgr3simp2 26234 . . . . . . . 8 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐵 𝑥) = (𝐸 𝑦))
1181, 2, 32, 8, 37, 16, 45, 19, 27, 48, 28, 69, 115, 117, 91hypcgr 26514 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐵 𝐶) = (𝐸 𝑓))
119 eqid 2818 . . . . . . . . 9 (pInvG‘𝐺) = (pInvG‘𝐺)
12051, 68eqbrtrd 5079 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑥𝐿𝐶))
1211, 2, 32, 33, 8, 12, 16, 49, 19, 120perprag 26439 . . . . . . . . 9 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → ⟨“𝐴𝑥𝐶”⟩ ∈ (∟G‘𝐺))
1221, 2, 32, 33, 119, 8, 12, 45, 19, 121ragcom 26411 . . . . . . . 8 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → ⟨“𝐶𝑥𝐴”⟩ ∈ (∟G‘𝐺))
123104, 114eqbrtrrd 5081 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐷𝐿𝐸)(⟂G‘𝐺)(𝑦𝐿𝑓))
1241, 2, 32, 33, 8, 23, 27, 88, 28, 123perprag 26439 . . . . . . . . 9 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → ⟨“𝐷𝑦𝑓”⟩ ∈ (∟G‘𝐺))
1251, 2, 32, 33, 119, 8, 23, 48, 28, 124ragcom 26411 . . . . . . . 8 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → ⟨“𝑓𝑦𝐷”⟩ ∈ (∟G‘𝐺))
1261, 2, 32, 8, 45, 19, 48, 28, 91tgcgrcomlr 26193 . . . . . . . 8 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐶 𝑥) = (𝑓 𝑦))
1271, 2, 32, 3, 8, 12, 16, 45, 23, 27, 48, 116cgr3simp3 26235 . . . . . . . 8 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝑥 𝐴) = (𝑦 𝐷))
1281, 2, 32, 8, 37, 19, 45, 12, 28, 48, 23, 122, 125, 126, 127hypcgr 26514 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐶 𝐴) = (𝑓 𝐷))
1291, 2, 3, 8, 12, 16, 19, 23, 27, 28, 31, 118, 128trgcgr 26229 . . . . . 6 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩)
1301, 32, 33, 4, 20, 24, 72tgelrnln 26343 . . . . . . . . 9 (𝜑 → (𝐷𝐿𝐸) ∈ ran 𝐿)
131130ad4antr 728 . . . . . . . 8 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐷𝐿𝐸) ∈ ran 𝐿)
132131ad3antrrr 726 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (𝐷𝐿𝐸) ∈ ran 𝐿)
133 simpl 483 . . . . . . . . . . 11 ((𝑤 = 𝑘𝑣 = 𝑙) → 𝑤 = 𝑘)
134133eleq1d 2894 . . . . . . . . . 10 ((𝑤 = 𝑘𝑣 = 𝑙) → (𝑤 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ↔ 𝑘 ∈ (𝑃 ∖ (𝐷𝐿𝐸))))
135 simpr 485 . . . . . . . . . . 11 ((𝑤 = 𝑘𝑣 = 𝑙) → 𝑣 = 𝑙)
136135eleq1d 2894 . . . . . . . . . 10 ((𝑤 = 𝑘𝑣 = 𝑙) → (𝑣 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ↔ 𝑙 ∈ (𝑃 ∖ (𝐷𝐿𝐸))))
137134, 136anbi12d 630 . . . . . . . . 9 ((𝑤 = 𝑘𝑣 = 𝑙) → ((𝑤 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ↔ (𝑘 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑙 ∈ (𝑃 ∖ (𝐷𝐿𝐸)))))
138 simpr 485 . . . . . . . . . . 11 (((𝑤 = 𝑘𝑣 = 𝑙) ∧ 𝑧 = 𝑗) → 𝑧 = 𝑗)
139 simpll 763 . . . . . . . . . . . 12 (((𝑤 = 𝑘𝑣 = 𝑙) ∧ 𝑧 = 𝑗) → 𝑤 = 𝑘)
140 simplr 765 . . . . . . . . . . . 12 (((𝑤 = 𝑘𝑣 = 𝑙) ∧ 𝑧 = 𝑗) → 𝑣 = 𝑙)
141139, 140oveq12d 7163 . . . . . . . . . . 11 (((𝑤 = 𝑘𝑣 = 𝑙) ∧ 𝑧 = 𝑗) → (𝑤𝐼𝑣) = (𝑘𝐼𝑙))
142138, 141eleq12d 2904 . . . . . . . . . 10 (((𝑤 = 𝑘𝑣 = 𝑙) ∧ 𝑧 = 𝑗) → (𝑧 ∈ (𝑤𝐼𝑣) ↔ 𝑗 ∈ (𝑘𝐼𝑙)))
143142cbvrexdva 3458 . . . . . . . . 9 ((𝑤 = 𝑘𝑣 = 𝑙) → (∃𝑧 ∈ (𝐷𝐿𝐸)𝑧 ∈ (𝑤𝐼𝑣) ↔ ∃𝑗 ∈ (𝐷𝐿𝐸)𝑗 ∈ (𝑘𝐼𝑙)))
144137, 143anbi12d 630 . . . . . . . 8 ((𝑤 = 𝑘𝑣 = 𝑙) → (((𝑤 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑧 ∈ (𝐷𝐿𝐸)𝑧 ∈ (𝑤𝐼𝑣)) ↔ ((𝑘 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑙 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑗 ∈ (𝐷𝐿𝐸)𝑗 ∈ (𝑘𝐼𝑙))))
145144cbvopabv 5129 . . . . . . 7 {⟨𝑤, 𝑣⟩ ∣ ((𝑤 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑧 ∈ (𝐷𝐿𝐸)𝑧 ∈ (𝑤𝐼𝑣))} = {⟨𝑘, 𝑙⟩ ∣ ((𝑘 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑙 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑗 ∈ (𝐷𝐿𝐸)𝑗 ∈ (𝑘𝐼𝑙))}
1468adantr 481 . . . . . . . . . 10 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → 𝐺 ∈ TarskiG)
14719adantr 481 . . . . . . . . . 10 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → 𝐶𝑃)
14816adantr 481 . . . . . . . . . 10 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → 𝐵𝑃)
14912adantr 481 . . . . . . . . . 10 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → 𝐴𝑃)
15023adantr 481 . . . . . . . . . . . 12 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → 𝐷𝑃)
15127adantr 481 . . . . . . . . . . . 12 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → 𝐸𝑃)
15228adantr 481 . . . . . . . . . . . 12 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → 𝑓𝑃)
15373ad8antr 736 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → 𝐸𝐷)
154 simpr 485 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → 𝑓 ∈ (𝐷𝐿𝐸))
1551, 32, 33, 146, 151, 150, 152, 153, 154lncom 26335 . . . . . . . . . . . . . 14 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → 𝑓 ∈ (𝐸𝐿𝐷))
156155orcd 869 . . . . . . . . . . . . 13 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → (𝑓 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
1571, 33, 32, 146, 151, 150, 152, 156colrot1 26272 . . . . . . . . . . . 12 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → (𝐸 ∈ (𝐷𝐿𝑓) ∨ 𝐷 = 𝑓))
158129adantr 481 . . . . . . . . . . . . 13 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩)
1591, 2, 32, 3, 146, 149, 148, 147, 150, 151, 152, 158trgcgrcom 26241 . . . . . . . . . . . 12 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → ⟨“𝐷𝐸𝑓”⟩(cgrG‘𝐺)⟨“𝐴𝐵𝐶”⟩)
1601, 33, 32, 146, 150, 151, 152, 3, 149, 148, 147, 157, 159lnxfr 26279 . . . . . . . . . . 11 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
1611, 33, 32, 146, 149, 147, 148, 160colrot1 26272 . . . . . . . . . 10 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
1621, 33, 32, 146, 147, 148, 149, 161colcom 26271 . . . . . . . . 9 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
16334ad8antr 736 . . . . . . . . 9 (((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) ∧ 𝑓 ∈ (𝐷𝐿𝐸)) → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
164162, 163pm2.65da 813 . . . . . . . 8 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → ¬ 𝑓 ∈ (𝐷𝐿𝐸))
1651, 32, 33, 8, 132, 48, 145, 108, 88, 28, 95, 164, 109hphl 26484 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝑞)
16670ad4antr 728 . . . . . . . . 9 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐹𝑃)
167166ad2antrr 722 . . . . . . . 8 (((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝐹𝑃)
168167adantr 481 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝐹𝑃)
169 simplrr 774 . . . . . . 7 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
1701, 32, 33, 8, 132, 28, 145, 95, 165, 168, 169hpgtr 26481 . . . . . 6 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
171129, 170jca 512 . . . . 5 ((((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (𝑓𝑃 ∧ (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
1721, 32, 108, 47, 44, 18, 7, 96, 2, 99, 64hlcgrex 26329 . . . . 5 (((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → ∃𝑓𝑃 (𝑓(𝐾𝑦)𝑞 ∧ (𝑦 𝑓) = (𝑥 𝐶)))
173171, 172reximddv 3272 . . . 4 (((((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) ∧ 𝑞𝑃) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → ∃𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
1741, 33, 32, 4, 24, 70, 20, 71ncolrot2 26276 . . . . . . . 8 (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
175 ioran 977 . . . . . . . 8 (¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸) ↔ (¬ 𝐹 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝐷 = 𝐸))
176174, 175sylib 219 . . . . . . 7 (𝜑 → (¬ 𝐹 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝐷 = 𝐸))
177176simpld 495 . . . . . 6 (𝜑 → ¬ 𝐹 ∈ (𝐷𝐿𝐸))
178177ad4antr 728 . . . . 5 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ¬ 𝐹 ∈ (𝐷𝐿𝐸))
1791, 2, 32, 33, 6, 36, 131, 145, 87, 166, 178lnperpex 26516 . . . 4 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ∃𝑞𝑃 ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑞𝐿𝑦) ∧ 𝑞((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
180173, 179r19.29a 3286 . . 3 (((((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ∃𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
1811, 33, 32, 5, 10, 14, 42, 3, 21, 25, 2, 79, 30lnext 26280 . . 3 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → ∃𝑦𝑃 ⟨“𝐴𝐵𝑥”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩)
182180, 181r19.29a 3286 . 2 (((𝜑𝑥 ∈ (𝐴𝐿𝐵)) ∧ (𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵)) → ∃𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
1831, 2, 32, 33, 4, 39, 17, 60footex 26434 . 2 (𝜑 → ∃𝑥 ∈ (𝐴𝐿𝐵)(𝐶𝐿𝑥)(⟂G‘𝐺)(𝐴𝐿𝐵))
184182, 183r19.29a 3286 1 (𝜑 → ∃𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  wne 3013  wrex 3136  cdif 3930   class class class wbr 5057  {copab 5119  ran crn 5549  cfv 6348  (class class class)co 7145  2c2 11680  ⟨“cs3 14192  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  DimTarskiGcstrkgld 26147  Itvcitv 26149  LineGclng 26150  cgrGccgrg 26223  hlGchlg 26313  pInvGcmir 26365  ⟂Gcperpg 26408  hpGchpg 26470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-hash 13679  df-word 13850  df-concat 13911  df-s1 13938  df-s2 14198  df-s3 14199  df-trkgc 26161  df-trkgb 26162  df-trkgcb 26163  df-trkgld 26165  df-trkg 26166  df-cgrg 26224  df-ismt 26246  df-leg 26296  df-hlg 26314  df-mir 26366  df-rag 26407  df-perpg 26409  df-hpg 26471  df-mid 26487  df-lmi 26488
This theorem is referenced by:  trgcopyeu  26519  acopy  26546  cgrg3col4  26566
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