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Theorem footexlem2 28742
Description: Lemma for footex 28743. (Contributed by Thierry Arnoux, 19-Oct-2019.) (Revised by Thierry Arnoux, 1-Jul-2023.)
Hypotheses
Ref Expression
isperp.p 𝑃 = (Base‘𝐺)
isperp.d = (dist‘𝐺)
isperp.i 𝐼 = (Itv‘𝐺)
isperp.l 𝐿 = (LineG‘𝐺)
isperp.g (𝜑𝐺 ∈ TarskiG)
isperp.a (𝜑𝐴 ∈ ran 𝐿)
foot.x (𝜑𝐶𝑃)
foot.y (𝜑 → ¬ 𝐶𝐴)
footexlem.e (𝜑𝐸𝑃)
footexlem.f (𝜑𝐹𝑃)
footexlem.r (𝜑𝑅𝑃)
footexlem.x (𝜑𝑋𝑃)
footexlem.y (𝜑𝑌𝑃)
footexlem.z (𝜑𝑍𝑃)
footexlem.d (𝜑𝐷𝑃)
footexlem.1 (𝜑𝐴 = (𝐸𝐿𝐹))
footexlem.2 (𝜑𝐸𝐹)
footexlem.3 (𝜑𝐸 ∈ (𝐹𝐼𝑌))
footexlem.4 (𝜑 → (𝐸 𝑌) = (𝐸 𝐶))
footexlem.5 (𝜑𝐶 = (((pInvG‘𝐺)‘𝑅)‘𝑌))
footexlem.6 (𝜑𝑌 ∈ (𝐸𝐼𝑍))
footexlem.7 (𝜑 → (𝑌 𝑍) = (𝑌 𝑅))
footexlem.q (𝜑𝑄𝑃)
footexlem.8 (𝜑𝑌 ∈ (𝑅𝐼𝑄))
footexlem.9 (𝜑 → (𝑌 𝑄) = (𝑌 𝐸))
footexlem.10 (𝜑𝑌 ∈ ((((pInvG‘𝐺)‘𝑍)‘𝑄)𝐼𝐷))
footexlem.11 (𝜑 → (𝑌 𝐷) = (𝑌 𝐶))
footexlem.12 (𝜑𝐷 = (((pInvG‘𝐺)‘𝑋)‘𝐶))
Assertion
Ref Expression
footexlem2 (𝜑 → (𝐶𝐿𝑋)(⟂G‘𝐺)𝐴)

Proof of Theorem footexlem2
StepHypRef Expression
1 isperp.p . 2 𝑃 = (Base‘𝐺)
2 isperp.d . 2 = (dist‘𝐺)
3 isperp.i . 2 𝐼 = (Itv‘𝐺)
4 isperp.l . 2 𝐿 = (LineG‘𝐺)
5 isperp.g . 2 (𝜑𝐺 ∈ TarskiG)
6 foot.x . . 3 (𝜑𝐶𝑃)
7 footexlem.x . . 3 (𝜑𝑋𝑃)
8 isperp.a . . . . . 6 (𝜑𝐴 ∈ ran 𝐿)
9 foot.y . . . . . 6 (𝜑 → ¬ 𝐶𝐴)
10 footexlem.e . . . . . 6 (𝜑𝐸𝑃)
11 footexlem.f . . . . . 6 (𝜑𝐹𝑃)
12 footexlem.r . . . . . 6 (𝜑𝑅𝑃)
13 footexlem.y . . . . . 6 (𝜑𝑌𝑃)
14 footexlem.z . . . . . 6 (𝜑𝑍𝑃)
15 footexlem.d . . . . . 6 (𝜑𝐷𝑃)
16 footexlem.1 . . . . . 6 (𝜑𝐴 = (𝐸𝐿𝐹))
17 footexlem.2 . . . . . 6 (𝜑𝐸𝐹)
18 footexlem.3 . . . . . 6 (𝜑𝐸 ∈ (𝐹𝐼𝑌))
19 footexlem.4 . . . . . 6 (𝜑 → (𝐸 𝑌) = (𝐸 𝐶))
20 footexlem.5 . . . . . 6 (𝜑𝐶 = (((pInvG‘𝐺)‘𝑅)‘𝑌))
21 footexlem.6 . . . . . 6 (𝜑𝑌 ∈ (𝐸𝐼𝑍))
22 footexlem.7 . . . . . 6 (𝜑 → (𝑌 𝑍) = (𝑌 𝑅))
23 footexlem.q . . . . . 6 (𝜑𝑄𝑃)
24 footexlem.8 . . . . . 6 (𝜑𝑌 ∈ (𝑅𝐼𝑄))
25 footexlem.9 . . . . . 6 (𝜑 → (𝑌 𝑄) = (𝑌 𝐸))
26 footexlem.10 . . . . . 6 (𝜑𝑌 ∈ ((((pInvG‘𝐺)‘𝑍)‘𝑄)𝐼𝐷))
27 footexlem.11 . . . . . 6 (𝜑 → (𝑌 𝐷) = (𝑌 𝐶))
28 footexlem.12 . . . . . 6 (𝜑𝐷 = (((pInvG‘𝐺)‘𝑋)‘𝐶))
291, 2, 3, 4, 5, 8, 6, 9, 10, 11, 12, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28footexlem1 28741 . . . . 5 (𝜑𝑋𝐴)
30 nelne2 3037 . . . . 5 ((𝑋𝐴 ∧ ¬ 𝐶𝐴) → 𝑋𝐶)
3129, 9, 30syl2anc 584 . . . 4 (𝜑𝑋𝐶)
3231necomd 2993 . . 3 (𝜑𝐶𝑋)
331, 3, 4, 5, 6, 7, 32tgelrnln 28652 . 2 (𝜑 → (𝐶𝐿𝑋) ∈ ran 𝐿)
341, 3, 4, 5, 6, 7, 32tglinerflx2 28656 . . 3 (𝜑𝑋 ∈ (𝐶𝐿𝑋))
3534, 29elind 4209 . 2 (𝜑𝑋 ∈ ((𝐶𝐿𝑋) ∩ 𝐴))
361, 3, 4, 5, 6, 7, 32tglinerflx1 28655 . 2 (𝜑𝐶 ∈ (𝐶𝐿𝑋))
3717necomd 2993 . . . . 5 (𝜑𝐹𝐸)
381, 3, 4, 5, 11, 10, 13, 37, 18btwnlng3 28643 . . . 4 (𝜑𝑌 ∈ (𝐹𝐿𝐸))
391, 3, 4, 5, 10, 11, 13, 17, 38lncom 28644 . . 3 (𝜑𝑌 ∈ (𝐸𝐿𝐹))
4039, 16eleqtrrd 2841 . 2 (𝜑𝑌𝐴)
41 eqid 2734 . . . . 5 (pInvG‘𝐺) = (pInvG‘𝐺)
425adantr 480 . . . . 5 ((𝜑𝑌 = 𝑋) → 𝐺 ∈ TarskiG)
4310adantr 480 . . . . 5 ((𝜑𝑌 = 𝑋) → 𝐸𝑃)
4413adantr 480 . . . . 5 ((𝜑𝑌 = 𝑋) → 𝑌𝑃)
4512adantr 480 . . . . 5 ((𝜑𝑌 = 𝑋) → 𝑅𝑃)
466adantr 480 . . . . . . 7 ((𝜑𝑌 = 𝑋) → 𝐶𝑃)
47 eqidd 2735 . . . . . . . . 9 ((𝜑𝑌 = 𝑋) → 𝐶 = 𝐶)
48 simpr 484 . . . . . . . . 9 ((𝜑𝑌 = 𝑋) → 𝑌 = 𝑋)
49 eqidd 2735 . . . . . . . . 9 ((𝜑𝑌 = 𝑋) → 𝐸 = 𝐸)
5047, 48, 49s3eqd 14899 . . . . . . . 8 ((𝜑𝑌 = 𝑋) → ⟨“𝐶𝑌𝐸”⟩ = ⟨“𝐶𝑋𝐸”⟩)
517adantr 480 . . . . . . . . 9 ((𝜑𝑌 = 𝑋) → 𝑋𝑃)
5214adantr 480 . . . . . . . . . 10 ((𝜑𝑌 = 𝑋) → 𝑍𝑃)
53 eqid 2734 . . . . . . . . . . . . . . . 16 ((pInvG‘𝐺)‘𝑍) = ((pInvG‘𝐺)‘𝑍)
541, 2, 3, 4, 41, 5, 14, 53, 23mircl 28683 . . . . . . . . . . . . . . 15 (𝜑 → (((pInvG‘𝐺)‘𝑍)‘𝑄) ∈ 𝑃)
551, 2, 3, 5, 10, 13, 10, 6, 19tgcgrcomlr 28502 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑌 𝐸) = (𝐶 𝐸))
5625, 55eqtr2d 2775 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐶 𝐸) = (𝑌 𝑄))
571, 3, 4, 5, 10, 11, 17tglinerflx1 28655 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐸 ∈ (𝐸𝐿𝐹))
5857, 16eleqtrrd 2841 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐸𝐴)
59 nelne2 3037 . . . . . . . . . . . . . . . . . . 19 ((𝐸𝐴 ∧ ¬ 𝐶𝐴) → 𝐸𝐶)
6058, 9, 59syl2anc 584 . . . . . . . . . . . . . . . . . 18 (𝜑𝐸𝐶)
6160necomd 2993 . . . . . . . . . . . . . . . . 17 (𝜑𝐶𝐸)
621, 2, 3, 5, 6, 10, 13, 23, 56, 61tgcgrneq 28505 . . . . . . . . . . . . . . . 16 (𝜑𝑌𝑄)
6362necomd 2993 . . . . . . . . . . . . . . 15 (𝜑𝑄𝑌)
64 nelne2 3037 . . . . . . . . . . . . . . . . . . . . 21 ((𝑌𝐴 ∧ ¬ 𝐶𝐴) → 𝑌𝐶)
6540, 9, 64syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑌𝐶)
6665necomd 2993 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐶𝑌)
6720, 66eqnetrrd 3006 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌)
68 eqid 2734 . . . . . . . . . . . . . . . . . . . 20 ((pInvG‘𝐺)‘𝑅) = ((pInvG‘𝐺)‘𝑅)
691, 2, 3, 4, 41, 5, 12, 68, 13mirinv 28688 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) = 𝑌𝑅 = 𝑌))
7069necon3bid 2982 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌𝑅𝑌))
7167, 70mpbid 232 . . . . . . . . . . . . . . . . 17 (𝜑𝑅𝑌)
721, 2, 3, 4, 41, 5, 12, 68, 13mirbtwn 28680 . . . . . . . . . . . . . . . . . 18 (𝜑𝑅 ∈ ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
7320oveq1d 7445 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐶𝐼𝑌) = ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
7472, 73eleqtrrd 2841 . . . . . . . . . . . . . . . . 17 (𝜑𝑅 ∈ (𝐶𝐼𝑌))
751, 2, 3, 5, 6, 12, 13, 23, 71, 74, 24tgbtwnouttr2 28517 . . . . . . . . . . . . . . . 16 (𝜑𝑌 ∈ (𝐶𝐼𝑄))
761, 2, 3, 5, 6, 13, 23, 75tgbtwncom 28510 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ (𝑄𝐼𝐶))
77 eqid 2734 . . . . . . . . . . . . . . . . . . 19 (cgrG‘𝐺) = (cgrG‘𝐺)
7820oveq2d 7446 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐸 𝐶) = (𝐸 (((pInvG‘𝐺)‘𝑅)‘𝑌)))
7919, 78eqtrd 2774 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐸 𝑌) = (𝐸 (((pInvG‘𝐺)‘𝑅)‘𝑌)))
801, 2, 3, 4, 41, 5, 10, 12, 13israg 28719 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺) ↔ (𝐸 𝑌) = (𝐸 (((pInvG‘𝐺)‘𝑅)‘𝑌))))
8179, 80mpbird 257 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺))
821, 2, 3, 5, 12, 13, 23, 24tgbtwncom 28510 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑌 ∈ (𝑄𝐼𝑅))
831, 2, 3, 5, 13, 23, 13, 10, 25tgcgrcomlr 28502 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑄 𝑌) = (𝐸 𝑌))
8422eqcomd 2740 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑌 𝑅) = (𝑌 𝑍))
851, 2, 3, 5, 23, 10axtgcgrrflx 28484 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑄 𝐸) = (𝐸 𝑄))
8625eqcomd 2740 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑌 𝐸) = (𝑌 𝑄))
871, 2, 3, 5, 23, 13, 12, 10, 13, 14, 10, 23, 63, 82, 21, 83, 84, 85, 86axtg5seg 28487 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑅 𝐸) = (𝑍 𝑄))
881, 2, 3, 5, 12, 10, 14, 23, 87tgcgrcomlr 28502 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐸 𝑅) = (𝑄 𝑍))
891, 2, 3, 5, 13, 12, 13, 14, 84tgcgrcomlr 28502 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑅 𝑌) = (𝑍 𝑌))
901, 2, 77, 5, 10, 12, 13, 23, 14, 13, 88, 89, 86trgcgr 28538 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ⟨“𝐸𝑅𝑌”⟩(cgrG‘𝐺)⟨“𝑄𝑍𝑌”⟩)
911, 2, 3, 4, 41, 5, 10, 12, 13, 77, 23, 14, 13, 81, 90ragcgr 28729 . . . . . . . . . . . . . . . . . 18 (𝜑 → ⟨“𝑄𝑍𝑌”⟩ ∈ (∟G‘𝐺))
921, 2, 3, 4, 41, 5, 23, 14, 13, 91ragcom 28720 . . . . . . . . . . . . . . . . 17 (𝜑 → ⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺))
931, 2, 3, 4, 41, 5, 13, 14, 23israg 28719 . . . . . . . . . . . . . . . . 17 (𝜑 → (⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝑌 𝑄) = (𝑌 (((pInvG‘𝐺)‘𝑍)‘𝑄))))
9492, 93mpbid 232 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑌 𝑄) = (𝑌 (((pInvG‘𝐺)‘𝑍)‘𝑄)))
951, 2, 3, 5, 13, 23, 13, 54, 94tgcgrcomlr 28502 . . . . . . . . . . . . . . 15 (𝜑 → (𝑄 𝑌) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) 𝑌))
9627eqcomd 2740 . . . . . . . . . . . . . . 15 (𝜑 → (𝑌 𝐶) = (𝑌 𝐷))
971, 2, 3, 4, 41, 5, 14, 53, 23mircgr 28679 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑍 (((pInvG‘𝐺)‘𝑍)‘𝑄)) = (𝑍 𝑄))
9897eqcomd 2740 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑍 𝑄) = (𝑍 (((pInvG‘𝐺)‘𝑍)‘𝑄)))
991, 2, 3, 5, 14, 23, 14, 54, 98tgcgrcomlr 28502 . . . . . . . . . . . . . . 15 (𝜑 → (𝑄 𝑍) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) 𝑍))
100 eqidd 2735 . . . . . . . . . . . . . . 15 (𝜑 → (𝑌 𝑍) = (𝑌 𝑍))
1011, 2, 3, 5, 23, 13, 6, 54, 13, 15, 14, 14, 63, 76, 26, 95, 96, 99, 100axtg5seg 28487 . . . . . . . . . . . . . 14 (𝜑 → (𝐶 𝑍) = (𝐷 𝑍))
1021, 2, 3, 5, 6, 14, 15, 14, 101tgcgrcomlr 28502 . . . . . . . . . . . . 13 (𝜑 → (𝑍 𝐶) = (𝑍 𝐷))
10328oveq2d 7446 . . . . . . . . . . . . 13 (𝜑 → (𝑍 𝐷) = (𝑍 (((pInvG‘𝐺)‘𝑋)‘𝐶)))
104102, 103eqtrd 2774 . . . . . . . . . . . 12 (𝜑 → (𝑍 𝐶) = (𝑍 (((pInvG‘𝐺)‘𝑋)‘𝐶)))
1051, 2, 3, 4, 41, 5, 14, 7, 6israg 28719 . . . . . . . . . . . 12 (𝜑 → (⟨“𝑍𝑋𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝑍 𝐶) = (𝑍 (((pInvG‘𝐺)‘𝑋)‘𝐶))))
106104, 105mpbird 257 . . . . . . . . . . 11 (𝜑 → ⟨“𝑍𝑋𝐶”⟩ ∈ (∟G‘𝐺))
107106adantr 480 . . . . . . . . . 10 ((𝜑𝑌 = 𝑋) → ⟨“𝑍𝑋𝐶”⟩ ∈ (∟G‘𝐺))
10871necomd 2993 . . . . . . . . . . . . . 14 (𝜑𝑌𝑅)
1091, 2, 3, 5, 13, 12, 13, 14, 84, 108tgcgrneq 28505 . . . . . . . . . . . . 13 (𝜑𝑌𝑍)
110109necomd 2993 . . . . . . . . . . . 12 (𝜑𝑍𝑌)
111110adantr 480 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑋) → 𝑍𝑌)
112111, 48neeqtrd 3007 . . . . . . . . . 10 ((𝜑𝑌 = 𝑋) → 𝑍𝑋)
11319eqcomd 2740 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐸 𝐶) = (𝐸 𝑌))
114113adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑋) → (𝐸 𝐶) = (𝐸 𝑌))
11560adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑋) → 𝐸𝐶)
1161, 2, 3, 42, 43, 46, 43, 44, 114, 115tgcgrneq 28505 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑋) → 𝐸𝑌)
117116necomd 2993 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑋) → 𝑌𝐸)
1181, 2, 3, 5, 10, 6, 10, 13, 113, 60tgcgrneq 28505 . . . . . . . . . . . . . . 15 (𝜑𝐸𝑌)
1191, 3, 4, 5, 10, 13, 14, 118, 21btwnlng3 28643 . . . . . . . . . . . . . 14 (𝜑𝑍 ∈ (𝐸𝐿𝑌))
120119adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑋) → 𝑍 ∈ (𝐸𝐿𝑌))
1211, 3, 4, 42, 44, 43, 52, 117, 120lncom 28644 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑋) → 𝑍 ∈ (𝑌𝐿𝐸))
12248oveq1d 7445 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑋) → (𝑌𝐿𝐸) = (𝑋𝐿𝐸))
123121, 122eleqtrd 2840 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑋) → 𝑍 ∈ (𝑋𝐿𝐸))
124123orcd 873 . . . . . . . . . 10 ((𝜑𝑌 = 𝑋) → (𝑍 ∈ (𝑋𝐿𝐸) ∨ 𝑋 = 𝐸))
1251, 2, 3, 4, 41, 42, 52, 51, 46, 43, 107, 112, 124ragcol 28721 . . . . . . . . 9 ((𝜑𝑌 = 𝑋) → ⟨“𝐸𝑋𝐶”⟩ ∈ (∟G‘𝐺))
1261, 2, 3, 4, 41, 42, 43, 51, 46, 125ragcom 28720 . . . . . . . 8 ((𝜑𝑌 = 𝑋) → ⟨“𝐶𝑋𝐸”⟩ ∈ (∟G‘𝐺))
12750, 126eqeltrd 2838 . . . . . . 7 ((𝜑𝑌 = 𝑋) → ⟨“𝐶𝑌𝐸”⟩ ∈ (∟G‘𝐺))
12866adantr 480 . . . . . . 7 ((𝜑𝑌 = 𝑋) → 𝐶𝑌)
1291, 2, 3, 5, 6, 12, 13, 74tgbtwncom 28510 . . . . . . . . 9 (𝜑𝑅 ∈ (𝑌𝐼𝐶))
1301, 4, 3, 5, 13, 12, 6, 129btwncolg3 28579 . . . . . . . 8 (𝜑 → (𝐶 ∈ (𝑌𝐿𝑅) ∨ 𝑌 = 𝑅))
131130adantr 480 . . . . . . 7 ((𝜑𝑌 = 𝑋) → (𝐶 ∈ (𝑌𝐿𝑅) ∨ 𝑌 = 𝑅))
1321, 2, 3, 4, 41, 42, 46, 44, 43, 45, 127, 128, 131ragcol 28721 . . . . . 6 ((𝜑𝑌 = 𝑋) → ⟨“𝑅𝑌𝐸”⟩ ∈ (∟G‘𝐺))
1331, 2, 3, 4, 41, 42, 45, 44, 43, 132ragcom 28720 . . . . 5 ((𝜑𝑌 = 𝑋) → ⟨“𝐸𝑌𝑅”⟩ ∈ (∟G‘𝐺))
13481adantr 480 . . . . 5 ((𝜑𝑌 = 𝑋) → ⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺))
1351, 2, 3, 4, 41, 42, 43, 44, 45, 133, 134ragflat 28726 . . . 4 ((𝜑𝑌 = 𝑋) → 𝑌 = 𝑅)
136108adantr 480 . . . . 5 ((𝜑𝑌 = 𝑋) → 𝑌𝑅)
137136neneqd 2942 . . . 4 ((𝜑𝑌 = 𝑋) → ¬ 𝑌 = 𝑅)
138135, 137pm2.65da 817 . . 3 (𝜑 → ¬ 𝑌 = 𝑋)
139138neqned 2944 . 2 (𝜑𝑌𝑋)
14028oveq2d 7446 . . . . 5 (𝜑 → (𝑌 𝐷) = (𝑌 (((pInvG‘𝐺)‘𝑋)‘𝐶)))
14196, 140eqtrd 2774 . . . 4 (𝜑 → (𝑌 𝐶) = (𝑌 (((pInvG‘𝐺)‘𝑋)‘𝐶)))
1421, 2, 3, 4, 41, 5, 13, 7, 6israg 28719 . . . 4 (𝜑 → (⟨“𝑌𝑋𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝑌 𝐶) = (𝑌 (((pInvG‘𝐺)‘𝑋)‘𝐶))))
143141, 142mpbird 257 . . 3 (𝜑 → ⟨“𝑌𝑋𝐶”⟩ ∈ (∟G‘𝐺))
1441, 2, 3, 4, 41, 5, 13, 7, 6, 143ragcom 28720 . 2 (𝜑 → ⟨“𝐶𝑋𝑌”⟩ ∈ (∟G‘𝐺))
1451, 2, 3, 4, 5, 33, 8, 35, 36, 40, 32, 139, 144ragperp 28739 1 (𝜑 → (𝐶𝐿𝑋)(⟂G‘𝐺)𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847   = wceq 1536  wcel 2105  wne 2937   class class class wbr 5147  ran crn 5689  cfv 6562  (class class class)co 7430  ⟨“cs3 14877  Basecbs 17244  distcds 17306  TarskiGcstrkg 28449  Itvcitv 28455  LineGclng 28456  cgrGccgrg 28532  pInvGcmir 28674  ∟Gcrag 28715  ⟂Gcperpg 28717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-concat 14605  df-s1 14630  df-s2 14883  df-s3 14884  df-trkgc 28470  df-trkgb 28471  df-trkgcb 28472  df-trkg 28475  df-cgrg 28533  df-leg 28605  df-mir 28675  df-rag 28716  df-perpg 28718
This theorem is referenced by:  footex  28743
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