MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  footexlem2 Structured version   Visualization version   GIF version

Theorem footexlem2 28728
Description: Lemma for footex 28729. (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 28727 . . . . 5 (𝜑𝑋𝐴)
30 nelne2 3040 . . . . 5 ((𝑋𝐴 ∧ ¬ 𝐶𝐴) → 𝑋𝐶)
3129, 9, 30syl2anc 584 . . . 4 (𝜑𝑋𝐶)
3231necomd 2996 . . 3 (𝜑𝐶𝑋)
331, 3, 4, 5, 6, 7, 32tgelrnln 28638 . 2 (𝜑 → (𝐶𝐿𝑋) ∈ ran 𝐿)
341, 3, 4, 5, 6, 7, 32tglinerflx2 28642 . . 3 (𝜑𝑋 ∈ (𝐶𝐿𝑋))
3534, 29elind 4200 . 2 (𝜑𝑋 ∈ ((𝐶𝐿𝑋) ∩ 𝐴))
361, 3, 4, 5, 6, 7, 32tglinerflx1 28641 . 2 (𝜑𝐶 ∈ (𝐶𝐿𝑋))
3717necomd 2996 . . . . 5 (𝜑𝐹𝐸)
381, 3, 4, 5, 11, 10, 13, 37, 18btwnlng3 28629 . . . 4 (𝜑𝑌 ∈ (𝐹𝐿𝐸))
391, 3, 4, 5, 10, 11, 13, 17, 38lncom 28630 . . 3 (𝜑𝑌 ∈ (𝐸𝐿𝐹))
4039, 16eleqtrrd 2844 . 2 (𝜑𝑌𝐴)
41 eqid 2737 . . . . 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 2738 . . . . . . . . 9 ((𝜑𝑌 = 𝑋) → 𝐶 = 𝐶)
48 simpr 484 . . . . . . . . 9 ((𝜑𝑌 = 𝑋) → 𝑌 = 𝑋)
49 eqidd 2738 . . . . . . . . 9 ((𝜑𝑌 = 𝑋) → 𝐸 = 𝐸)
5047, 48, 49s3eqd 14903 . . . . . . . 8 ((𝜑𝑌 = 𝑋) → ⟨“𝐶𝑌𝐸”⟩ = ⟨“𝐶𝑋𝐸”⟩)
517adantr 480 . . . . . . . . 9 ((𝜑𝑌 = 𝑋) → 𝑋𝑃)
5214adantr 480 . . . . . . . . . 10 ((𝜑𝑌 = 𝑋) → 𝑍𝑃)
53 eqid 2737 . . . . . . . . . . . . . . . 16 ((pInvG‘𝐺)‘𝑍) = ((pInvG‘𝐺)‘𝑍)
541, 2, 3, 4, 41, 5, 14, 53, 23mircl 28669 . . . . . . . . . . . . . . 15 (𝜑 → (((pInvG‘𝐺)‘𝑍)‘𝑄) ∈ 𝑃)
551, 2, 3, 5, 10, 13, 10, 6, 19tgcgrcomlr 28488 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑌 𝐸) = (𝐶 𝐸))
5625, 55eqtr2d 2778 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐶 𝐸) = (𝑌 𝑄))
571, 3, 4, 5, 10, 11, 17tglinerflx1 28641 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐸 ∈ (𝐸𝐿𝐹))
5857, 16eleqtrrd 2844 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐸𝐴)
59 nelne2 3040 . . . . . . . . . . . . . . . . . . 19 ((𝐸𝐴 ∧ ¬ 𝐶𝐴) → 𝐸𝐶)
6058, 9, 59syl2anc 584 . . . . . . . . . . . . . . . . . 18 (𝜑𝐸𝐶)
6160necomd 2996 . . . . . . . . . . . . . . . . 17 (𝜑𝐶𝐸)
621, 2, 3, 5, 6, 10, 13, 23, 56, 61tgcgrneq 28491 . . . . . . . . . . . . . . . 16 (𝜑𝑌𝑄)
6362necomd 2996 . . . . . . . . . . . . . . 15 (𝜑𝑄𝑌)
64 nelne2 3040 . . . . . . . . . . . . . . . . . . . . 21 ((𝑌𝐴 ∧ ¬ 𝐶𝐴) → 𝑌𝐶)
6540, 9, 64syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑌𝐶)
6665necomd 2996 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐶𝑌)
6720, 66eqnetrrd 3009 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌)
68 eqid 2737 . . . . . . . . . . . . . . . . . . . 20 ((pInvG‘𝐺)‘𝑅) = ((pInvG‘𝐺)‘𝑅)
691, 2, 3, 4, 41, 5, 12, 68, 13mirinv 28674 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) = 𝑌𝑅 = 𝑌))
7069necon3bid 2985 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌𝑅𝑌))
7167, 70mpbid 232 . . . . . . . . . . . . . . . . 17 (𝜑𝑅𝑌)
721, 2, 3, 4, 41, 5, 12, 68, 13mirbtwn 28666 . . . . . . . . . . . . . . . . . 18 (𝜑𝑅 ∈ ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
7320oveq1d 7446 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐶𝐼𝑌) = ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
7472, 73eleqtrrd 2844 . . . . . . . . . . . . . . . . 17 (𝜑𝑅 ∈ (𝐶𝐼𝑌))
751, 2, 3, 5, 6, 12, 13, 23, 71, 74, 24tgbtwnouttr2 28503 . . . . . . . . . . . . . . . 16 (𝜑𝑌 ∈ (𝐶𝐼𝑄))
761, 2, 3, 5, 6, 13, 23, 75tgbtwncom 28496 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ (𝑄𝐼𝐶))
77 eqid 2737 . . . . . . . . . . . . . . . . . . 19 (cgrG‘𝐺) = (cgrG‘𝐺)
7820oveq2d 7447 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐸 𝐶) = (𝐸 (((pInvG‘𝐺)‘𝑅)‘𝑌)))
7919, 78eqtrd 2777 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐸 𝑌) = (𝐸 (((pInvG‘𝐺)‘𝑅)‘𝑌)))
801, 2, 3, 4, 41, 5, 10, 12, 13israg 28705 . . . . . . . . . . . . . . . . . . . 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 28496 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑌 ∈ (𝑄𝐼𝑅))
831, 2, 3, 5, 13, 23, 13, 10, 25tgcgrcomlr 28488 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑄 𝑌) = (𝐸 𝑌))
8422eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑌 𝑅) = (𝑌 𝑍))
851, 2, 3, 5, 23, 10axtgcgrrflx 28470 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑄 𝐸) = (𝐸 𝑄))
8625eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . 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 28473 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑅 𝐸) = (𝑍 𝑄))
881, 2, 3, 5, 12, 10, 14, 23, 87tgcgrcomlr 28488 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐸 𝑅) = (𝑄 𝑍))
891, 2, 3, 5, 13, 12, 13, 14, 84tgcgrcomlr 28488 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑅 𝑌) = (𝑍 𝑌))
901, 2, 77, 5, 10, 12, 13, 23, 14, 13, 88, 89, 86trgcgr 28524 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ⟨“𝐸𝑅𝑌”⟩(cgrG‘𝐺)⟨“𝑄𝑍𝑌”⟩)
911, 2, 3, 4, 41, 5, 10, 12, 13, 77, 23, 14, 13, 81, 90ragcgr 28715 . . . . . . . . . . . . . . . . . 18 (𝜑 → ⟨“𝑄𝑍𝑌”⟩ ∈ (∟G‘𝐺))
921, 2, 3, 4, 41, 5, 23, 14, 13, 91ragcom 28706 . . . . . . . . . . . . . . . . 17 (𝜑 → ⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺))
931, 2, 3, 4, 41, 5, 13, 14, 23israg 28705 . . . . . . . . . . . . . . . . 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 28488 . . . . . . . . . . . . . . 15 (𝜑 → (𝑄 𝑌) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) 𝑌))
9627eqcomd 2743 . . . . . . . . . . . . . . 15 (𝜑 → (𝑌 𝐶) = (𝑌 𝐷))
971, 2, 3, 4, 41, 5, 14, 53, 23mircgr 28665 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑍 (((pInvG‘𝐺)‘𝑍)‘𝑄)) = (𝑍 𝑄))
9897eqcomd 2743 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑍 𝑄) = (𝑍 (((pInvG‘𝐺)‘𝑍)‘𝑄)))
991, 2, 3, 5, 14, 23, 14, 54, 98tgcgrcomlr 28488 . . . . . . . . . . . . . . 15 (𝜑 → (𝑄 𝑍) = ((((pInvG‘𝐺)‘𝑍)‘𝑄) 𝑍))
100 eqidd 2738 . . . . . . . . . . . . . . 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 28473 . . . . . . . . . . . . . 14 (𝜑 → (𝐶 𝑍) = (𝐷 𝑍))
1021, 2, 3, 5, 6, 14, 15, 14, 101tgcgrcomlr 28488 . . . . . . . . . . . . 13 (𝜑 → (𝑍 𝐶) = (𝑍 𝐷))
10328oveq2d 7447 . . . . . . . . . . . . 13 (𝜑 → (𝑍 𝐷) = (𝑍 (((pInvG‘𝐺)‘𝑋)‘𝐶)))
104102, 103eqtrd 2777 . . . . . . . . . . . 12 (𝜑 → (𝑍 𝐶) = (𝑍 (((pInvG‘𝐺)‘𝑋)‘𝐶)))
1051, 2, 3, 4, 41, 5, 14, 7, 6israg 28705 . . . . . . . . . . . 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 2996 . . . . . . . . . . . . . 14 (𝜑𝑌𝑅)
1091, 2, 3, 5, 13, 12, 13, 14, 84, 108tgcgrneq 28491 . . . . . . . . . . . . 13 (𝜑𝑌𝑍)
110109necomd 2996 . . . . . . . . . . . 12 (𝜑𝑍𝑌)
111110adantr 480 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑋) → 𝑍𝑌)
112111, 48neeqtrd 3010 . . . . . . . . . 10 ((𝜑𝑌 = 𝑋) → 𝑍𝑋)
11319eqcomd 2743 . . . . . . . . . . . . . . . 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 28491 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑋) → 𝐸𝑌)
117116necomd 2996 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑋) → 𝑌𝐸)
1181, 2, 3, 5, 10, 6, 10, 13, 113, 60tgcgrneq 28491 . . . . . . . . . . . . . . 15 (𝜑𝐸𝑌)
1191, 3, 4, 5, 10, 13, 14, 118, 21btwnlng3 28629 . . . . . . . . . . . . . 14 (𝜑𝑍 ∈ (𝐸𝐿𝑌))
120119adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑋) → 𝑍 ∈ (𝐸𝐿𝑌))
1211, 3, 4, 42, 44, 43, 52, 117, 120lncom 28630 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑋) → 𝑍 ∈ (𝑌𝐿𝐸))
12248oveq1d 7446 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑋) → (𝑌𝐿𝐸) = (𝑋𝐿𝐸))
123121, 122eleqtrd 2843 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑋) → 𝑍 ∈ (𝑋𝐿𝐸))
124123orcd 874 . . . . . . . . . 10 ((𝜑𝑌 = 𝑋) → (𝑍 ∈ (𝑋𝐿𝐸) ∨ 𝑋 = 𝐸))
1251, 2, 3, 4, 41, 42, 52, 51, 46, 43, 107, 112, 124ragcol 28707 . . . . . . . . 9 ((𝜑𝑌 = 𝑋) → ⟨“𝐸𝑋𝐶”⟩ ∈ (∟G‘𝐺))
1261, 2, 3, 4, 41, 42, 43, 51, 46, 125ragcom 28706 . . . . . . . 8 ((𝜑𝑌 = 𝑋) → ⟨“𝐶𝑋𝐸”⟩ ∈ (∟G‘𝐺))
12750, 126eqeltrd 2841 . . . . . . 7 ((𝜑𝑌 = 𝑋) → ⟨“𝐶𝑌𝐸”⟩ ∈ (∟G‘𝐺))
12866adantr 480 . . . . . . 7 ((𝜑𝑌 = 𝑋) → 𝐶𝑌)
1291, 2, 3, 5, 6, 12, 13, 74tgbtwncom 28496 . . . . . . . . 9 (𝜑𝑅 ∈ (𝑌𝐼𝐶))
1301, 4, 3, 5, 13, 12, 6, 129btwncolg3 28565 . . . . . . . 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 28707 . . . . . 6 ((𝜑𝑌 = 𝑋) → ⟨“𝑅𝑌𝐸”⟩ ∈ (∟G‘𝐺))
1331, 2, 3, 4, 41, 42, 45, 44, 43, 132ragcom 28706 . . . . 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 28712 . . . 4 ((𝜑𝑌 = 𝑋) → 𝑌 = 𝑅)
136108adantr 480 . . . . 5 ((𝜑𝑌 = 𝑋) → 𝑌𝑅)
137136neneqd 2945 . . . 4 ((𝜑𝑌 = 𝑋) → ¬ 𝑌 = 𝑅)
138135, 137pm2.65da 817 . . 3 (𝜑 → ¬ 𝑌 = 𝑋)
139138neqned 2947 . 2 (𝜑𝑌𝑋)
14028oveq2d 7447 . . . . 5 (𝜑 → (𝑌 𝐷) = (𝑌 (((pInvG‘𝐺)‘𝑋)‘𝐶)))
14196, 140eqtrd 2777 . . . 4 (𝜑 → (𝑌 𝐶) = (𝑌 (((pInvG‘𝐺)‘𝑋)‘𝐶)))
1421, 2, 3, 4, 41, 5, 13, 7, 6israg 28705 . . . 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 28706 . 2 (𝜑 → ⟨“𝐶𝑋𝑌”⟩ ∈ (∟G‘𝐺))
1451, 2, 3, 4, 5, 33, 8, 35, 36, 40, 32, 139, 144ragperp 28725 1 (𝜑 → (𝐶𝐿𝑋)(⟂G‘𝐺)𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  ran crn 5686  cfv 6561  (class class class)co 7431  ⟨“cs3 14881  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442  cgrGccgrg 28518  pInvGcmir 28660  ∟Gcrag 28701  ⟂Gcperpg 28703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-s2 14887  df-s3 14888  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461  df-cgrg 28519  df-leg 28591  df-mir 28661  df-rag 28702  df-perpg 28704
This theorem is referenced by:  footex  28729
  Copyright terms: Public domain W3C validator