MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  footexlem1 Structured version   Visualization version   GIF version

Theorem footexlem1 28727
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
footexlem1 (𝜑𝑋𝐴)

Proof of Theorem footexlem1
StepHypRef Expression
1 isperp.p . . 3 𝑃 = (Base‘𝐺)
2 isperp.i . . 3 𝐼 = (Itv‘𝐺)
3 isperp.l . . 3 𝐿 = (LineG‘𝐺)
4 isperp.g . . 3 (𝜑𝐺 ∈ TarskiG)
5 footexlem.y . . 3 (𝜑𝑌𝑃)
6 footexlem.z . . 3 (𝜑𝑍𝑃)
7 footexlem.x . . 3 (𝜑𝑋𝑃)
8 isperp.d . . . 4 = (dist‘𝐺)
9 footexlem.r . . . 4 (𝜑𝑅𝑃)
10 footexlem.7 . . . . 5 (𝜑 → (𝑌 𝑍) = (𝑌 𝑅))
1110eqcomd 2743 . . . 4 (𝜑 → (𝑌 𝑅) = (𝑌 𝑍))
12 footexlem.5 . . . . . . 7 (𝜑𝐶 = (((pInvG‘𝐺)‘𝑅)‘𝑌))
13 footexlem.e . . . . . . . . . . 11 (𝜑𝐸𝑃)
14 footexlem.f . . . . . . . . . . 11 (𝜑𝐹𝑃)
15 footexlem.2 . . . . . . . . . . 11 (𝜑𝐸𝐹)
1615necomd 2996 . . . . . . . . . . . 12 (𝜑𝐹𝐸)
17 footexlem.3 . . . . . . . . . . . 12 (𝜑𝐸 ∈ (𝐹𝐼𝑌))
181, 2, 3, 4, 14, 13, 5, 16, 17btwnlng3 28629 . . . . . . . . . . 11 (𝜑𝑌 ∈ (𝐹𝐿𝐸))
191, 2, 3, 4, 13, 14, 5, 15, 18lncom 28630 . . . . . . . . . 10 (𝜑𝑌 ∈ (𝐸𝐿𝐹))
20 footexlem.1 . . . . . . . . . 10 (𝜑𝐴 = (𝐸𝐿𝐹))
2119, 20eleqtrrd 2844 . . . . . . . . 9 (𝜑𝑌𝐴)
22 foot.y . . . . . . . . 9 (𝜑 → ¬ 𝐶𝐴)
23 nelne2 3040 . . . . . . . . 9 ((𝑌𝐴 ∧ ¬ 𝐶𝐴) → 𝑌𝐶)
2421, 22, 23syl2anc 584 . . . . . . . 8 (𝜑𝑌𝐶)
2524necomd 2996 . . . . . . 7 (𝜑𝐶𝑌)
2612, 25eqnetrrd 3009 . . . . . 6 (𝜑 → (((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌)
27 eqid 2737 . . . . . . . 8 (pInvG‘𝐺) = (pInvG‘𝐺)
28 eqid 2737 . . . . . . . 8 ((pInvG‘𝐺)‘𝑅) = ((pInvG‘𝐺)‘𝑅)
291, 8, 2, 3, 27, 4, 9, 28, 5mirinv 28674 . . . . . . 7 (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) = 𝑌𝑅 = 𝑌))
3029necon3bid 2985 . . . . . 6 (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌𝑅𝑌))
3126, 30mpbid 232 . . . . 5 (𝜑𝑅𝑌)
3231necomd 2996 . . . 4 (𝜑𝑌𝑅)
331, 8, 2, 4, 5, 9, 5, 6, 11, 32tgcgrneq 28491 . . 3 (𝜑𝑌𝑍)
3433necomd 2996 . . . 4 (𝜑𝑍𝑌)
35 eqid 2737 . . . . 5 ((pInvG‘𝐺)‘𝑍) = ((pInvG‘𝐺)‘𝑍)
36 eqid 2737 . . . . 5 ((pInvG‘𝐺)‘𝑋) = ((pInvG‘𝐺)‘𝑋)
37 footexlem.q . . . . 5 (𝜑𝑄𝑃)
381, 8, 2, 3, 27, 4, 6, 35, 37mircl 28669 . . . . 5 (𝜑 → (((pInvG‘𝐺)‘𝑍)‘𝑄) ∈ 𝑃)
39 foot.x . . . . 5 (𝜑𝐶𝑃)
40 footexlem.d . . . . 5 (𝜑𝐷𝑃)
411, 8, 2, 3, 27, 4, 9, 28, 5mirbtwn 28666 . . . . . . . 8 (𝜑𝑅 ∈ ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
4212oveq1d 7446 . . . . . . . 8 (𝜑 → (𝐶𝐼𝑌) = ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
4341, 42eleqtrrd 2844 . . . . . . 7 (𝜑𝑅 ∈ (𝐶𝐼𝑌))
44 footexlem.8 . . . . . . 7 (𝜑𝑌 ∈ (𝑅𝐼𝑄))
451, 8, 2, 4, 39, 9, 5, 37, 31, 43, 44tgbtwnouttr2 28503 . . . . . 6 (𝜑𝑌 ∈ (𝐶𝐼𝑄))
461, 8, 2, 4, 39, 5, 37, 45tgbtwncom 28496 . . . . 5 (𝜑𝑌 ∈ (𝑄𝐼𝐶))
47 footexlem.10 . . . . 5 (𝜑𝑌 ∈ ((((pInvG‘𝐺)‘𝑍)‘𝑄)𝐼𝐷))
48 eqid 2737 . . . . . . . 8 (cgrG‘𝐺) = (cgrG‘𝐺)
49 footexlem.4 . . . . . . . . . 10 (𝜑 → (𝐸 𝑌) = (𝐸 𝐶))
5012oveq2d 7447 . . . . . . . . . 10 (𝜑 → (𝐸 𝐶) = (𝐸 (((pInvG‘𝐺)‘𝑅)‘𝑌)))
5149, 50eqtrd 2777 . . . . . . . . 9 (𝜑 → (𝐸 𝑌) = (𝐸 (((pInvG‘𝐺)‘𝑅)‘𝑌)))
521, 8, 2, 3, 27, 4, 13, 9, 5israg 28705 . . . . . . . . 9 (𝜑 → (⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺) ↔ (𝐸 𝑌) = (𝐸 (((pInvG‘𝐺)‘𝑅)‘𝑌))))
5351, 52mpbird 257 . . . . . . . 8 (𝜑 → ⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺))
54 footexlem.9 . . . . . . . . . . . . . 14 (𝜑 → (𝑌 𝑄) = (𝑌 𝐸))
551, 8, 2, 4, 13, 5, 13, 39, 49tgcgrcomlr 28488 . . . . . . . . . . . . . 14 (𝜑 → (𝑌 𝐸) = (𝐶 𝐸))
5654, 55eqtr2d 2778 . . . . . . . . . . . . 13 (𝜑 → (𝐶 𝐸) = (𝑌 𝑄))
571, 2, 3, 4, 13, 14, 15tglinerflx1 28641 . . . . . . . . . . . . . . . 16 (𝜑𝐸 ∈ (𝐸𝐿𝐹))
5857, 20eleqtrrd 2844 . . . . . . . . . . . . . . 15 (𝜑𝐸𝐴)
59 nelne2 3040 . . . . . . . . . . . . . . 15 ((𝐸𝐴 ∧ ¬ 𝐶𝐴) → 𝐸𝐶)
6058, 22, 59syl2anc 584 . . . . . . . . . . . . . 14 (𝜑𝐸𝐶)
6160necomd 2996 . . . . . . . . . . . . 13 (𝜑𝐶𝐸)
621, 8, 2, 4, 39, 13, 5, 37, 56, 61tgcgrneq 28491 . . . . . . . . . . . 12 (𝜑𝑌𝑄)
6362necomd 2996 . . . . . . . . . . 11 (𝜑𝑄𝑌)
641, 8, 2, 4, 9, 5, 37, 44tgbtwncom 28496 . . . . . . . . . . 11 (𝜑𝑌 ∈ (𝑄𝐼𝑅))
65 footexlem.6 . . . . . . . . . . 11 (𝜑𝑌 ∈ (𝐸𝐼𝑍))
661, 8, 2, 4, 5, 37, 5, 13, 54tgcgrcomlr 28488 . . . . . . . . . . 11 (𝜑 → (𝑄 𝑌) = (𝐸 𝑌))
671, 8, 2, 4, 37, 13axtgcgrrflx 28470 . . . . . . . . . . 11 (𝜑 → (𝑄 𝐸) = (𝐸 𝑄))
6854eqcomd 2743 . . . . . . . . . . 11 (𝜑 → (𝑌 𝐸) = (𝑌 𝑄))
691, 8, 2, 4, 37, 5, 9, 13, 5, 6, 13, 37, 63, 64, 65, 66, 11, 67, 68axtg5seg 28473 . . . . . . . . . 10 (𝜑 → (𝑅 𝐸) = (𝑍 𝑄))
701, 8, 2, 4, 9, 13, 6, 37, 69tgcgrcomlr 28488 . . . . . . . . 9 (𝜑 → (𝐸 𝑅) = (𝑄 𝑍))
711, 8, 2, 4, 5, 9, 5, 6, 11tgcgrcomlr 28488 . . . . . . . . 9 (𝜑 → (𝑅 𝑌) = (𝑍 𝑌))
721, 8, 48, 4, 13, 9, 5, 37, 6, 5, 70, 71, 68trgcgr 28524 . . . . . . . 8 (𝜑 → ⟨“𝐸𝑅𝑌”⟩(cgrG‘𝐺)⟨“𝑄𝑍𝑌”⟩)
731, 8, 2, 3, 27, 4, 13, 9, 5, 48, 37, 6, 5, 53, 72ragcgr 28715 . . . . . . 7 (𝜑 → ⟨“𝑄𝑍𝑌”⟩ ∈ (∟G‘𝐺))
741, 8, 2, 3, 27, 4, 37, 6, 5, 73ragcom 28706 . . . . . 6 (𝜑 → ⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺))
751, 8, 2, 3, 27, 4, 5, 6, 37israg 28705 . . . . . 6 (𝜑 → (⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝑌 𝑄) = (𝑌 (((pInvG‘𝐺)‘𝑍)‘𝑄))))
7674, 75mpbid 232 . . . . 5 (𝜑 → (𝑌 𝑄) = (𝑌 (((pInvG‘𝐺)‘𝑍)‘𝑄)))
77 footexlem.11 . . . . . 6 (𝜑 → (𝑌 𝐷) = (𝑌 𝐶))
7877eqcomd 2743 . . . . 5 (𝜑 → (𝑌 𝐶) = (𝑌 𝐷))
79 eqidd 2738 . . . . 5 (𝜑 → (((pInvG‘𝐺)‘𝑍)‘𝑄) = (((pInvG‘𝐺)‘𝑍)‘𝑄))
80 footexlem.12 . . . . 5 (𝜑𝐷 = (((pInvG‘𝐺)‘𝑋)‘𝐶))
811, 8, 2, 3, 27, 4, 35, 36, 37, 38, 5, 39, 40, 6, 7, 46, 47, 76, 78, 79, 80krippen 28699 . . . 4 (𝜑𝑌 ∈ (𝑍𝐼𝑋))
821, 2, 3, 4, 6, 5, 7, 34, 81btwnlng3 28629 . . 3 (𝜑𝑋 ∈ (𝑍𝐿𝑌))
831, 2, 3, 4, 5, 6, 7, 33, 82lncom 28630 . 2 (𝜑𝑋 ∈ (𝑌𝐿𝑍))
84 isperp.a . . 3 (𝜑𝐴 ∈ ran 𝐿)
8549eqcomd 2743 . . . . . 6 (𝜑 → (𝐸 𝐶) = (𝐸 𝑌))
861, 8, 2, 4, 13, 39, 13, 5, 85, 60tgcgrneq 28491 . . . . 5 (𝜑𝐸𝑌)
871, 2, 3, 4, 13, 5, 6, 86, 65btwnlng3 28629 . . . 4 (𝜑𝑍 ∈ (𝐸𝐿𝑌))
881, 2, 3, 4, 13, 5, 86, 86, 84, 58, 21tglinethru 28644 . . . 4 (𝜑𝐴 = (𝐸𝐿𝑌))
8987, 88eleqtrrd 2844 . . 3 (𝜑𝑍𝐴)
901, 2, 3, 4, 5, 6, 33, 33, 84, 21, 89tglinethru 28644 . 2 (𝜑𝐴 = (𝑌𝐿𝑍))
9183, 90eleqtrrd 2844 1 (𝜑𝑋𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  wne 2940  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
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
This theorem is referenced by:  footexlem2  28728  footex  28729
  Copyright terms: Public domain W3C validator