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Theorem footexlem1 26513
Description: Lemma for footex 26515. (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 2804 . . . 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 3042 . . . . . . . . . . . 12 (𝜑𝐹𝐸)
17 footexlem.3 . . . . . . . . . . . 12 (𝜑𝐸 ∈ (𝐹𝐼𝑌))
181, 2, 3, 4, 14, 13, 5, 16, 17btwnlng3 26415 . . . . . . . . . . 11 (𝜑𝑌 ∈ (𝐹𝐿𝐸))
191, 2, 3, 4, 13, 14, 5, 15, 18lncom 26416 . . . . . . . . . 10 (𝜑𝑌 ∈ (𝐸𝐿𝐹))
20 footexlem.1 . . . . . . . . . 10 (𝜑𝐴 = (𝐸𝐿𝐹))
2119, 20eleqtrrd 2893 . . . . . . . . 9 (𝜑𝑌𝐴)
22 foot.y . . . . . . . . 9 (𝜑 → ¬ 𝐶𝐴)
23 nelne2 3084 . . . . . . . . 9 ((𝑌𝐴 ∧ ¬ 𝐶𝐴) → 𝑌𝐶)
2421, 22, 23syl2anc 587 . . . . . . . 8 (𝜑𝑌𝐶)
2524necomd 3042 . . . . . . 7 (𝜑𝐶𝑌)
2612, 25eqnetrrd 3055 . . . . . 6 (𝜑 → (((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌)
27 eqid 2798 . . . . . . . 8 (pInvG‘𝐺) = (pInvG‘𝐺)
28 eqid 2798 . . . . . . . 8 ((pInvG‘𝐺)‘𝑅) = ((pInvG‘𝐺)‘𝑅)
291, 8, 2, 3, 27, 4, 9, 28, 5mirinv 26460 . . . . . . 7 (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) = 𝑌𝑅 = 𝑌))
3029necon3bid 3031 . . . . . 6 (𝜑 → ((((pInvG‘𝐺)‘𝑅)‘𝑌) ≠ 𝑌𝑅𝑌))
3126, 30mpbid 235 . . . . 5 (𝜑𝑅𝑌)
3231necomd 3042 . . . 4 (𝜑𝑌𝑅)
331, 8, 2, 4, 5, 9, 5, 6, 11, 32tgcgrneq 26277 . . 3 (𝜑𝑌𝑍)
3433necomd 3042 . . . 4 (𝜑𝑍𝑌)
35 eqid 2798 . . . . 5 ((pInvG‘𝐺)‘𝑍) = ((pInvG‘𝐺)‘𝑍)
36 eqid 2798 . . . . 5 ((pInvG‘𝐺)‘𝑋) = ((pInvG‘𝐺)‘𝑋)
37 footexlem.q . . . . 5 (𝜑𝑄𝑃)
381, 8, 2, 3, 27, 4, 6, 35, 37mircl 26455 . . . . 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 26452 . . . . . . . 8 (𝜑𝑅 ∈ ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
4212oveq1d 7150 . . . . . . . 8 (𝜑 → (𝐶𝐼𝑌) = ((((pInvG‘𝐺)‘𝑅)‘𝑌)𝐼𝑌))
4341, 42eleqtrrd 2893 . . . . . . 7 (𝜑𝑅 ∈ (𝐶𝐼𝑌))
44 footexlem.8 . . . . . . 7 (𝜑𝑌 ∈ (𝑅𝐼𝑄))
451, 8, 2, 4, 39, 9, 5, 37, 31, 43, 44tgbtwnouttr2 26289 . . . . . 6 (𝜑𝑌 ∈ (𝐶𝐼𝑄))
461, 8, 2, 4, 39, 5, 37, 45tgbtwncom 26282 . . . . 5 (𝜑𝑌 ∈ (𝑄𝐼𝐶))
47 footexlem.10 . . . . 5 (𝜑𝑌 ∈ ((((pInvG‘𝐺)‘𝑍)‘𝑄)𝐼𝐷))
48 eqid 2798 . . . . . . . 8 (cgrG‘𝐺) = (cgrG‘𝐺)
49 footexlem.4 . . . . . . . . . 10 (𝜑 → (𝐸 𝑌) = (𝐸 𝐶))
5012oveq2d 7151 . . . . . . . . . 10 (𝜑 → (𝐸 𝐶) = (𝐸 (((pInvG‘𝐺)‘𝑅)‘𝑌)))
5149, 50eqtrd 2833 . . . . . . . . 9 (𝜑 → (𝐸 𝑌) = (𝐸 (((pInvG‘𝐺)‘𝑅)‘𝑌)))
521, 8, 2, 3, 27, 4, 13, 9, 5israg 26491 . . . . . . . . 9 (𝜑 → (⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺) ↔ (𝐸 𝑌) = (𝐸 (((pInvG‘𝐺)‘𝑅)‘𝑌))))
5351, 52mpbird 260 . . . . . . . 8 (𝜑 → ⟨“𝐸𝑅𝑌”⟩ ∈ (∟G‘𝐺))
54 footexlem.9 . . . . . . . . . . . . . 14 (𝜑 → (𝑌 𝑄) = (𝑌 𝐸))
551, 8, 2, 4, 13, 5, 13, 39, 49tgcgrcomlr 26274 . . . . . . . . . . . . . 14 (𝜑 → (𝑌 𝐸) = (𝐶 𝐸))
5654, 55eqtr2d 2834 . . . . . . . . . . . . 13 (𝜑 → (𝐶 𝐸) = (𝑌 𝑄))
571, 2, 3, 4, 13, 14, 15tglinerflx1 26427 . . . . . . . . . . . . . . . 16 (𝜑𝐸 ∈ (𝐸𝐿𝐹))
5857, 20eleqtrrd 2893 . . . . . . . . . . . . . . 15 (𝜑𝐸𝐴)
59 nelne2 3084 . . . . . . . . . . . . . . 15 ((𝐸𝐴 ∧ ¬ 𝐶𝐴) → 𝐸𝐶)
6058, 22, 59syl2anc 587 . . . . . . . . . . . . . 14 (𝜑𝐸𝐶)
6160necomd 3042 . . . . . . . . . . . . 13 (𝜑𝐶𝐸)
621, 8, 2, 4, 39, 13, 5, 37, 56, 61tgcgrneq 26277 . . . . . . . . . . . 12 (𝜑𝑌𝑄)
6362necomd 3042 . . . . . . . . . . 11 (𝜑𝑄𝑌)
641, 8, 2, 4, 9, 5, 37, 44tgbtwncom 26282 . . . . . . . . . . 11 (𝜑𝑌 ∈ (𝑄𝐼𝑅))
65 footexlem.6 . . . . . . . . . . 11 (𝜑𝑌 ∈ (𝐸𝐼𝑍))
661, 8, 2, 4, 5, 37, 5, 13, 54tgcgrcomlr 26274 . . . . . . . . . . 11 (𝜑 → (𝑄 𝑌) = (𝐸 𝑌))
671, 8, 2, 4, 37, 13axtgcgrrflx 26256 . . . . . . . . . . 11 (𝜑 → (𝑄 𝐸) = (𝐸 𝑄))
6854eqcomd 2804 . . . . . . . . . . 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 26259 . . . . . . . . . 10 (𝜑 → (𝑅 𝐸) = (𝑍 𝑄))
701, 8, 2, 4, 9, 13, 6, 37, 69tgcgrcomlr 26274 . . . . . . . . 9 (𝜑 → (𝐸 𝑅) = (𝑄 𝑍))
711, 8, 2, 4, 5, 9, 5, 6, 11tgcgrcomlr 26274 . . . . . . . . 9 (𝜑 → (𝑅 𝑌) = (𝑍 𝑌))
721, 8, 48, 4, 13, 9, 5, 37, 6, 5, 70, 71, 68trgcgr 26310 . . . . . . . 8 (𝜑 → ⟨“𝐸𝑅𝑌”⟩(cgrG‘𝐺)⟨“𝑄𝑍𝑌”⟩)
731, 8, 2, 3, 27, 4, 13, 9, 5, 48, 37, 6, 5, 53, 72ragcgr 26501 . . . . . . 7 (𝜑 → ⟨“𝑄𝑍𝑌”⟩ ∈ (∟G‘𝐺))
741, 8, 2, 3, 27, 4, 37, 6, 5, 73ragcom 26492 . . . . . 6 (𝜑 → ⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺))
751, 8, 2, 3, 27, 4, 5, 6, 37israg 26491 . . . . . 6 (𝜑 → (⟨“𝑌𝑍𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝑌 𝑄) = (𝑌 (((pInvG‘𝐺)‘𝑍)‘𝑄))))
7674, 75mpbid 235 . . . . 5 (𝜑 → (𝑌 𝑄) = (𝑌 (((pInvG‘𝐺)‘𝑍)‘𝑄)))
77 footexlem.11 . . . . . 6 (𝜑 → (𝑌 𝐷) = (𝑌 𝐶))
7877eqcomd 2804 . . . . 5 (𝜑 → (𝑌 𝐶) = (𝑌 𝐷))
79 eqidd 2799 . . . . 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 26485 . . . 4 (𝜑𝑌 ∈ (𝑍𝐼𝑋))
821, 2, 3, 4, 6, 5, 7, 34, 81btwnlng3 26415 . . 3 (𝜑𝑋 ∈ (𝑍𝐿𝑌))
831, 2, 3, 4, 5, 6, 7, 33, 82lncom 26416 . 2 (𝜑𝑋 ∈ (𝑌𝐿𝑍))
84 isperp.a . . 3 (𝜑𝐴 ∈ ran 𝐿)
8549eqcomd 2804 . . . . . 6 (𝜑 → (𝐸 𝐶) = (𝐸 𝑌))
861, 8, 2, 4, 13, 39, 13, 5, 85, 60tgcgrneq 26277 . . . . 5 (𝜑𝐸𝑌)
871, 2, 3, 4, 13, 5, 6, 86, 65btwnlng3 26415 . . . 4 (𝜑𝑍 ∈ (𝐸𝐿𝑌))
881, 2, 3, 4, 13, 5, 86, 86, 84, 58, 21tglinethru 26430 . . . 4 (𝜑𝐴 = (𝐸𝐿𝑌))
8987, 88eleqtrrd 2893 . . 3 (𝜑𝑍𝐴)
901, 2, 3, 4, 5, 6, 33, 33, 84, 21, 89tglinethru 26430 . 2 (𝜑𝐴 = (𝑌𝐿𝑍))
9183, 90eleqtrrd 2893 1 (𝜑𝑋𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2111  wne 2987  ran crn 5520  cfv 6324  (class class class)co 7135  ⟨“cs3 14195  Basecbs 16475  distcds 16566  TarskiGcstrkg 26224  Itvcitv 26230  LineGclng 26231  cgrGccgrg 26304  pInvGcmir 26446  ∟Gcrag 26487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-concat 13914  df-s1 13941  df-s2 14201  df-s3 14202  df-trkgc 26242  df-trkgb 26243  df-trkgcb 26244  df-trkg 26247  df-cgrg 26305  df-leg 26377  df-mir 26447  df-rag 26488
This theorem is referenced by:  footexlem2  26514  footex  26515
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