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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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