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Theorem perpln1 25505
Description: Derive a line from perpendicularity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
Hypotheses
Ref Expression
perpln.l 𝐿 = (LineG‘𝐺)
perpln.1 (𝜑𝐺 ∈ TarskiG)
perpln.2 (𝜑𝐴(⟂G‘𝐺)𝐵)
Assertion
Ref Expression
perpln1 (𝜑𝐴 ∈ ran 𝐿)

Proof of Theorem perpln1
Dummy variables 𝑎 𝑏 𝑔 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-perpg 25491 . . . . . . 7 ⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
21a1i 11 . . . . . 6 (𝜑 → ⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))}))
3 simpr 477 . . . . . . . . . . . . 13 ((𝜑𝑔 = 𝐺) → 𝑔 = 𝐺)
43fveq2d 6152 . . . . . . . . . . . 12 ((𝜑𝑔 = 𝐺) → (LineG‘𝑔) = (LineG‘𝐺))
5 perpln.l . . . . . . . . . . . 12 𝐿 = (LineG‘𝐺)
64, 5syl6eqr 2673 . . . . . . . . . . 11 ((𝜑𝑔 = 𝐺) → (LineG‘𝑔) = 𝐿)
76rneqd 5313 . . . . . . . . . 10 ((𝜑𝑔 = 𝐺) → ran (LineG‘𝑔) = ran 𝐿)
87eleq2d 2684 . . . . . . . . 9 ((𝜑𝑔 = 𝐺) → (𝑎 ∈ ran (LineG‘𝑔) ↔ 𝑎 ∈ ran 𝐿))
97eleq2d 2684 . . . . . . . . 9 ((𝜑𝑔 = 𝐺) → (𝑏 ∈ ran (LineG‘𝑔) ↔ 𝑏 ∈ ran 𝐿))
108, 9anbi12d 746 . . . . . . . 8 ((𝜑𝑔 = 𝐺) → ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ↔ (𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿)))
113fveq2d 6152 . . . . . . . . . . 11 ((𝜑𝑔 = 𝐺) → (∟G‘𝑔) = (∟G‘𝐺))
1211eleq2d 2684 . . . . . . . . . 10 ((𝜑𝑔 = 𝐺) → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
1312ralbidv 2980 . . . . . . . . 9 ((𝜑𝑔 = 𝐺) → (∀𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∀𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
1413rexralbidv 3051 . . . . . . . 8 ((𝜑𝑔 = 𝐺) → (∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
1510, 14anbi12d 746 . . . . . . 7 ((𝜑𝑔 = 𝐺) → (((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔)) ↔ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))))
1615opabbidv 4678 . . . . . 6 ((𝜑𝑔 = 𝐺) → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))})
17 perpln.1 . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
18 elex 3198 . . . . . . 7 (𝐺 ∈ TarskiG → 𝐺 ∈ V)
1917, 18syl 17 . . . . . 6 (𝜑𝐺 ∈ V)
20 fvex 6158 . . . . . . . . . 10 (LineG‘𝐺) ∈ V
215, 20eqeltri 2694 . . . . . . . . 9 𝐿 ∈ V
22 rnexg 7045 . . . . . . . . 9 (𝐿 ∈ V → ran 𝐿 ∈ V)
2321, 22mp1i 13 . . . . . . . 8 (𝜑 → ran 𝐿 ∈ V)
24 xpexg 6913 . . . . . . . 8 ((ran 𝐿 ∈ V ∧ ran 𝐿 ∈ V) → (ran 𝐿 × ran 𝐿) ∈ V)
2523, 23, 24syl2anc 692 . . . . . . 7 (𝜑 → (ran 𝐿 × ran 𝐿) ∈ V)
26 opabssxp 5154 . . . . . . . 8 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ⊆ (ran 𝐿 × ran 𝐿)
2726a1i 11 . . . . . . 7 (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ⊆ (ran 𝐿 × ran 𝐿))
2825, 27ssexd 4765 . . . . . 6 (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ∈ V)
292, 16, 19, 28fvmptd 6245 . . . . 5 (𝜑 → (⟂G‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))})
30 anass 680 . . . . . 6 (((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) ↔ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))))
3130opabbii 4679 . . . . 5 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))}
3229, 31syl6eq 2671 . . . 4 (𝜑 → (⟂G‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))})
3332dmeqd 5286 . . 3 (𝜑 → dom (⟂G‘𝐺) = dom {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))})
34 dmopabss 5296 . . 3 dom {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))} ⊆ ran 𝐿
3533, 34syl6eqss 3634 . 2 (𝜑 → dom (⟂G‘𝐺) ⊆ ran 𝐿)
36 relopab 5207 . . . . . 6 Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}
3729releqd 5164 . . . . . 6 (𝜑 → (Rel (⟂G‘𝐺) ↔ Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}))
3836, 37mpbiri 248 . . . . 5 (𝜑 → Rel (⟂G‘𝐺))
39 perpln.2 . . . . 5 (𝜑𝐴(⟂G‘𝐺)𝐵)
40 brrelex12 5115 . . . . 5 ((Rel (⟂G‘𝐺) ∧ 𝐴(⟂G‘𝐺)𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4138, 39, 40syl2anc 692 . . . 4 (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4241simpld 475 . . 3 (𝜑𝐴 ∈ V)
4341simprd 479 . . 3 (𝜑𝐵 ∈ V)
44 breldmg 5290 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴(⟂G‘𝐺)𝐵) → 𝐴 ∈ dom (⟂G‘𝐺))
4542, 43, 39, 44syl3anc 1323 . 2 (𝜑𝐴 ∈ dom (⟂G‘𝐺))
4635, 45sseldd 3584 1 (𝜑𝐴 ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cin 3554  wss 3555   class class class wbr 4613  {copab 4672  cmpt 4673   × cxp 5072  dom cdm 5074  ran crn 5075  Rel wrel 5079  cfv 5847  ⟨“cs3 13524  TarskiGcstrkg 25229  LineGclng 25236  ∟Gcrag 25488  ⟂Gcperpg 25490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-perpg 25491
This theorem is referenced by:  footne  25515  footeq  25516  perpdragALT  25519  perpdrag  25520  colperp  25521  midex  25529  opphl  25546  lmieu  25576  lnperpex  25595  trgcopy  25596
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