MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  perpln1 Structured version   Visualization version   GIF version

Theorem perpln1 25825
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 25811 . . . . . . 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 471 . . . . . . . . . . . . 13 ((𝜑𝑔 = 𝐺) → 𝑔 = 𝐺)
43fveq2d 6336 . . . . . . . . . . . 12 ((𝜑𝑔 = 𝐺) → (LineG‘𝑔) = (LineG‘𝐺))
5 perpln.l . . . . . . . . . . . 12 𝐿 = (LineG‘𝐺)
64, 5syl6eqr 2822 . . . . . . . . . . 11 ((𝜑𝑔 = 𝐺) → (LineG‘𝑔) = 𝐿)
76rneqd 5491 . . . . . . . . . 10 ((𝜑𝑔 = 𝐺) → ran (LineG‘𝑔) = ran 𝐿)
87eleq2d 2835 . . . . . . . . 9 ((𝜑𝑔 = 𝐺) → (𝑎 ∈ ran (LineG‘𝑔) ↔ 𝑎 ∈ ran 𝐿))
97eleq2d 2835 . . . . . . . . 9 ((𝜑𝑔 = 𝐺) → (𝑏 ∈ ran (LineG‘𝑔) ↔ 𝑏 ∈ ran 𝐿))
108, 9anbi12d 608 . . . . . . . 8 ((𝜑𝑔 = 𝐺) → ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ↔ (𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿)))
113fveq2d 6336 . . . . . . . . . . 11 ((𝜑𝑔 = 𝐺) → (∟G‘𝑔) = (∟G‘𝐺))
1211eleq2d 2835 . . . . . . . . . 10 ((𝜑𝑔 = 𝐺) → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
1312ralbidv 3134 . . . . . . . . 9 ((𝜑𝑔 = 𝐺) → (∀𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∀𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
1413rexralbidv 3205 . . . . . . . 8 ((𝜑𝑔 = 𝐺) → (∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
1510, 14anbi12d 608 . . . . . . 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 4848 . . . . . 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 3361 . . . . . . 7 (𝐺 ∈ TarskiG → 𝐺 ∈ V)
1917, 18syl 17 . . . . . 6 (𝜑𝐺 ∈ V)
20 fvex 6342 . . . . . . . . . 10 (LineG‘𝐺) ∈ V
215, 20eqeltri 2845 . . . . . . . . 9 𝐿 ∈ V
22 rnexg 7244 . . . . . . . . 9 (𝐿 ∈ V → ran 𝐿 ∈ V)
2321, 22mp1i 13 . . . . . . . 8 (𝜑 → ran 𝐿 ∈ V)
24 xpexg 7106 . . . . . . . 8 ((ran 𝐿 ∈ V ∧ ran 𝐿 ∈ V) → (ran 𝐿 × ran 𝐿) ∈ V)
2523, 23, 24syl2anc 565 . . . . . . 7 (𝜑 → (ran 𝐿 × ran 𝐿) ∈ V)
26 opabssxp 5333 . . . . . . . 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 4936 . . . . . 6 (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ∈ V)
292, 16, 19, 28fvmptd 6430 . . . . 5 (𝜑 → (⟂G‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))})
30 anass 459 . . . . . 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 4849 . . . . 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 2820 . . . 4 (𝜑 → (⟂G‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))})
3332dmeqd 5464 . . 3 (𝜑 → dom (⟂G‘𝐺) = dom {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))})
34 dmopabss 5474 . . 3 dom {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))} ⊆ ran 𝐿
3533, 34syl6eqss 3802 . 2 (𝜑 → dom (⟂G‘𝐺) ⊆ ran 𝐿)
36 relopab 5386 . . . . . 6 Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}
3729releqd 5343 . . . . . 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 5295 . . . . 5 ((Rel (⟂G‘𝐺) ∧ 𝐴(⟂G‘𝐺)𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4138, 39, 40syl2anc 565 . . . 4 (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4241simpld 476 . . 3 (𝜑𝐴 ∈ V)
4341simprd 477 . . 3 (𝜑𝐵 ∈ V)
44 breldmg 5468 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴(⟂G‘𝐺)𝐵) → 𝐴 ∈ dom (⟂G‘𝐺))
4542, 43, 39, 44syl3anc 1475 . 2 (𝜑𝐴 ∈ dom (⟂G‘𝐺))
4635, 45sseldd 3751 1 (𝜑𝐴 ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wral 3060  wrex 3061  Vcvv 3349  cin 3720  wss 3721   class class class wbr 4784  {copab 4844  cmpt 4861   × cxp 5247  dom cdm 5249  ran crn 5250  Rel wrel 5254  cfv 6031  ⟨“cs3 13795  TarskiGcstrkg 25549  LineGclng 25556  ∟Gcrag 25808  ⟂Gcperpg 25810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-perpg 25811
This theorem is referenced by:  footne  25835  footeq  25836  perpdragALT  25839  perpdrag  25840  colperp  25841  midex  25849  opphl  25866  lmieu  25896  lnperpex  25915  trgcopy  25916
  Copyright terms: Public domain W3C validator