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Theorem perpln1 28507
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 28493 . . . . . 6 ⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
2 simpr 484 . . . . . . . . . . . . 13 ((𝜑𝑔 = 𝐺) → 𝑔 = 𝐺)
32fveq2d 6895 . . . . . . . . . . . 12 ((𝜑𝑔 = 𝐺) → (LineG‘𝑔) = (LineG‘𝐺))
4 perpln.l . . . . . . . . . . . 12 𝐿 = (LineG‘𝐺)
53, 4eqtr4di 2785 . . . . . . . . . . 11 ((𝜑𝑔 = 𝐺) → (LineG‘𝑔) = 𝐿)
65rneqd 5934 . . . . . . . . . 10 ((𝜑𝑔 = 𝐺) → ran (LineG‘𝑔) = ran 𝐿)
76eleq2d 2814 . . . . . . . . 9 ((𝜑𝑔 = 𝐺) → (𝑎 ∈ ran (LineG‘𝑔) ↔ 𝑎 ∈ ran 𝐿))
86eleq2d 2814 . . . . . . . . 9 ((𝜑𝑔 = 𝐺) → (𝑏 ∈ ran (LineG‘𝑔) ↔ 𝑏 ∈ ran 𝐿))
97, 8anbi12d 630 . . . . . . . 8 ((𝜑𝑔 = 𝐺) → ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ↔ (𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿)))
102fveq2d 6895 . . . . . . . . . . 11 ((𝜑𝑔 = 𝐺) → (∟G‘𝑔) = (∟G‘𝐺))
1110eleq2d 2814 . . . . . . . . . 10 ((𝜑𝑔 = 𝐺) → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
1211ralbidv 3172 . . . . . . . . 9 ((𝜑𝑔 = 𝐺) → (∀𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∀𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
1312rexralbidv 3215 . . . . . . . 8 ((𝜑𝑔 = 𝐺) → (∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
149, 13anbi12d 630 . . . . . . 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))))
1514opabbidv 5208 . . . . . 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))})
16 perpln.1 . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
1716elexd 3490 . . . . . 6 (𝜑𝐺 ∈ V)
184fvexi 6905 . . . . . . . . 9 𝐿 ∈ V
19 rnexg 7904 . . . . . . . . 9 (𝐿 ∈ V → ran 𝐿 ∈ V)
2018, 19mp1i 13 . . . . . . . 8 (𝜑 → ran 𝐿 ∈ V)
2120, 20xpexd 7747 . . . . . . 7 (𝜑 → (ran 𝐿 × ran 𝐿) ∈ V)
22 opabssxp 5764 . . . . . . . 8 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ⊆ (ran 𝐿 × ran 𝐿)
2322a1i 11 . . . . . . 7 (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ⊆ (ran 𝐿 × ran 𝐿))
2421, 23ssexd 5318 . . . . . 6 (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ∈ V)
251, 15, 17, 24fvmptd2 7007 . . . . 5 (𝜑 → (⟂G‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))})
26 anass 468 . . . . . 6 (((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) ↔ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))))
2726opabbii 5209 . . . . 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)))}
2825, 27eqtrdi 2783 . . . 4 (𝜑 → (⟂G‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))})
2928dmeqd 5902 . . 3 (𝜑 → dom (⟂G‘𝐺) = dom {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))})
30 dmopabss 5915 . . 3 dom {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))} ⊆ ran 𝐿
3129, 30eqsstrdi 4032 . 2 (𝜑 → dom (⟂G‘𝐺) ⊆ ran 𝐿)
32 relopabv 5817 . . . . . 6 Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}
3325releqd 5774 . . . . . 6 (𝜑 → (Rel (⟂G‘𝐺) ↔ Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}))
3432, 33mpbiri 258 . . . . 5 (𝜑 → Rel (⟂G‘𝐺))
35 perpln.2 . . . . 5 (𝜑𝐴(⟂G‘𝐺)𝐵)
36 brrelex12 5724 . . . . 5 ((Rel (⟂G‘𝐺) ∧ 𝐴(⟂G‘𝐺)𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3734, 35, 36syl2anc 583 . . . 4 (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3837simpld 494 . . 3 (𝜑𝐴 ∈ V)
3937simprd 495 . . 3 (𝜑𝐵 ∈ V)
40 breldmg 5906 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴(⟂G‘𝐺)𝐵) → 𝐴 ∈ dom (⟂G‘𝐺))
4138, 39, 35, 40syl3anc 1369 . 2 (𝜑𝐴 ∈ dom (⟂G‘𝐺))
4231, 41sseldd 3979 1 (𝜑𝐴 ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3056  wrex 3065  Vcvv 3469  cin 3943  wss 3944   class class class wbr 5142  {copab 5204   × cxp 5670  dom cdm 5672  ran crn 5673  Rel wrel 5677  cfv 6542  ⟨“cs3 14819  TarskiGcstrkg 28224  LineGclng 28231  ∟Gcrag 28490  ⟂Gcperpg 28492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fv 6550  df-perpg 28493
This theorem is referenced by:  footne  28520  footeq  28521  perpdragALT  28524  perpdrag  28525  colperp  28526  midex  28534  opphl  28551  lmieu  28581  lnperpex  28600  trgcopy  28601
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