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Theorem perpln2 26802
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
perpln2 (𝜑𝐵 ∈ ran 𝐿)

Proof of Theorem perpln2
Dummy variables 𝑎 𝑏 𝑔 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-perpg 26787 . . . . . 6 ⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
2 simpr 488 . . . . . . . . . . . . 13 ((𝜑𝑔 = 𝐺) → 𝑔 = 𝐺)
32fveq2d 6721 . . . . . . . . . . . 12 ((𝜑𝑔 = 𝐺) → (LineG‘𝑔) = (LineG‘𝐺))
4 perpln.l . . . . . . . . . . . 12 𝐿 = (LineG‘𝐺)
53, 4eqtr4di 2796 . . . . . . . . . . 11 ((𝜑𝑔 = 𝐺) → (LineG‘𝑔) = 𝐿)
65rneqd 5807 . . . . . . . . . 10 ((𝜑𝑔 = 𝐺) → ran (LineG‘𝑔) = ran 𝐿)
76eleq2d 2823 . . . . . . . . 9 ((𝜑𝑔 = 𝐺) → (𝑎 ∈ ran (LineG‘𝑔) ↔ 𝑎 ∈ ran 𝐿))
86eleq2d 2823 . . . . . . . . 9 ((𝜑𝑔 = 𝐺) → (𝑏 ∈ ran (LineG‘𝑔) ↔ 𝑏 ∈ ran 𝐿))
97, 8anbi12d 634 . . . . . . . 8 ((𝜑𝑔 = 𝐺) → ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ↔ (𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿)))
102fveq2d 6721 . . . . . . . . . . 11 ((𝜑𝑔 = 𝐺) → (∟G‘𝑔) = (∟G‘𝐺))
1110eleq2d 2823 . . . . . . . . . 10 ((𝜑𝑔 = 𝐺) → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
1211ralbidv 3118 . . . . . . . . 9 ((𝜑𝑔 = 𝐺) → (∀𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∀𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
1312rexralbidv 3220 . . . . . . . 8 ((𝜑𝑔 = 𝐺) → (∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔) ↔ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
149, 13anbi12d 634 . . . . . . 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 5119 . . . . . 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 3428 . . . . . 6 (𝜑𝐺 ∈ V)
184fvexi 6731 . . . . . . . . 9 𝐿 ∈ V
19 rnexg 7682 . . . . . . . . 9 (𝐿 ∈ V → ran 𝐿 ∈ V)
2018, 19mp1i 13 . . . . . . . 8 (𝜑 → ran 𝐿 ∈ V)
2120, 20xpexd 7536 . . . . . . 7 (𝜑 → (ran 𝐿 × ran 𝐿) ∈ V)
22 opabssxp 5640 . . . . . . . 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 5217 . . . . . 6 (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ∈ V)
251, 15, 17, 24fvmptd2 6826 . . . . 5 (𝜑 → (⟂G‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))})
2625rneqd 5807 . . . 4 (𝜑 → ran (⟂G‘𝐺) = ran {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))})
2722rnssi 5809 . . . 4 ran {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))} ⊆ ran (ran 𝐿 × ran 𝐿)
2826, 27eqsstrdi 3955 . . 3 (𝜑 → ran (⟂G‘𝐺) ⊆ ran (ran 𝐿 × ran 𝐿))
29 rnxpss 6035 . . 3 ran (ran 𝐿 × ran 𝐿) ⊆ ran 𝐿
3028, 29sstrdi 3913 . 2 (𝜑 → ran (⟂G‘𝐺) ⊆ ran 𝐿)
31 relopabv 5691 . . . . . 6 Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}
3225releqd 5650 . . . . . 6 (𝜑 → (Rel (⟂G‘𝐺) ↔ Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran 𝐿𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))}))
3331, 32mpbiri 261 . . . . 5 (𝜑 → Rel (⟂G‘𝐺))
34 perpln.2 . . . . 5 (𝜑𝐴(⟂G‘𝐺)𝐵)
35 brrelex12 5601 . . . . 5 ((Rel (⟂G‘𝐺) ∧ 𝐴(⟂G‘𝐺)𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3633, 34, 35syl2anc 587 . . . 4 (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3736simpld 498 . . 3 (𝜑𝐴 ∈ V)
3836simprd 499 . . 3 (𝜑𝐵 ∈ V)
39 brelrng 5810 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴(⟂G‘𝐺)𝐵) → 𝐵 ∈ ran (⟂G‘𝐺))
4037, 38, 34, 39syl3anc 1373 . 2 (𝜑𝐵 ∈ ran (⟂G‘𝐺))
4130, 40sseldd 3902 1 (𝜑𝐵 ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wral 3061  wrex 3062  Vcvv 3408  cin 3865  wss 3866   class class class wbr 5053  {copab 5115   × cxp 5549  ran crn 5552  Rel wrel 5556  cfv 6380  ⟨“cs3 14407  TarskiGcstrkg 26521  LineGclng 26528  ∟Gcrag 26784  ⟂Gcperpg 26786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fv 6388  df-perpg 26787
This theorem is referenced by:  hlperpnel  26816  mideulem2  26825  opphllem  26826  opphllem3  26840  opphllem5  26842  opphllem6  26843  trgcopy  26895
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