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Theorem perpln1 28513
Description: Derive a line from perpendicularity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
Hypotheses
Ref Expression
perpln.l 𝐿 = (LineGβ€˜πΊ)
perpln.1 (πœ‘ β†’ 𝐺 ∈ TarskiG)
perpln.2 (πœ‘ β†’ 𝐴(βŸ‚Gβ€˜πΊ)𝐡)
Assertion
Ref Expression
perpln1 (πœ‘ β†’ 𝐴 ∈ ran 𝐿)

Proof of Theorem perpln1
Dummy variables π‘Ž 𝑏 𝑔 𝑒 𝑣 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-perpg 28499 . . . . . 6 βŸ‚G = (𝑔 ∈ V ↦ {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ ran (LineGβ€˜π‘”) ∧ 𝑏 ∈ ran (LineGβ€˜π‘”)) ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜π‘”))})
2 simpr 484 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ 𝑔 = 𝐺)
32fveq2d 6901 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (LineGβ€˜π‘”) = (LineGβ€˜πΊ))
4 perpln.l . . . . . . . . . . . 12 𝐿 = (LineGβ€˜πΊ)
53, 4eqtr4di 2786 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (LineGβ€˜π‘”) = 𝐿)
65rneqd 5940 . . . . . . . . . 10 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ ran (LineGβ€˜π‘”) = ran 𝐿)
76eleq2d 2815 . . . . . . . . 9 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (π‘Ž ∈ ran (LineGβ€˜π‘”) ↔ π‘Ž ∈ ran 𝐿))
86eleq2d 2815 . . . . . . . . 9 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (𝑏 ∈ ran (LineGβ€˜π‘”) ↔ 𝑏 ∈ ran 𝐿))
97, 8anbi12d 631 . . . . . . . 8 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ ((π‘Ž ∈ ran (LineGβ€˜π‘”) ∧ 𝑏 ∈ ran (LineGβ€˜π‘”)) ↔ (π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿)))
102fveq2d 6901 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (∟Gβ€˜π‘”) = (∟Gβ€˜πΊ))
1110eleq2d 2815 . . . . . . . . . 10 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜π‘”) ↔ βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))
1211ralbidv 3174 . . . . . . . . 9 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜π‘”) ↔ βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))
1312rexralbidv 3217 . . . . . . . 8 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜π‘”) ↔ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))
149, 13anbi12d 631 . . . . . . 7 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (((π‘Ž ∈ ran (LineGβ€˜π‘”) ∧ 𝑏 ∈ ran (LineGβ€˜π‘”)) ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜π‘”)) ↔ ((π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ))))
1514opabbidv 5214 . . . . . 6 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ ran (LineGβ€˜π‘”) ∧ 𝑏 ∈ ran (LineGβ€˜π‘”)) ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜π‘”))} = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ))})
16 perpln.1 . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ TarskiG)
1716elexd 3492 . . . . . 6 (πœ‘ β†’ 𝐺 ∈ V)
184fvexi 6911 . . . . . . . . 9 𝐿 ∈ V
19 rnexg 7910 . . . . . . . . 9 (𝐿 ∈ V β†’ ran 𝐿 ∈ V)
2018, 19mp1i 13 . . . . . . . 8 (πœ‘ β†’ ran 𝐿 ∈ V)
2120, 20xpexd 7753 . . . . . . 7 (πœ‘ β†’ (ran 𝐿 Γ— ran 𝐿) ∈ V)
22 opabssxp 5770 . . . . . . . 8 {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ))} βŠ† (ran 𝐿 Γ— ran 𝐿)
2322a1i 11 . . . . . . 7 (πœ‘ β†’ {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ))} βŠ† (ran 𝐿 Γ— ran 𝐿))
2421, 23ssexd 5324 . . . . . 6 (πœ‘ β†’ {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ))} ∈ V)
251, 15, 17, 24fvmptd2 7013 . . . . 5 (πœ‘ β†’ (βŸ‚Gβ€˜πΊ) = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ))})
26 anass 468 . . . . . 6 (((π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)) ↔ (π‘Ž ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ))))
2726opabbii 5215 . . . . 5 {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ))} = {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))}
2825, 27eqtrdi 2784 . . . 4 (πœ‘ β†’ (βŸ‚Gβ€˜πΊ) = {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))})
2928dmeqd 5908 . . 3 (πœ‘ β†’ dom (βŸ‚Gβ€˜πΊ) = dom {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))})
30 dmopabss 5921 . . 3 dom {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž ∈ ran 𝐿 ∧ (𝑏 ∈ ran 𝐿 ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))} βŠ† ran 𝐿
3129, 30eqsstrdi 4034 . 2 (πœ‘ β†’ dom (βŸ‚Gβ€˜πΊ) βŠ† ran 𝐿)
32 relopabv 5823 . . . . . 6 Rel {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ))}
3325releqd 5780 . . . . . 6 (πœ‘ β†’ (Rel (βŸ‚Gβ€˜πΊ) ↔ Rel {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ))}))
3432, 33mpbiri 258 . . . . 5 (πœ‘ β†’ Rel (βŸ‚Gβ€˜πΊ))
35 perpln.2 . . . . 5 (πœ‘ β†’ 𝐴(βŸ‚Gβ€˜πΊ)𝐡)
36 brrelex12 5730 . . . . 5 ((Rel (βŸ‚Gβ€˜πΊ) ∧ 𝐴(βŸ‚Gβ€˜πΊ)𝐡) β†’ (𝐴 ∈ V ∧ 𝐡 ∈ V))
3734, 35, 36syl2anc 583 . . . 4 (πœ‘ β†’ (𝐴 ∈ V ∧ 𝐡 ∈ V))
3837simpld 494 . . 3 (πœ‘ β†’ 𝐴 ∈ V)
3937simprd 495 . . 3 (πœ‘ β†’ 𝐡 ∈ V)
40 breldmg 5912 . . 3 ((𝐴 ∈ V ∧ 𝐡 ∈ V ∧ 𝐴(βŸ‚Gβ€˜πΊ)𝐡) β†’ 𝐴 ∈ dom (βŸ‚Gβ€˜πΊ))
4138, 39, 35, 40syl3anc 1369 . 2 (πœ‘ β†’ 𝐴 ∈ dom (βŸ‚Gβ€˜πΊ))
4231, 41sseldd 3981 1 (πœ‘ β†’ 𝐴 ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  βˆƒwrex 3067  Vcvv 3471   ∩ cin 3946   βŠ† wss 3947   class class class wbr 5148  {copab 5210   Γ— cxp 5676  dom cdm 5678  ran crn 5679  Rel wrel 5683  β€˜cfv 6548  βŸ¨β€œcs3 14825  TarskiGcstrkg 28230  LineGclng 28237  βˆŸGcrag 28496  βŸ‚Gcperpg 28498
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 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
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 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-perpg 28499
This theorem is referenced by:  footne  28526  footeq  28527  perpdragALT  28530  perpdrag  28531  colperp  28532  midex  28540  opphl  28557  lmieu  28587  lnperpex  28606  trgcopy  28607
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