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

Theorem perpln1 28454
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 28440 . . . . . 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 6886 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (LineGβ€˜π‘”) = (LineGβ€˜πΊ))
4 perpln.l . . . . . . . . . . . 12 𝐿 = (LineGβ€˜πΊ)
53, 4eqtr4di 2782 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (LineGβ€˜π‘”) = 𝐿)
65rneqd 5928 . . . . . . . . . 10 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ ran (LineGβ€˜π‘”) = ran 𝐿)
76eleq2d 2811 . . . . . . . . 9 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (π‘Ž ∈ ran (LineGβ€˜π‘”) ↔ π‘Ž ∈ ran 𝐿))
86eleq2d 2811 . . . . . . . . 9 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (𝑏 ∈ ran (LineGβ€˜π‘”) ↔ 𝑏 ∈ ran 𝐿))
97, 8anbi12d 630 . . . . . . . 8 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ ((π‘Ž ∈ ran (LineGβ€˜π‘”) ∧ 𝑏 ∈ ran (LineGβ€˜π‘”)) ↔ (π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿)))
102fveq2d 6886 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (∟Gβ€˜π‘”) = (∟Gβ€˜πΊ))
1110eleq2d 2811 . . . . . . . . . 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 3169 . . . . . . . . 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 3212 . . . . . . . 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 630 . . . . . . 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 5205 . . . . . 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 3487 . . . . . 6 (πœ‘ β†’ 𝐺 ∈ V)
184fvexi 6896 . . . . . . . . 9 𝐿 ∈ V
19 rnexg 7889 . . . . . . . . 9 (𝐿 ∈ V β†’ ran 𝐿 ∈ V)
2018, 19mp1i 13 . . . . . . . 8 (πœ‘ β†’ ran 𝐿 ∈ V)
2120, 20xpexd 7732 . . . . . . 7 (πœ‘ β†’ (ran 𝐿 Γ— ran 𝐿) ∈ V)
22 opabssxp 5759 . . . . . . . 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 5315 . . . . . 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 6997 . . . . 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 5206 . . . . 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 2780 . . . 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 5896 . . 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 5909 . . 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 4029 . 2 (πœ‘ β†’ dom (βŸ‚Gβ€˜πΊ) βŠ† ran 𝐿)
32 relopabv 5812 . . . . . 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 5769 . . . . . 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 5719 . . . . 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 5900 . . 3 ((𝐴 ∈ V ∧ 𝐡 ∈ V ∧ 𝐴(βŸ‚Gβ€˜πΊ)𝐡) β†’ 𝐴 ∈ dom (βŸ‚Gβ€˜πΊ))
4138, 39, 35, 40syl3anc 1368 . 2 (πœ‘ β†’ 𝐴 ∈ dom (βŸ‚Gβ€˜πΊ))
4231, 41sseldd 3976 1 (πœ‘ β†’ 𝐴 ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βˆƒwrex 3062  Vcvv 3466   ∩ cin 3940   βŠ† wss 3941   class class class wbr 5139  {copab 5201   Γ— cxp 5665  dom cdm 5667  ran crn 5668  Rel wrel 5672  β€˜cfv 6534  βŸ¨β€œcs3 14795  TarskiGcstrkg 28171  LineGclng 28178  βˆŸGcrag 28437  βŸ‚Gcperpg 28439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-perpg 28440
This theorem is referenced by:  footne  28467  footeq  28468  perpdragALT  28471  perpdrag  28472  colperp  28473  midex  28481  opphl  28498  lmieu  28528  lnperpex  28547  trgcopy  28548
  Copyright terms: Public domain W3C validator