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Theorem perpln2 27962
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
perpln2 (πœ‘ β†’ 𝐡 ∈ ran 𝐿)

Proof of Theorem perpln2
Dummy variables π‘Ž 𝑏 𝑔 𝑒 𝑣 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-perpg 27947 . . . . . 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 486 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ 𝑔 = 𝐺)
32fveq2d 6896 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (LineGβ€˜π‘”) = (LineGβ€˜πΊ))
4 perpln.l . . . . . . . . . . . 12 𝐿 = (LineGβ€˜πΊ)
53, 4eqtr4di 2791 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (LineGβ€˜π‘”) = 𝐿)
65rneqd 5938 . . . . . . . . . 10 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ ran (LineGβ€˜π‘”) = ran 𝐿)
76eleq2d 2820 . . . . . . . . 9 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (π‘Ž ∈ ran (LineGβ€˜π‘”) ↔ π‘Ž ∈ ran 𝐿))
86eleq2d 2820 . . . . . . . . 9 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (𝑏 ∈ ran (LineGβ€˜π‘”) ↔ 𝑏 ∈ ran 𝐿))
97, 8anbi12d 632 . . . . . . . 8 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ ((π‘Ž ∈ ran (LineGβ€˜π‘”) ∧ 𝑏 ∈ ran (LineGβ€˜π‘”)) ↔ (π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿)))
102fveq2d 6896 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (∟Gβ€˜π‘”) = (∟Gβ€˜πΊ))
1110eleq2d 2820 . . . . . . . . . 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 3178 . . . . . . . . 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 3221 . . . . . . . 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 632 . . . . . . 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 5215 . . . . . 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 3495 . . . . . 6 (πœ‘ β†’ 𝐺 ∈ V)
184fvexi 6906 . . . . . . . . 9 𝐿 ∈ V
19 rnexg 7895 . . . . . . . . 9 (𝐿 ∈ V β†’ ran 𝐿 ∈ V)
2018, 19mp1i 13 . . . . . . . 8 (πœ‘ β†’ ran 𝐿 ∈ V)
2120, 20xpexd 7738 . . . . . . 7 (πœ‘ β†’ (ran 𝐿 Γ— ran 𝐿) ∈ V)
22 opabssxp 5769 . . . . . . . 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 5325 . . . . . 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 7007 . . . . 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))})
2625rneqd 5938 . . . 4 (πœ‘ β†’ ran (βŸ‚Gβ€˜πΊ) = ran {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))})
2722rnssi 5940 . . . 4 ran {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 (ran 𝐿 Γ— ran 𝐿)
2826, 27eqsstrdi 4037 . . 3 (πœ‘ β†’ ran (βŸ‚Gβ€˜πΊ) βŠ† ran (ran 𝐿 Γ— ran 𝐿))
29 rnxpss 6172 . . 3 ran (ran 𝐿 Γ— ran 𝐿) βŠ† ran 𝐿
3028, 29sstrdi 3995 . 2 (πœ‘ β†’ ran (βŸ‚Gβ€˜πΊ) βŠ† ran 𝐿)
31 relopabv 5822 . . . . . 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))}
3225releqd 5779 . . . . . 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))}))
3331, 32mpbiri 258 . . . . 5 (πœ‘ β†’ Rel (βŸ‚Gβ€˜πΊ))
34 perpln.2 . . . . 5 (πœ‘ β†’ 𝐴(βŸ‚Gβ€˜πΊ)𝐡)
35 brrelex12 5729 . . . . 5 ((Rel (βŸ‚Gβ€˜πΊ) ∧ 𝐴(βŸ‚Gβ€˜πΊ)𝐡) β†’ (𝐴 ∈ V ∧ 𝐡 ∈ V))
3633, 34, 35syl2anc 585 . . . 4 (πœ‘ β†’ (𝐴 ∈ V ∧ 𝐡 ∈ V))
3736simpld 496 . . 3 (πœ‘ β†’ 𝐴 ∈ V)
3836simprd 497 . . 3 (πœ‘ β†’ 𝐡 ∈ V)
39 brelrng 5941 . . 3 ((𝐴 ∈ V ∧ 𝐡 ∈ V ∧ 𝐴(βŸ‚Gβ€˜πΊ)𝐡) β†’ 𝐡 ∈ ran (βŸ‚Gβ€˜πΊ))
4037, 38, 34, 39syl3anc 1372 . 2 (πœ‘ β†’ 𝐡 ∈ ran (βŸ‚Gβ€˜πΊ))
4130, 40sseldd 3984 1 (πœ‘ β†’ 𝐡 ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   ∩ cin 3948   βŠ† wss 3949   class class class wbr 5149  {copab 5211   Γ— cxp 5675  ran crn 5678  Rel wrel 5682  β€˜cfv 6544  βŸ¨β€œcs3 14793  TarskiGcstrkg 27678  LineGclng 27685  βˆŸGcrag 27944  βŸ‚Gcperpg 27946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-perpg 27947
This theorem is referenced by:  hlperpnel  27976  mideulem2  27985  opphllem  27986  opphllem3  28000  opphllem5  28002  opphllem6  28003  trgcopy  28055
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