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Theorem perpln1 27701
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 27687 . . . . . 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 6850 . . . . . . . . . . . 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 5897 . . . . . . . . . 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 6850 . . . . . . . . . . 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 3171 . . . . . . . . 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 3211 . . . . . . . 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 5175 . . . . . 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 3467 . . . . . 6 (πœ‘ β†’ 𝐺 ∈ V)
184fvexi 6860 . . . . . . . . 9 𝐿 ∈ V
19 rnexg 7845 . . . . . . . . 9 (𝐿 ∈ V β†’ ran 𝐿 ∈ V)
2018, 19mp1i 13 . . . . . . . 8 (πœ‘ β†’ ran 𝐿 ∈ V)
2120, 20xpexd 7689 . . . . . . 7 (πœ‘ β†’ (ran 𝐿 Γ— ran 𝐿) ∈ V)
22 opabssxp 5728 . . . . . . . 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 5285 . . . . . 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 6960 . . . . 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 470 . . . . . 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 5176 . . . . 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 2789 . . . 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 5865 . . 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 5878 . . 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 4002 . 2 (πœ‘ β†’ dom (βŸ‚Gβ€˜πΊ) βŠ† ran 𝐿)
32 relopabv 5781 . . . . . 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 5738 . . . . . 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 5688 . . . . 5 ((Rel (βŸ‚Gβ€˜πΊ) ∧ 𝐴(βŸ‚Gβ€˜πΊ)𝐡) β†’ (𝐴 ∈ V ∧ 𝐡 ∈ V))
3734, 35, 36syl2anc 585 . . . 4 (πœ‘ β†’ (𝐴 ∈ V ∧ 𝐡 ∈ V))
3837simpld 496 . . 3 (πœ‘ β†’ 𝐴 ∈ V)
3937simprd 497 . . 3 (πœ‘ β†’ 𝐡 ∈ V)
40 breldmg 5869 . . 3 ((𝐴 ∈ V ∧ 𝐡 ∈ V ∧ 𝐴(βŸ‚Gβ€˜πΊ)𝐡) β†’ 𝐴 ∈ dom (βŸ‚Gβ€˜πΊ))
4138, 39, 35, 40syl3anc 1372 . 2 (πœ‘ β†’ 𝐴 ∈ dom (βŸ‚Gβ€˜πΊ))
4231, 41sseldd 3949 1 (πœ‘ β†’ 𝐴 ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447   ∩ cin 3913   βŠ† wss 3914   class class class wbr 5109  {copab 5171   Γ— cxp 5635  dom cdm 5637  ran crn 5638  Rel wrel 5642  β€˜cfv 6500  βŸ¨β€œcs3 14740  TarskiGcstrkg 27418  LineGclng 27425  βˆŸGcrag 27684  βŸ‚Gcperpg 27686
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fv 6508  df-perpg 27687
This theorem is referenced by:  footne  27714  footeq  27715  perpdragALT  27718  perpdrag  27719  colperp  27720  midex  27728  opphl  27745  lmieu  27775  lnperpex  27794  trgcopy  27795
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