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Theorem isperp 27703
Description: Property for 2 lines A, B to be perpendicular. Item (ii) of definition 8.11 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 16-Oct-2019.)
Hypotheses
Ref Expression
isperp.p 𝑃 = (Baseβ€˜πΊ)
isperp.d βˆ’ = (distβ€˜πΊ)
isperp.i 𝐼 = (Itvβ€˜πΊ)
isperp.l 𝐿 = (LineGβ€˜πΊ)
isperp.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
isperp.a (πœ‘ β†’ 𝐴 ∈ ran 𝐿)
isperp.b (πœ‘ β†’ 𝐡 ∈ ran 𝐿)
Assertion
Ref Expression
isperp (πœ‘ β†’ (𝐴(βŸ‚Gβ€˜πΊ)𝐡 ↔ βˆƒπ‘₯ ∈ (𝐴 ∩ 𝐡)βˆ€π‘’ ∈ 𝐴 βˆ€π‘£ ∈ 𝐡 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))
Distinct variable groups:   𝑣,𝑒,π‘₯,𝐴   𝑒,𝐡,𝑣,π‘₯   𝑒,𝐺,𝑣,π‘₯   πœ‘,𝑒,𝑣,π‘₯
Allowed substitution hints:   𝑃(π‘₯,𝑣,𝑒)   𝐼(π‘₯,𝑣,𝑒)   𝐿(π‘₯,𝑣,𝑒)   βˆ’ (π‘₯,𝑣,𝑒)

Proof of Theorem isperp
Dummy variables π‘Ž 𝑏 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 5110 . . 3 (𝐴(βŸ‚Gβ€˜πΊ)𝐡 ↔ ⟨𝐴, 𝐡⟩ ∈ (βŸ‚Gβ€˜πΊ))
2 df-perpg 27687 . . . . 5 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‘”))})
3 simpr 486 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ 𝑔 = 𝐺)
43fveq2d 6850 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (LineGβ€˜π‘”) = (LineGβ€˜πΊ))
5 isperp.l . . . . . . . . . . 11 𝐿 = (LineGβ€˜πΊ)
64, 5eqtr4di 2791 . . . . . . . . . 10 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (LineGβ€˜π‘”) = 𝐿)
76rneqd 5897 . . . . . . . . 9 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ ran (LineGβ€˜π‘”) = ran 𝐿)
87eleq2d 2820 . . . . . . . 8 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (π‘Ž ∈ ran (LineGβ€˜π‘”) ↔ π‘Ž ∈ ran 𝐿))
97eleq2d 2820 . . . . . . . 8 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (𝑏 ∈ ran (LineGβ€˜π‘”) ↔ 𝑏 ∈ ran 𝐿))
108, 9anbi12d 632 . . . . . . 7 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ ((π‘Ž ∈ ran (LineGβ€˜π‘”) ∧ 𝑏 ∈ ran (LineGβ€˜π‘”)) ↔ (π‘Ž ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿)))
113fveq2d 6850 . . . . . . . . . 10 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (∟Gβ€˜π‘”) = (∟Gβ€˜πΊ))
1211eleq2d 2820 . . . . . . . . 9 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜π‘”) ↔ βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))
1312ralbidv 3171 . . . . . . . 8 ((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)))
1413rexralbidv 3211 . . . . . . 7 ((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)))
1510, 14anbi12d 632 . . . . . 6 ((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))))
1615opabbidv 5175 . . . . 5 ((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))})
17 isperp.g . . . . . 6 (πœ‘ β†’ 𝐺 ∈ TarskiG)
1817elexd 3467 . . . . 5 (πœ‘ β†’ 𝐺 ∈ V)
195fvexi 6860 . . . . . . . 8 𝐿 ∈ V
20 rnexg 7845 . . . . . . . 8 (𝐿 ∈ V β†’ ran 𝐿 ∈ V)
2119, 20mp1i 13 . . . . . . 7 (πœ‘ β†’ ran 𝐿 ∈ V)
2221, 21xpexd 7689 . . . . . 6 (πœ‘ β†’ (ran 𝐿 Γ— ran 𝐿) ∈ V)
23 opabssxp 5728 . . . . . . 7 {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)
2423a1i 11 . . . . . 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))} βŠ† (ran 𝐿 Γ— ran 𝐿))
2522, 24ssexd 5285 . . . . 5 (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)
262, 16, 18, 25fvmptd2 6960 . . . 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))})
2726eleq2d 2820 . . 3 (πœ‘ β†’ (⟨𝐴, 𝐡⟩ ∈ (βŸ‚Gβ€˜πΊ) ↔ ⟨𝐴, 𝐡⟩ ∈ {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))}))
281, 27bitrid 283 . 2 (πœ‘ β†’ (𝐴(βŸ‚Gβ€˜πΊ)𝐡 ↔ ⟨𝐴, 𝐡⟩ ∈ {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))}))
29 isperp.a . . 3 (πœ‘ β†’ 𝐴 ∈ ran 𝐿)
30 isperp.b . . 3 (πœ‘ β†’ 𝐡 ∈ ran 𝐿)
31 ineq12 4171 . . . . 5 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ (π‘Ž ∩ 𝑏) = (𝐴 ∩ 𝐡))
32 simpll 766 . . . . . 6 (((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) ∧ π‘₯ ∈ (π‘Ž ∩ 𝑏)) β†’ π‘Ž = 𝐴)
33 simpllr 775 . . . . . . 7 ((((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) ∧ π‘₯ ∈ (π‘Ž ∩ 𝑏)) ∧ 𝑒 ∈ π‘Ž) β†’ 𝑏 = 𝐡)
3433raleqdv 3312 . . . . . 6 ((((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) ∧ π‘₯ ∈ (π‘Ž ∩ 𝑏)) ∧ 𝑒 ∈ π‘Ž) β†’ (βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ) ↔ βˆ€π‘£ ∈ 𝐡 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))
3532, 34raleqbidva 3320 . . . . 5 (((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) ∧ π‘₯ ∈ (π‘Ž ∩ 𝑏)) β†’ (βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ) ↔ βˆ€π‘’ ∈ 𝐴 βˆ€π‘£ ∈ 𝐡 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))
3631, 35rexeqbidva 3321 . . . 4 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ (βˆƒπ‘₯ ∈ (π‘Ž ∩ 𝑏)βˆ€π‘’ ∈ π‘Ž βˆ€π‘£ ∈ 𝑏 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ) ↔ βˆƒπ‘₯ ∈ (𝐴 ∩ 𝐡)βˆ€π‘’ ∈ 𝐴 βˆ€π‘£ ∈ 𝐡 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))
3736opelopab2a 5496 . . 3 ((𝐴 ∈ ran 𝐿 ∧ 𝐡 ∈ ran 𝐿) β†’ (⟨𝐴, 𝐡⟩ ∈ {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))} ↔ βˆƒπ‘₯ ∈ (𝐴 ∩ 𝐡)βˆ€π‘’ ∈ 𝐴 βˆ€π‘£ ∈ 𝐡 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))
3829, 30, 37syl2anc 585 . 2 (πœ‘ β†’ (⟨𝐴, 𝐡⟩ ∈ {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))} ↔ βˆƒπ‘₯ ∈ (𝐴 ∩ 𝐡)βˆ€π‘’ ∈ 𝐴 βˆ€π‘£ ∈ 𝐡 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))
3928, 38bitrd 279 1 (πœ‘ β†’ (𝐴(βŸ‚Gβ€˜πΊ)𝐡 ↔ βˆƒπ‘₯ ∈ (𝐴 ∩ 𝐡)βˆ€π‘’ ∈ 𝐴 βˆ€π‘£ ∈ 𝐡 βŸ¨β€œπ‘’π‘₯π‘£β€βŸ© ∈ (∟Gβ€˜πΊ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447   ∩ cin 3913   βŠ† wss 3914  βŸ¨cop 4596   class class class wbr 5109  {copab 5171   Γ— cxp 5635  ran crn 5638  β€˜cfv 6500  βŸ¨β€œcs3 14740  Basecbs 17091  distcds 17150  TarskiGcstrkg 27418  Itvcitv 27424  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:  perpcom  27704  perpneq  27705  isperp2  27706
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