Theorem isperp 25607

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 = ( Base ` G )
isperp.d	|- .- = ( dist ` G )
isperp.i	|- I = (Itv ` G )
isperp.l	|- L = (LineG ` G )
isperp.g	|- ( ph -> G e. TarskiG )
isperp.a	|- ( ph -> A e. ran L )
isperp.b	|- ( ph -> B e. ran L )

Assertion

Ref	Expression
isperp	|- ( ph -> ( A (⟂G ` G ) B <-> E. x e. ( A i^i B ) A. u e. A A. v e. B <" u x v "> e. (∟G ` G ) ) )

Distinct variable groups:   v, u, x, A   u, B, v, x   u, G, v, x   ph, u, v, x

Allowed substitution hints:   P( x, v, u)   I( x, v, u)   L( x, v, u)   .- ( x, v, u)

Proof of Theorem isperp

Dummy variables a b g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 4654	. . 3 |- ( A (⟂G ` G ) B <-> <. A , B >. e. (⟂G ` G ) )
2		df-perpg 25591	. . . . . 6 |- ⟂G = ( g e. _V |-> { <. a , b >. | ( ( a e. ran (LineG ` g ) /\ b e. ran (LineG ` g ) ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` g ) ) } )
3	2	a1i 11	. . . . 5 |- ( ph -> ⟂G = ( g e. _V |-> { <. a , b >. | ( ( a e. ran (LineG ` g ) /\ b e. ran (LineG ` g ) ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` g ) ) } ) )
4		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ g = G ) -> g = G )
5	4	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ph /\ g = G ) -> (LineG ` g ) = (LineG ` G ) )
6		isperp.l	. . . . . . . . . . 11 |- L = (LineG ` G )
7	5, 6	syl6eqr 2674	. . . . . . . . . 10 |- ( ( ph /\ g = G ) -> (LineG ` g ) = L )
8	7	rneqd 5353	. . . . . . . . 9 |- ( ( ph /\ g = G ) -> ran (LineG ` g ) = ran L )
9	8	eleq2d 2687	. . . . . . . 8 |- ( ( ph /\ g = G ) -> ( a e. ran (LineG ` g ) <-> a e. ran L ) )
10	8	eleq2d 2687	. . . . . . . 8 |- ( ( ph /\ g = G ) -> ( b e. ran (LineG ` g ) <-> b e. ran L ) )
11	9, 10	anbi12d 747	. . . . . . 7 |- ( ( ph /\ g = G ) -> ( ( a e. ran (LineG ` g ) /\ b e. ran (LineG ` g ) ) <-> ( a e. ran L /\ b e. ran L ) ) )
12	4	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ g = G ) -> (∟G ` g ) = (∟G ` G ) )
13	12	eleq2d 2687	. . . . . . . . 9 |- ( ( ph /\ g = G ) -> ( <" u x v "> e. (∟G ` g ) <-> <" u x v "> e. (∟G ` G ) ) )
14	13	ralbidv 2986	. . . . . . . 8 |- ( ( ph /\ g = G ) -> ( A. v e. b <" u x v "> e. (∟G ` g ) <-> A. v e. b <" u x v "> e. (∟G ` G ) ) )
15	14	rexralbidv 3058	. . . . . . 7 |- ( ( ph /\ g = G ) -> ( E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` g ) <-> E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` G ) ) )
16	11, 15	anbi12d 747	. . . . . 6 |- ( ( ph /\ g = G ) -> ( ( ( a e. ran (LineG ` g ) /\ b e. ran (LineG ` g ) ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` g ) ) <-> ( ( a e. ran L /\ b e. ran L ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` G ) ) ) )
17	16	opabbidv 4716	. . . . 5 |- ( ( ph /\ g = G ) -> { <. a , b >. | ( ( a e. ran (LineG ` g ) /\ b e. ran (LineG ` g ) ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` g ) ) } = { <. a , b >. | ( ( a e. ran L /\ b e. ran L ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` G ) ) } )
18		isperp.g	. . . . . 6 |- ( ph -> G e. TarskiG )
19		elex 3212	. . . . . 6 |- ( G e. TarskiG -> G e. _V )
20	18, 19	syl 17	. . . . 5 |- ( ph -> G e. _V )
21		fvex 6201	. . . . . . . . 9 |- (LineG ` G ) e. _V
22	6, 21	eqeltri 2697	. . . . . . . 8 |- L e. _V
23		rnexg 7098	. . . . . . . 8 |- ( L e. _V -> ran L e. _V )
24	22, 23	mp1i 13	. . . . . . 7 |- ( ph -> ran L e. _V )
25		xpexg 6960	. . . . . . 7 |- ( ( ran L e. _V /\ ran L e. _V ) -> ( ran L X. ran L ) e. _V )
26	24, 24, 25	syl2anc 693	. . . . . 6 |- ( ph -> ( ran L X. ran L ) e. _V )
27		opabssxp 5193	. . . . . . 7 |- { <. a , b >. | ( ( a e. ran L /\ b e. ran L ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` G ) ) } C_ ( ran L X. ran L )
28	27	a1i 11	. . . . . 6 |- ( ph -> { <. a , b >. | ( ( a e. ran L /\ b e. ran L ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` G ) ) } C_ ( ran L X. ran L ) )
29	26, 28	ssexd 4805	. . . . 5 |- ( ph -> { <. a , b >. | ( ( a e. ran L /\ b e. ran L ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` G ) ) } e. _V )
30	3, 17, 20, 29	fvmptd 6288	. . . 4 |- ( ph -> (⟂G ` G ) = { <. a , b >. | ( ( a e. ran L /\ b e. ran L ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` G ) ) } )
31	30	eleq2d 2687	. . 3 |- ( ph -> ( <. A , B >. e. (⟂G ` G ) <-> <. A , B >. e. { <. a , b >. | ( ( a e. ran L /\ b e. ran L ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` G ) ) } ) )
32	1, 31	syl5bb 272	. 2 |- ( ph -> ( A (⟂G ` G ) B <-> <. A , B >. e. { <. a , b >. | ( ( a e. ran L /\ b e. ran L ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` G ) ) } ) )
33		isperp.a	. . 3 |- ( ph -> A e. ran L )
34		isperp.b	. . 3 |- ( ph -> B e. ran L )
35		ineq12 3809	. . . . 5 |- ( ( a = A /\ b = B ) -> ( a i^i b ) = ( A i^i B ) )
36		simpll 790	. . . . . 6 |- ( ( ( a = A /\ b = B ) /\ x e. ( a i^i b ) ) -> a = A )
37		simpllr 799	. . . . . . 7 |- ( ( ( ( a = A /\ b = B ) /\ x e. ( a i^i b ) ) /\ u e. a ) -> b = B )
38	37	raleqdv 3144	. . . . . 6 |- ( ( ( ( a = A /\ b = B ) /\ x e. ( a i^i b ) ) /\ u e. a ) -> ( A. v e. b <" u x v "> e. (∟G ` G ) <-> A. v e. B <" u x v "> e. (∟G ` G ) ) )
39	36, 38	raleqbidva 3154	. . . . 5 |- ( ( ( a = A /\ b = B ) /\ x e. ( a i^i b ) ) -> ( A. u e. a A. v e. b <" u x v "> e. (∟G ` G ) <-> A. u e. A A. v e. B <" u x v "> e. (∟G ` G ) ) )
40	35, 39	rexeqbidva 3155	. . . 4 |- ( ( a = A /\ b = B ) -> ( E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` G ) <-> E. x e. ( A i^i B ) A. u e. A A. v e. B <" u x v "> e. (∟G ` G ) ) )
41	40	opelopab2a 4990	. . 3 |- ( ( A e. ran L /\ B e. ran L ) -> ( <. A , B >. e. { <. a , b >. | ( ( a e. ran L /\ b e. ran L ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` G ) ) } <-> E. x e. ( A i^i B ) A. u e. A A. v e. B <" u x v "> e. (∟G ` G ) ) )
42	33, 34, 41	syl2anc 693	. 2 |- ( ph -> ( <. A , B >. e. { <. a , b >. | ( ( a e. ran L /\ b e. ran L ) /\ E. x e. ( a i^i b ) A. u e. a A. v e. b <" u x v "> e. (∟G ` G ) ) } <-> E. x e. ( A i^i B ) A. u e. A A. v e. B <" u x v "> e. (∟G ` G ) ) )
43	32, 42	bitrd 268	1 |- ( ph -> ( A (⟂G ` G ) B <-> E. x e. ( A i^i B ) A. u e. A A. v e. B <" u x v "> e. (∟G ` G ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   <.cop 4183   class class class wbr 4653   {copab 4712   |-> cmpt 4729   X. cxp 5112   ran crn 5115   ` cfv 5888   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  ∟Gcrag 25588  ⟂Gcperpg 25590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-perpg 25591

This theorem is referenced by:  perpcom  25608  perpneq  25609  isperp2  25610