Theorem perpln1 25605

Description: Derive a line from perpendicularity. (Contributed by Thierry Arnoux, 27-Nov-2019.)

Hypotheses

Ref	Expression
perpln.l	|- L = (LineG ` G )
perpln.1	|- ( ph -> G e. TarskiG )
perpln.2	|- ( ph -> A (⟂G ` G ) B )

Assertion

Ref	Expression
perpln1	|- ( ph -> A e. ran L )

Proof of Theorem perpln1

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

Step	Hyp	Ref	Expression
1		df-perpg 25591	. . . . . . 7 |- ⟂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 ) ) } )
2	1	a1i 11	. . . . . 6 |- ( 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 ) ) } ) )
3		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ g = G ) -> g = G )
4	3	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ph /\ g = G ) -> (LineG ` g ) = (LineG ` G ) )
5		perpln.l	. . . . . . . . . . . 12 |- L = (LineG ` G )
6	4, 5	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( ph /\ g = G ) -> (LineG ` g ) = L )
7	6	rneqd 5353	. . . . . . . . . 10 |- ( ( ph /\ g = G ) -> ran (LineG ` g ) = ran L )
8	7	eleq2d 2687	. . . . . . . . 9 |- ( ( ph /\ g = G ) -> ( a e. ran (LineG ` g ) <-> a e. ran L ) )
9	7	eleq2d 2687	. . . . . . . . 9 |- ( ( ph /\ g = G ) -> ( b e. ran (LineG ` g ) <-> b e. ran L ) )
10	8, 9	anbi12d 747	. . . . . . . 8 |- ( ( ph /\ g = G ) -> ( ( a e. ran (LineG ` g ) /\ b e. ran (LineG ` g ) ) <-> ( a e. ran L /\ b e. ran L ) ) )
11	3	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ph /\ g = G ) -> (∟G ` g ) = (∟G ` G ) )
12	11	eleq2d 2687	. . . . . . . . . 10 |- ( ( ph /\ g = G ) -> ( <" u x v "> e. (∟G ` g ) <-> <" u x v "> e. (∟G ` G ) ) )
13	12	ralbidv 2986	. . . . . . . . 9 |- ( ( ph /\ g = G ) -> ( A. v e. b <" u x v "> e. (∟G ` g ) <-> A. v e. b <" u x v "> e. (∟G ` G ) ) )
14	13	rexralbidv 3058	. . . . . . . 8 |- ( ( 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 ) ) )
15	10, 14	anbi12d 747	. . . . . . 7 |- ( ( 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 ) ) ) )
16	15	opabbidv 4716	. . . . . 6 |- ( ( 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 ) ) } )
17		perpln.1	. . . . . . 7 |- ( ph -> G e. TarskiG )
18		elex 3212	. . . . . . 7 |- ( G e. TarskiG -> G e. _V )
19	17, 18	syl 17	. . . . . 6 |- ( ph -> G e. _V )
20		fvex 6201	. . . . . . . . . 10 |- (LineG ` G ) e. _V
21	5, 20	eqeltri 2697	. . . . . . . . 9 |- L e. _V
22		rnexg 7098	. . . . . . . . 9 |- ( L e. _V -> ran L e. _V )
23	21, 22	mp1i 13	. . . . . . . 8 |- ( ph -> ran L e. _V )
24		xpexg 6960	. . . . . . . 8 |- ( ( ran L e. _V /\ ran L e. _V ) -> ( ran L X. ran L ) e. _V )
25	23, 23, 24	syl2anc 693	. . . . . . 7 |- ( ph -> ( ran L X. ran L ) e. _V )
26		opabssxp 5193	. . . . . . . 8 |- { <. 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 )
27	26	a1i 11	. . . . . . 7 |- ( 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 ) )
28	25, 27	ssexd 4805	. . . . . 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 ) ) } e. _V )
29	2, 16, 19, 28	fvmptd 6288	. . . . 5 |- ( 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 ) ) } )
30		anass 681	. . . . . 6 |- ( ( ( 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 ) ) <-> ( 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	opabbii 4717	. . . . 5 |- { <. 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 ) ) } = { <. 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	29, 31	syl6eq 2672	. . . 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 ) ) ) } )
33	32	dmeqd 5326	. . 3 |- ( ph -> dom (⟂G ` G ) = dom { <. 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 ) ) ) } )
34		dmopabss 5336	. . 3 |- dom { <. 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
35	33, 34	syl6eqss 3655	. 2 |- ( ph -> dom (⟂G ` G ) C_ ran L )
36		relopab 5247	. . . . . 6 |- Rel { <. 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 ) ) }
37	29	releqd 5203	. . . . . 6 |- ( ph -> ( Rel (⟂G ` G ) <-> Rel { <. 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 ) ) } ) )
38	36, 37	mpbiri 248	. . . . 5 |- ( ph -> Rel (⟂G ` G ) )
39		perpln.2	. . . . 5 |- ( ph -> A (⟂G ` G ) B )
40		brrelex12 5155	. . . . 5 |- ( ( Rel (⟂G ` G ) /\ A (⟂G ` G ) B ) -> ( A e. _V /\ B e. _V ) )
41	38, 39, 40	syl2anc 693	. . . 4 |- ( ph -> ( A e. _V /\ B e. _V ) )
42	41	simpld 475	. . 3 |- ( ph -> A e. _V )
43	41	simprd 479	. . 3 |- ( ph -> B e. _V )
44		breldmg 5330	. . 3 |- ( ( A e. _V /\ B e. _V /\ A (⟂G ` G ) B ) -> A e. dom (⟂G ` G ) )
45	42, 43, 39, 44	syl3anc 1326	. 2 |- ( ph -> A e. dom (⟂G ` G ) )
46	35, 45	sseldd 3604	1 |- ( ph -> A e. ran L )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   {copab 4712   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   Rel wrel 5119   ` cfv 5888   <"cs3 13587  TarskiGcstrkg 25329  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:  footne  25615  footeq  25616  perpdragALT  25619  perpdrag  25620  colperp  25621  midex  25629  opphl  25646  lmieu  25676  lnperpex  25695  trgcopy  25696