Theorem perpneq 25609

Description: Two perpendicular lines are different. Theorem 8.14 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 18-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 )
perpcom.1	|- ( ph -> A (⟂G ` G ) B )

Assertion

Ref	Expression
perpneq	|- ( ph -> A =/= B )

Proof of Theorem perpneq

Dummy variables u v x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isperp.p	. . . . . . 7 |- P = ( Base ` G )
2		isperp.i	. . . . . . 7 |- I = (Itv ` G )
3		isperp.l	. . . . . . 7 |- L = (LineG ` G )
4		isperp.g	. . . . . . . . 9 |- ( ph -> G e. TarskiG )
5	4	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( A i^i B ) ) -> G e. TarskiG )
6	5	ad5antr 770	. . . . . . 7 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> G e. TarskiG )
7	4	ad5antr 770	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> G e. TarskiG )
8		isperp.a	. . . . . . . . . 10 |- ( ph -> A e. ran L )
9	8	ad5antr 770	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> A e. ran L )
10		inss1 3833	. . . . . . . . . . 11 |- ( A i^i B ) C_ A
11		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A i^i B ) ) -> x e. ( A i^i B ) )
12	10, 11	sseldi 3601	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A i^i B ) ) -> x e. A )
13	12	ad4antr 768	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> x e. A )
14	1, 3, 2, 7, 9, 13	tglnpt 25444	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> x e. P )
15	14	adantl4r 787	. . . . . . 7 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> x e. P )
16		isperp.b	. . . . . . . . . 10 |- ( ph -> B e. ran L )
17	16	ad5antr 770	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> B e. ran L )
18		simplr 792	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> v e. B )
19	1, 3, 2, 7, 17, 18	tglnpt 25444	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> v e. P )
20	19	adantl4r 787	. . . . . . 7 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> v e. P )
21		simp-4r 807	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> u e. A )
22	1, 3, 2, 7, 9, 21	tglnpt 25444	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> u e. P )
23	22	adantl4r 787	. . . . . . 7 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> u e. P )
24		isperp.d	. . . . . . . . 9 |- .- = ( dist ` G )
25		eqid 2622	. . . . . . . . 9 |- (pInvG ` G ) = (pInvG ` G )
26		simp-4r 807	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> u e. A )
27		simplr 792	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> v e. B )
28		simp-5r 809	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) )
29		id 22	. . . . . . . . . . . . 13 |- ( y = u -> y = u )
30		eqidd 2623	. . . . . . . . . . . . 13 |- ( y = u -> x = x )
31		eqidd 2623	. . . . . . . . . . . . 13 |- ( y = u -> z = z )
32	29, 30, 31	s3eqd 13609	. . . . . . . . . . . 12 |- ( y = u -> <" y x z "> = <" u x z "> )
33	32	eleq1d 2686	. . . . . . . . . . 11 |- ( y = u -> ( <" y x z "> e. (∟G ` G ) <-> <" u x z "> e. (∟G ` G ) ) )
34		eqidd 2623	. . . . . . . . . . . . 13 |- ( z = v -> u = u )
35		eqidd 2623	. . . . . . . . . . . . 13 |- ( z = v -> x = x )
36		id 22	. . . . . . . . . . . . 13 |- ( z = v -> z = v )
37	34, 35, 36	s3eqd 13609	. . . . . . . . . . . 12 |- ( z = v -> <" u x z "> = <" u x v "> )
38	37	eleq1d 2686	. . . . . . . . . . 11 |- ( z = v -> ( <" u x z "> e. (∟G ` G ) <-> <" u x v "> e. (∟G ` G ) ) )
39	33, 38	rspc2va 3323	. . . . . . . . . 10 |- ( ( ( u e. A /\ v e. B ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) -> <" u x v "> e. (∟G ` G ) )
40	26, 27, 28, 39	syl21anc 1325	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> <" u x v "> e. (∟G ` G ) )
41		simpllr 799	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> x =/= u )
42	41	necomd 2849	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> u =/= x )
43	42	adantl4r 787	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> u =/= x )
44		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> x =/= v )
45	44	necomd 2849	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> v =/= x )
46	45	adantl4r 787	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> v =/= x )
47	1, 24, 2, 3, 25, 6, 23, 15, 20, 40, 43, 46	ragncol 25604	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> -. ( v e. ( u L x ) \/ u = x ) )
48	1, 3, 2, 6, 23, 15, 20, 47	ncolrot2 25458	. . . . . . 7 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> -. ( x e. ( v L u ) \/ v = u ) )
49	1, 2, 3, 6, 15, 20, 23, 15, 48	tglineneq 25539	. . . . . 6 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> ( x L v ) =/= ( u L x ) )
50	49	necomd 2849	. . . . 5 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> ( u L x ) =/= ( x L v ) )
51	1, 2, 3, 7, 22, 14, 42, 42, 9, 21, 13	tglinethru 25531	. . . . . 6 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> A = ( u L x ) )
52	51	adantl4r 787	. . . . 5 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> A = ( u L x ) )
53		inss2 3834	. . . . . . . . 9 |- ( A i^i B ) C_ B
54	53, 11	sseldi 3601	. . . . . . . 8 |- ( ( ph /\ x e. ( A i^i B ) ) -> x e. B )
55	54	ad4antr 768	. . . . . . 7 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> x e. B )
56	1, 2, 3, 7, 14, 19, 44, 44, 17, 55, 18	tglinethru 25531	. . . . . 6 |- ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> B = ( x L v ) )
57	56	adantl4r 787	. . . . 5 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> B = ( x L v ) )
58	50, 52, 57	3netr4d 2871	. . . 4 |- ( ( ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) /\ v e. B ) /\ x =/= v ) -> A =/= B )
59	16	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( A i^i B ) ) -> B e. ran L )
60	1, 2, 3, 5, 59, 54	tglnpt2 25536	. . . . 5 |- ( ( ph /\ x e. ( A i^i B ) ) -> E. v e. B x =/= v )
61	60	ad3antrrr 766	. . . 4 |- ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) -> E. v e. B x =/= v )
62	58, 61	r19.29a 3078	. . 3 |- ( ( ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) /\ u e. A ) /\ x =/= u ) -> A =/= B )
63	8	adantr 481	. . . . 5 |- ( ( ph /\ x e. ( A i^i B ) ) -> A e. ran L )
64	1, 2, 3, 5, 63, 12	tglnpt2 25536	. . . 4 |- ( ( ph /\ x e. ( A i^i B ) ) -> E. u e. A x =/= u )
65	64	adantr 481	. . 3 |- ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) -> E. u e. A x =/= u )
66	62, 65	r19.29a 3078	. 2 |- ( ( ( ph /\ x e. ( A i^i B ) ) /\ A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) -> A =/= B )
67		perpcom.1	. . 3 |- ( ph -> A (⟂G ` G ) B )
68	1, 24, 2, 3, 4, 8, 16	isperp 25607	. . 3 |- ( ph -> ( A (⟂G ` G ) B <-> E. x e. ( A i^i B ) A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) ) )
69	67, 68	mpbid 222	. 2 |- ( ph -> E. x e. ( A i^i B ) A. y e. A A. z e. B <" y x z "> e. (∟G ` G ) )
70	66, 69	r19.29a 3078	1 |- ( ph -> A =/= B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   class class class wbr 4653   ran crn 5115   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547  ∟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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406  df-mir 25548  df-rag 25589  df-perpg 25591

This theorem is referenced by:  isperp2  25610  footne  25615  lmieu  25676