Theorem opphllem6 25644

Description: First part of Lemma 9.4 of [Schwabhauser] p. 68. (Contributed by Thierry Arnoux, 3-Mar-2020.)

Hypotheses

Ref	Expression
hpg.p	|- P = ( Base ` G )
hpg.d	|- .- = ( dist ` G )
hpg.i	|- I = (Itv ` G )
hpg.o	|- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
opphl.l	|- L = (LineG ` G )
opphl.d	|- ( ph -> D e. ran L )
opphl.g	|- ( ph -> G e. TarskiG )
opphl.k	|- K = (hlG ` G )
opphllem5.n	|- N = ( (pInvG ` G ) ` M )
opphllem5.a	|- ( ph -> A e. P )
opphllem5.c	|- ( ph -> C e. P )
opphllem5.r	|- ( ph -> R e. D )
opphllem5.s	|- ( ph -> S e. D )
opphllem5.m	|- ( ph -> M e. P )
opphllem5.o	|- ( ph -> A O C )
opphllem5.p	|- ( ph -> D (⟂G ` G ) ( A L R ) )
opphllem5.q	|- ( ph -> D (⟂G ` G ) ( C L S ) )
opphllem5.u	|- ( ph -> U e. P )
opphllem6.v	|- ( ph -> ( N ` R ) = S )

Assertion

Ref	Expression
opphllem6	|- ( ph -> ( U ( K ` R ) A <-> ( N ` U ) ( K ` S ) C ) )

Distinct variable groups:   D, a, b   I, a, b   P, a, b   t, A   t, D   t, R   t, C   t, G   t, L   t, U   t, I   t, K   t, M   t, O   t, N   t, P   t, S   ph, t   t, .-   t, a, b

Allowed substitution hints:   ph( a, b)   A( a, b)   C( a, b)   R( a, b)   S( a, b)   U( a, b)   G( a, b)   K( a, b)   L( a, b)   M( a, b)   .- ( a, b)   N( a, b)   O( a, b)

Proof of Theorem opphllem6

Step	Hyp	Ref	Expression
1		hpg.p	. . . 4 |- P = ( Base ` G )
2		hpg.d	. . . 4 |- .- = ( dist ` G )
3		hpg.i	. . . 4 |- I = (Itv ` G )
4		opphl.l	. . . 4 |- L = (LineG ` G )
5		eqid 2622	. . . 4 |- (pInvG ` G ) = (pInvG ` G )
6		opphl.g	. . . . 5 |- ( ph -> G e. TarskiG )
7	6	adantr 481	. . . 4 |- ( ( ph /\ R = S ) -> G e. TarskiG )
8		opphllem5.n	. . . 4 |- N = ( (pInvG ` G ) ` M )
9		opphl.k	. . . 4 |- K = (hlG ` G )
10		opphllem5.m	. . . . 5 |- ( ph -> M e. P )
11	10	adantr 481	. . . 4 |- ( ( ph /\ R = S ) -> M e. P )
12		opphllem5.a	. . . . 5 |- ( ph -> A e. P )
13	12	adantr 481	. . . 4 |- ( ( ph /\ R = S ) -> A e. P )
14		opphllem5.c	. . . . 5 |- ( ph -> C e. P )
15	14	adantr 481	. . . 4 |- ( ( ph /\ R = S ) -> C e. P )
16		opphllem5.u	. . . . 5 |- ( ph -> U e. P )
17	16	adantr 481	. . . 4 |- ( ( ph /\ R = S ) -> U e. P )
18		opphl.d	. . . . . . . 8 |- ( ph -> D e. ran L )
19		opphllem5.r	. . . . . . . 8 |- ( ph -> R e. D )
20	1, 4, 3, 6, 18, 19	tglnpt 25444	. . . . . . 7 |- ( ph -> R e. P )
21		opphllem5.p	. . . . . . . 8 |- ( ph -> D (⟂G ` G ) ( A L R ) )
22	4, 6, 21	perpln2 25606	. . . . . . 7 |- ( ph -> ( A L R ) e. ran L )
23	1, 3, 4, 6, 12, 20, 22	tglnne 25523	. . . . . 6 |- ( ph -> A =/= R )
24	23	adantr 481	. . . . 5 |- ( ( ph /\ R = S ) -> A =/= R )
25		opphllem6.v	. . . . . . . 8 |- ( ph -> ( N ` R ) = S )
26	25	adantr 481	. . . . . . 7 |- ( ( ph /\ R = S ) -> ( N ` R ) = S )
27		simpr 477	. . . . . . 7 |- ( ( ph /\ R = S ) -> R = S )
28	26, 27	eqtr4d 2659	. . . . . 6 |- ( ( ph /\ R = S ) -> ( N ` R ) = R )
29	1, 2, 3, 4, 5, 6, 10, 8, 20	mirinv 25561	. . . . . . 7 |- ( ph -> ( ( N ` R ) = R <-> M = R ) )
30	29	adantr 481	. . . . . 6 |- ( ( ph /\ R = S ) -> ( ( N ` R ) = R <-> M = R ) )
31	28, 30	mpbid 222	. . . . 5 |- ( ( ph /\ R = S ) -> M = R )
32	24, 31	neeqtrrd 2868	. . . 4 |- ( ( ph /\ R = S ) -> A =/= M )
33		opphllem5.s	. . . . . . . 8 |- ( ph -> S e. D )
34	1, 4, 3, 6, 18, 33	tglnpt 25444	. . . . . . 7 |- ( ph -> S e. P )
35		opphllem5.q	. . . . . . . 8 |- ( ph -> D (⟂G ` G ) ( C L S ) )
36	4, 6, 35	perpln2 25606	. . . . . . 7 |- ( ph -> ( C L S ) e. ran L )
37	1, 3, 4, 6, 14, 34, 36	tglnne 25523	. . . . . 6 |- ( ph -> C =/= S )
38	37	adantr 481	. . . . 5 |- ( ( ph /\ R = S ) -> C =/= S )
39	31, 27	eqtrd 2656	. . . . 5 |- ( ( ph /\ R = S ) -> M = S )
40	38, 39	neeqtrrd 2868	. . . 4 |- ( ( ph /\ R = S ) -> C =/= M )
41		simpr 477	. . . . . . . 8 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R = t ) -> R = t )
42	6	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> G e. TarskiG )
43	42	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> G e. TarskiG )
44	14	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> C e. P )
45	44	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> C e. P )
46	20	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. P )
47	46	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> R e. P )
48	18	ad3antrrr 766	. . . . . . . . . . 11 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> D e. ran L )
49		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> t e. D )
50	1, 4, 3, 42, 48, 49	tglnpt 25444	. . . . . . . . . 10 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> t e. P )
51	50	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> t e. P )
52	12	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> A e. P )
53	52	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> A e. P )
54	34	ad3antrrr 766	. . . . . . . . . . 11 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> S e. P )
55	54	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> S e. P )
56		simpllr 799	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R = S )
57	1, 3, 4, 6, 14, 34, 37	tglinerflx2 25529	. . . . . . . . . . . . 13 |- ( ph -> S e. ( C L S ) )
58	57	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> S e. ( C L S ) )
59	56, 58	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. ( C L S ) )
60	59	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> R e. ( C L S ) )
61	1, 3, 4, 6, 14, 34, 37	tgelrnln 25525	. . . . . . . . . . . . 13 |- ( ph -> ( C L S ) e. ran L )
62	1, 2, 3, 4, 6, 18, 61, 35	perpcom 25608	. . . . . . . . . . . 12 |- ( ph -> ( C L S ) (⟂G ` G ) D )
63	62	ad4antr 768	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> ( C L S ) (⟂G ` G ) D )
64		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> R =/= t )
65	48	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> D e. ran L )
66	19	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. D )
67	66	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> R e. D )
68	49	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> t e. D )
69	1, 3, 4, 43, 47, 51, 64, 64, 65, 67, 68	tglinethru 25531	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> D = ( R L t ) )
70	63, 69	breqtrd 4679	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> ( C L S ) (⟂G ` G ) ( R L t ) )
71	1, 2, 3, 4, 43, 45, 55, 60, 51, 70	perprag 25618	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> <" C R t "> e. (∟G ` G ) )
72	1, 3, 4, 6, 12, 20, 23	tglinerflx2 25529	. . . . . . . . . . . 12 |- ( ph -> R e. ( A L R ) )
73	72	ad3antrrr 766	. . . . . . . . . . 11 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. ( A L R ) )
74	73	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> R e. ( A L R ) )
75	1, 3, 4, 6, 12, 20, 23	tgelrnln 25525	. . . . . . . . . . . . 13 |- ( ph -> ( A L R ) e. ran L )
76	1, 2, 3, 4, 6, 18, 75, 21	perpcom 25608	. . . . . . . . . . . 12 |- ( ph -> ( A L R ) (⟂G ` G ) D )
77	76	ad4antr 768	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> ( A L R ) (⟂G ` G ) D )
78	77, 69	breqtrd 4679	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> ( A L R ) (⟂G ` G ) ( R L t ) )
79	1, 2, 3, 4, 43, 53, 47, 74, 51, 78	perprag 25618	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> <" A R t "> e. (∟G ` G ) )
80		simplr 792	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> t e. ( A I C ) )
81	1, 2, 3, 43, 53, 51, 45, 80	tgbtwncom 25383	. . . . . . . . 9 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> t e. ( C I A ) )
82	1, 2, 3, 4, 5, 43, 45, 47, 51, 53, 71, 79, 81	ragflat2 25598	. . . . . . . 8 |- ( ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) /\ R =/= t ) -> R = t )
83	41, 82	pm2.61dane 2881	. . . . . . 7 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R = t )
84		simpr 477	. . . . . . 7 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> t e. ( A I C ) )
85	83, 84	eqeltrd 2701	. . . . . 6 |- ( ( ( ( ph /\ R = S ) /\ t e. D ) /\ t e. ( A I C ) ) -> R e. ( A I C ) )
86		opphllem5.o	. . . . . . . . 9 |- ( ph -> A O C )
87		hpg.o	. . . . . . . . . 10 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
88	1, 2, 3, 87, 12, 14	islnopp 25631	. . . . . . . . 9 |- ( ph -> ( A O C <-> ( ( -. A e. D /\ -. C e. D ) /\ E. t e. D t e. ( A I C ) ) ) )
89	86, 88	mpbid 222	. . . . . . . 8 |- ( ph -> ( ( -. A e. D /\ -. C e. D ) /\ E. t e. D t e. ( A I C ) ) )
90	89	simprd 479	. . . . . . 7 |- ( ph -> E. t e. D t e. ( A I C ) )
91	90	adantr 481	. . . . . 6 |- ( ( ph /\ R = S ) -> E. t e. D t e. ( A I C ) )
92	85, 91	r19.29a 3078	. . . . 5 |- ( ( ph /\ R = S ) -> R e. ( A I C ) )
93	31, 92	eqeltrd 2701	. . . 4 |- ( ( ph /\ R = S ) -> M e. ( A I C ) )
94	1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 17, 32, 40, 93	mirbtwnhl 25575	. . 3 |- ( ( ph /\ R = S ) -> ( U ( K ` M ) A <-> ( N ` U ) ( K ` M ) C ) )
95	31	fveq2d 6195	. . . 4 |- ( ( ph /\ R = S ) -> ( K ` M ) = ( K ` R ) )
96	95	breqd 4664	. . 3 |- ( ( ph /\ R = S ) -> ( U ( K ` M ) A <-> U ( K ` R ) A ) )
97	39	fveq2d 6195	. . . 4 |- ( ( ph /\ R = S ) -> ( K ` M ) = ( K ` S ) )
98	97	breqd 4664	. . 3 |- ( ( ph /\ R = S ) -> ( ( N ` U ) ( K ` M ) C <-> ( N ` U ) ( K ` S ) C ) )
99	94, 96, 98	3bitr3d 298	. 2 |- ( ( ph /\ R = S ) -> ( U ( K ` R ) A <-> ( N ` U ) ( K ` S ) C ) )
100	18	ad2antrr 762	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> D e. ran L )
101	6	ad2antrr 762	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> G e. TarskiG )
102	12	ad2antrr 762	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> A e. P )
103	14	ad2antrr 762	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> C e. P )
104	19	ad2antrr 762	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> R e. D )
105	33	ad2antrr 762	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> S e. D )
106	10	ad2antrr 762	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> M e. P )
107	86	ad2antrr 762	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> A O C )
108	21	ad2antrr 762	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> D (⟂G ` G ) ( A L R ) )
109	35	ad2antrr 762	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> D (⟂G ` G ) ( C L S ) )
110		simpr 477	. . . . 5 |- ( ( ph /\ R =/= S ) -> R =/= S )
111	110	adantr 481	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> R =/= S )
112		simpr 477	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> ( S .- C ) (≤G ` G ) ( R .- A ) )
113	16	ad2antrr 762	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> U e. P )
114	25	ad2antrr 762	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> ( N ` R ) = S )
115	1, 2, 3, 87, 4, 100, 101, 9, 8, 102, 103, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114	opphllem3 25641	. . 3 |- ( ( ( ph /\ R =/= S ) /\ ( S .- C ) (≤G ` G ) ( R .- A ) ) -> ( U ( K ` R ) A <-> ( N ` U ) ( K ` S ) C ) )
116	18	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> D e. ran L )
117	6	adantr 481	. . . . . 6 |- ( ( ph /\ R =/= S ) -> G e. TarskiG )
118	117	adantr 481	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> G e. TarskiG )
119	14	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> C e. P )
120	12	adantr 481	. . . . . 6 |- ( ( ph /\ R =/= S ) -> A e. P )
121	120	adantr 481	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> A e. P )
122	33	adantr 481	. . . . . 6 |- ( ( ph /\ R =/= S ) -> S e. D )
123	122	adantr 481	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> S e. D )
124	19	adantr 481	. . . . . 6 |- ( ( ph /\ R =/= S ) -> R e. D )
125	124	adantr 481	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> R e. D )
126	10	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> M e. P )
127	86	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> A O C )
128	1, 2, 3, 87, 4, 116, 118, 121, 119, 127	oppcom 25636	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> C O A )
129	35	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> D (⟂G ` G ) ( C L S ) )
130	21	adantr 481	. . . . . 6 |- ( ( ph /\ R =/= S ) -> D (⟂G ` G ) ( A L R ) )
131	130	adantr 481	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> D (⟂G ` G ) ( A L R ) )
132	110	necomd 2849	. . . . . 6 |- ( ( ph /\ R =/= S ) -> S =/= R )
133	132	adantr 481	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> S =/= R )
134		simpr 477	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> ( R .- A ) (≤G ` G ) ( S .- C ) )
135	16	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> U e. P )
136	1, 2, 3, 4, 5, 118, 126, 8, 135	mircl 25556	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> ( N ` U ) e. P )
137	20	adantr 481	. . . . . . 7 |- ( ( ph /\ R =/= S ) -> R e. P )
138	137	adantr 481	. . . . . 6 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> R e. P )
139	25	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> ( N ` R ) = S )
140	1, 2, 3, 4, 5, 118, 126, 8, 138, 139	mircom 25558	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> ( N ` S ) = R )
141	1, 2, 3, 87, 4, 116, 118, 9, 8, 119, 121, 123, 125, 126, 128, 129, 131, 133, 134, 136, 140	opphllem3 25641	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> ( ( N ` U ) ( K ` S ) C <-> ( N ` ( N ` U ) ) ( K ` R ) A ) )
142	1, 2, 3, 4, 5, 118, 126, 8, 135	mirmir 25557	. . . . 5 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> ( N ` ( N ` U ) ) = U )
143	142	breq1d 4663	. . . 4 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> ( ( N ` ( N ` U ) ) ( K ` R ) A <-> U ( K ` R ) A ) )
144	141, 143	bitr2d 269	. . 3 |- ( ( ( ph /\ R =/= S ) /\ ( R .- A ) (≤G ` G ) ( S .- C ) ) -> ( U ( K ` R ) A <-> ( N ` U ) ( K ` S ) C ) )
145		eqid 2622	. . . . 5 |- (≤G ` G ) = (≤G ` G )
146	1, 2, 3, 145, 6, 34, 14, 20, 12	legtrid 25486	. . . 4 |- ( ph -> ( ( S .- C ) (≤G ` G ) ( R .- A ) \/ ( R .- A ) (≤G ` G ) ( S .- C ) ) )
147	146	adantr 481	. . 3 |- ( ( ph /\ R =/= S ) -> ( ( S .- C ) (≤G ` G ) ( R .- A ) \/ ( R .- A ) (≤G ` G ) ( S .- C ) ) )
148	115, 144, 147	mpjaodan 827	. 2 |- ( ( ph /\ R =/= S ) -> ( U ( K ` R ) A <-> ( N ` U ) ( K ` S ) C ) )
149	99, 148	pm2.61dane 2881	1 |- ( ph -> ( U ( K ` R ) A <-> ( N ` U ) ( K ` S ) C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   class class class wbr 4653   {copab 4712   ran crn 5115   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  ≤Gcleg 25477  hlGchlg 25495  pInvGcmir 25547  ⟂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-leg 25478  df-hlg 25496  df-mir 25548  df-rag 25589  df-perpg 25591

This theorem is referenced by:  opphl  25646