Theorem mideulem2 25626

Description: Lemma for opphllem 25627, which is itself used for mideu 25630. (Contributed by Thierry Arnoux, 19-Feb-2020.)

Hypotheses

Ref	Expression
colperpex.p	|- P = ( Base ` G )
colperpex.d	|- .- = ( dist ` G )
colperpex.i	|- I = (Itv ` G )
colperpex.l	|- L = (LineG ` G )
colperpex.g	|- ( ph -> G e. TarskiG )
mideu.s	|- S = (pInvG ` G )
mideu.1	|- ( ph -> A e. P )
mideu.2	|- ( ph -> B e. P )
mideulem.1	|- ( ph -> A =/= B )
mideulem.2	|- ( ph -> Q e. P )
mideulem.3	|- ( ph -> O e. P )
mideulem.4	|- ( ph -> T e. P )
mideulem.5	|- ( ph -> ( A L B ) (⟂G ` G ) ( Q L B ) )
mideulem.6	|- ( ph -> ( A L B ) (⟂G ` G ) ( A L O ) )
mideulem.7	|- ( ph -> T e. ( A L B ) )
mideulem.8	|- ( ph -> T e. ( Q I O ) )
opphllem.1	|- ( ph -> R e. P )
opphllem.2	|- ( ph -> R e. ( B I Q ) )
opphllem.3	|- ( ph -> ( A .- O ) = ( B .- R ) )
mideulem2.1	|- ( ph -> X e. P )
mideulem2.2	|- ( ph -> X e. ( T I B ) )
mideulem2.3	|- ( ph -> X e. ( R I O ) )
mideulem2.4	|- ( ph -> Z e. P )
mideulem2.5	|- ( ph -> X e. ( ( ( S ` A ) ` O ) I Z ) )
mideulem2.6	|- ( ph -> ( X .- Z ) = ( X .- R ) )
mideulem2.7	|- ( ph -> M e. P )
mideulem2.8	|- ( ph -> R = ( ( S ` M ) ` Z ) )

Assertion

Ref	Expression
mideulem2	|- ( ph -> B = M )

Proof of Theorem mideulem2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . 3 |- ( y = B -> ( R L y ) = ( R L B ) )
2	1	breq1d 4663	. 2 |- ( y = B -> ( ( R L y ) (⟂G ` G ) ( A L B ) <-> ( R L B ) (⟂G ` G ) ( A L B ) ) )
3		oveq2 6658	. . 3 |- ( y = M -> ( R L y ) = ( R L M ) )
4	3	breq1d 4663	. 2 |- ( y = M -> ( ( R L y ) (⟂G ` G ) ( A L B ) <-> ( R L M ) (⟂G ` G ) ( A L B ) ) )
5		colperpex.p	. . 3 |- P = ( Base ` G )
6		colperpex.d	. . 3 |- .- = ( dist ` G )
7		colperpex.i	. . 3 |- I = (Itv ` G )
8		colperpex.l	. . 3 |- L = (LineG ` G )
9		colperpex.g	. . 3 |- ( ph -> G e. TarskiG )
10		mideu.1	. . . 4 |- ( ph -> A e. P )
11		mideu.2	. . . 4 |- ( ph -> B e. P )
12		mideulem.1	. . . 4 |- ( ph -> A =/= B )
13	5, 7, 8, 9, 10, 11, 12	tgelrnln 25525	. . 3 |- ( ph -> ( A L B ) e. ran L )
14		opphllem.1	. . 3 |- ( ph -> R e. P )
15	12	adantr 481	. . . . . 6 |- ( ( ph /\ R e. ( A L B ) ) -> A =/= B )
16	15	neneqd 2799	. . . . 5 |- ( ( ph /\ R e. ( A L B ) ) -> -. A = B )
17		mideulem.3	. . . . . . . . 9 |- ( ph -> O e. P )
18		opphllem.3	. . . . . . . . 9 |- ( ph -> ( A .- O ) = ( B .- R ) )
19		mideulem.6	. . . . . . . . . . 11 |- ( ph -> ( A L B ) (⟂G ` G ) ( A L O ) )
20	8, 9, 19	perpln2 25606	. . . . . . . . . 10 |- ( ph -> ( A L O ) e. ran L )
21	5, 7, 8, 9, 10, 17, 20	tglnne 25523	. . . . . . . . 9 |- ( ph -> A =/= O )
22	5, 6, 7, 9, 10, 17, 11, 14, 18, 21	tgcgrneq 25378	. . . . . . . 8 |- ( ph -> B =/= R )
23	22	adantr 481	. . . . . . 7 |- ( ( ph /\ R e. ( A L B ) ) -> B =/= R )
24	23	necomd 2849	. . . . . 6 |- ( ( ph /\ R e. ( A L B ) ) -> R =/= B )
25	24	neneqd 2799	. . . . 5 |- ( ( ph /\ R e. ( A L B ) ) -> -. R = B )
26	16, 25	jca 554	. . . 4 |- ( ( ph /\ R e. ( A L B ) ) -> ( -. A = B /\ -. R = B ) )
27		mideu.s	. . . . . 6 |- S = (pInvG ` G )
28	9	adantr 481	. . . . . 6 |- ( ( ph /\ R e. ( A L B ) ) -> G e. TarskiG )
29	10	adantr 481	. . . . . 6 |- ( ( ph /\ R e. ( A L B ) ) -> A e. P )
30	11	adantr 481	. . . . . 6 |- ( ( ph /\ R e. ( A L B ) ) -> B e. P )
31	14	adantr 481	. . . . . 6 |- ( ( ph /\ R e. ( A L B ) ) -> R e. P )
32		mideulem.2	. . . . . . . . 9 |- ( ph -> Q e. P )
33		mideulem.5	. . . . . . . . . . . . 13 |- ( ph -> ( A L B ) (⟂G ` G ) ( Q L B ) )
34	8, 9, 33	perpln2 25606	. . . . . . . . . . . 12 |- ( ph -> ( Q L B ) e. ran L )
35	5, 7, 8, 9, 32, 11, 34	tglnne 25523	. . . . . . . . . . 11 |- ( ph -> Q =/= B )
36	5, 7, 8, 9, 32, 11, 35	tglinerflx2 25529	. . . . . . . . . 10 |- ( ph -> B e. ( Q L B ) )
37	5, 6, 7, 8, 9, 13, 34, 33	perpcom 25608	. . . . . . . . . . 11 |- ( ph -> ( Q L B ) (⟂G ` G ) ( A L B ) )
38	5, 7, 8, 9, 10, 11, 12	tglinecom 25530	. . . . . . . . . . 11 |- ( ph -> ( A L B ) = ( B L A ) )
39	37, 38	breqtrd 4679	. . . . . . . . . 10 |- ( ph -> ( Q L B ) (⟂G ` G ) ( B L A ) )
40	5, 6, 7, 8, 9, 32, 11, 36, 10, 39	perprag 25618	. . . . . . . . 9 |- ( ph -> <" Q B A "> e. (∟G ` G ) )
41		opphllem.2	. . . . . . . . . 10 |- ( ph -> R e. ( B I Q ) )
42	5, 8, 7, 9, 11, 14, 32, 41	btwncolg3 25452	. . . . . . . . 9 |- ( ph -> ( Q e. ( B L R ) \/ B = R ) )
43	5, 6, 7, 8, 27, 9, 32, 11, 10, 14, 40, 35, 42	ragcol 25594	. . . . . . . 8 |- ( ph -> <" R B A "> e. (∟G ` G ) )
44	5, 6, 7, 8, 27, 9, 14, 11, 10, 43	ragcom 25593	. . . . . . 7 |- ( ph -> <" A B R "> e. (∟G ` G ) )
45	44	adantr 481	. . . . . 6 |- ( ( ph /\ R e. ( A L B ) ) -> <" A B R "> e. (∟G ` G ) )
46		simpr 477	. . . . . . 7 |- ( ( ph /\ R e. ( A L B ) ) -> R e. ( A L B ) )
47	46	orcd 407	. . . . . 6 |- ( ( ph /\ R e. ( A L B ) ) -> ( R e. ( A L B ) \/ A = B ) )
48	5, 6, 7, 8, 27, 28, 29, 30, 31, 45, 47	ragflat3 25601	. . . . 5 |- ( ( ph /\ R e. ( A L B ) ) -> ( A = B \/ R = B ) )
49		oran 517	. . . . 5 |- ( ( A = B \/ R = B ) <-> -. ( -. A = B /\ -. R = B ) )
50	48, 49	sylib 208	. . . 4 |- ( ( ph /\ R e. ( A L B ) ) -> -. ( -. A = B /\ -. R = B ) )
51	26, 50	pm2.65da 600	. . 3 |- ( ph -> -. R e. ( A L B ) )
52	5, 6, 7, 8, 9, 13, 14, 51	foot 25614	. 2 |- ( ph -> E! y e. ( A L B ) ( R L y ) (⟂G ` G ) ( A L B ) )
53	5, 7, 8, 9, 10, 11, 12	tglinerflx2 25529	. 2 |- ( ph -> B e. ( A L B ) )
54		mideulem2.1	. . 3 |- ( ph -> X e. P )
55	12	neneqd 2799	. . . . 5 |- ( ph -> -. A = B )
56		oveq2 6658	. . . . . . 7 |- ( y = A -> ( R L y ) = ( R L A ) )
57	56	breq1d 4663	. . . . . 6 |- ( y = A -> ( ( R L y ) (⟂G ` G ) ( A L B ) <-> ( R L A ) (⟂G ` G ) ( A L B ) ) )
58	52	adantr 481	. . . . . 6 |- ( ( ph /\ X = A ) -> E! y e. ( A L B ) ( R L y ) (⟂G ` G ) ( A L B ) )
59	5, 7, 8, 9, 10, 11, 12	tglinerflx1 25528	. . . . . . 7 |- ( ph -> A e. ( A L B ) )
60	59	adantr 481	. . . . . 6 |- ( ( ph /\ X = A ) -> A e. ( A L B ) )
61	53	adantr 481	. . . . . 6 |- ( ( ph /\ X = A ) -> B e. ( A L B ) )
62	9	adantr 481	. . . . . . . . 9 |- ( ( ph /\ X = A ) -> G e. TarskiG )
63	14	adantr 481	. . . . . . . . 9 |- ( ( ph /\ X = A ) -> R e. P )
64	10	adantr 481	. . . . . . . . 9 |- ( ( ph /\ X = A ) -> A e. P )
65	51, 55	jca 554	. . . . . . . . . . . 12 |- ( ph -> ( -. R e. ( A L B ) /\ -. A = B ) )
66		pm4.56 516	. . . . . . . . . . . 12 |- ( ( -. R e. ( A L B ) /\ -. A = B ) <-> -. ( R e. ( A L B ) \/ A = B ) )
67	65, 66	sylib 208	. . . . . . . . . . 11 |- ( ph -> -. ( R e. ( A L B ) \/ A = B ) )
68	5, 7, 8, 9, 14, 10, 11, 67	ncolne1 25520	. . . . . . . . . 10 |- ( ph -> R =/= A )
69	68	adantr 481	. . . . . . . . 9 |- ( ( ph /\ X = A ) -> R =/= A )
70	5, 7, 8, 62, 63, 64, 69	tglinecom 25530	. . . . . . . 8 |- ( ( ph /\ X = A ) -> ( R L A ) = ( A L R ) )
71	69	necomd 2849	. . . . . . . . 9 |- ( ( ph /\ X = A ) -> A =/= R )
72	17	adantr 481	. . . . . . . . 9 |- ( ( ph /\ X = A ) -> O e. P )
73	21	necomd 2849	. . . . . . . . . 10 |- ( ph -> O =/= A )
74	73	adantr 481	. . . . . . . . 9 |- ( ( ph /\ X = A ) -> O =/= A )
75	54	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ X = A ) -> X e. P )
76		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ X = A ) -> X = A )
77	76, 71	eqnetrd 2861	. . . . . . . . . . 11 |- ( ( ph /\ X = A ) -> X =/= R )
78		mideulem2.3	. . . . . . . . . . . . . . . 16 |- ( ph -> X e. ( R I O ) )
79	5, 6, 7, 9, 14, 54, 17, 78	tgbtwncom 25383	. . . . . . . . . . . . . . 15 |- ( ph -> X e. ( O I R ) )
80		mideulem.4	. . . . . . . . . . . . . . . . 17 |- ( ph -> T e. P )
81		mideulem.7	. . . . . . . . . . . . . . . . 17 |- ( ph -> T e. ( A L B ) )
82		mideulem2.2	. . . . . . . . . . . . . . . . 17 |- ( ph -> X e. ( T I B ) )
83	5, 7, 8, 9, 80, 10, 11, 54, 81, 82	coltr3 25543	. . . . . . . . . . . . . . . 16 |- ( ph -> X e. ( A L B ) )
84	12	necomd 2849	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> B =/= A )
85	84	neneqd 2799	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> -. B = A )
86	85	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ O e. ( B L A ) ) -> -. B = A )
87	73	neneqd 2799	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> -. O = A )
88	87	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ O e. ( B L A ) ) -> -. O = A )
89	86, 88	jca 554	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ O e. ( B L A ) ) -> ( -. B = A /\ -. O = A ) )
90	9	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ O e. ( B L A ) ) -> G e. TarskiG )
91	11	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ O e. ( B L A ) ) -> B e. P )
92	10	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ O e. ( B L A ) ) -> A e. P )
93	17	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ O e. ( B L A ) ) -> O e. P )
94	5, 7, 8, 9, 11, 10, 84	tglinerflx2 25529	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> A e. ( B L A ) )
95	38, 19	eqbrtrrd 4677	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> ( B L A ) (⟂G ` G ) ( A L O ) )
96	5, 6, 7, 8, 9, 11, 10, 94, 17, 95	perprag 25618	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> <" B A O "> e. (∟G ` G ) )
97	96	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ O e. ( B L A ) ) -> <" B A O "> e. (∟G ` G ) )
98		simpr 477	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ O e. ( B L A ) ) -> O e. ( B L A ) )
99	98	orcd 407	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ O e. ( B L A ) ) -> ( O e. ( B L A ) \/ B = A ) )
100	5, 6, 7, 8, 27, 90, 91, 92, 93, 97, 99	ragflat3 25601	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ O e. ( B L A ) ) -> ( B = A \/ O = A ) )
101		oran 517	. . . . . . . . . . . . . . . . . . 19 |- ( ( B = A \/ O = A ) <-> -. ( -. B = A /\ -. O = A ) )
102	100, 101	sylib 208	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ O e. ( B L A ) ) -> -. ( -. B = A /\ -. O = A ) )
103	89, 102	pm2.65da 600	. . . . . . . . . . . . . . . . 17 |- ( ph -> -. O e. ( B L A ) )
104	103, 38	neleqtrrd 2723	. . . . . . . . . . . . . . . 16 |- ( ph -> -. O e. ( A L B ) )
105		nelne2 2891	. . . . . . . . . . . . . . . 16 |- ( ( X e. ( A L B ) /\ -. O e. ( A L B ) ) -> X =/= O )
106	83, 104, 105	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ph -> X =/= O )
107	5, 6, 7, 9, 17, 54, 14, 79, 106	tgbtwnne 25385	. . . . . . . . . . . . . 14 |- ( ph -> O =/= R )
108	107	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ X = A ) -> O =/= R )
109	108	necomd 2849	. . . . . . . . . . . 12 |- ( ( ph /\ X = A ) -> R =/= O )
110	78	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ X = A ) -> X e. ( R I O ) )
111	5, 7, 8, 62, 63, 72, 75, 109, 110	btwnlng1 25514	. . . . . . . . . . 11 |- ( ( ph /\ X = A ) -> X e. ( R L O ) )
112	5, 7, 8, 62, 75, 63, 72, 77, 111, 109	lnrot2 25519	. . . . . . . . . 10 |- ( ( ph /\ X = A ) -> O e. ( X L R ) )
113	76	oveq1d 6665	. . . . . . . . . 10 |- ( ( ph /\ X = A ) -> ( X L R ) = ( A L R ) )
114	112, 113	eleqtrd 2703	. . . . . . . . 9 |- ( ( ph /\ X = A ) -> O e. ( A L R ) )
115	5, 7, 8, 62, 64, 63, 71, 72, 74, 114	tglineelsb2 25527	. . . . . . . 8 |- ( ( ph /\ X = A ) -> ( A L R ) = ( A L O ) )
116	70, 115	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ X = A ) -> ( R L A ) = ( A L O ) )
117	5, 6, 7, 8, 9, 13, 20, 19	perpcom 25608	. . . . . . . 8 |- ( ph -> ( A L O ) (⟂G ` G ) ( A L B ) )
118	117	adantr 481	. . . . . . 7 |- ( ( ph /\ X = A ) -> ( A L O ) (⟂G ` G ) ( A L B ) )
119	116, 118	eqbrtrd 4675	. . . . . 6 |- ( ( ph /\ X = A ) -> ( R L A ) (⟂G ` G ) ( A L B ) )
120	13	adantr 481	. . . . . . 7 |- ( ( ph /\ X = A ) -> ( A L B ) e. ran L )
121	22	necomd 2849	. . . . . . . . 9 |- ( ph -> R =/= B )
122	5, 7, 8, 9, 14, 11, 121	tgelrnln 25525	. . . . . . . 8 |- ( ph -> ( R L B ) e. ran L )
123	122	adantr 481	. . . . . . 7 |- ( ( ph /\ X = A ) -> ( R L B ) e. ran L )
124	5, 7, 8, 9, 14, 11, 121	tglinerflx2 25529	. . . . . . . . . 10 |- ( ph -> B e. ( R L B ) )
125	53, 124	elind 3798	. . . . . . . . 9 |- ( ph -> B e. ( ( A L B ) i^i ( R L B ) ) )
126	5, 7, 8, 9, 14, 11, 121	tglinerflx1 25528	. . . . . . . . 9 |- ( ph -> R e. ( R L B ) )
127	5, 6, 7, 8, 9, 13, 122, 125, 59, 126, 12, 121, 44	ragperp 25612	. . . . . . . 8 |- ( ph -> ( A L B ) (⟂G ` G ) ( R L B ) )
128	127	adantr 481	. . . . . . 7 |- ( ( ph /\ X = A ) -> ( A L B ) (⟂G ` G ) ( R L B ) )
129	5, 6, 7, 8, 62, 120, 123, 128	perpcom 25608	. . . . . 6 |- ( ( ph /\ X = A ) -> ( R L B ) (⟂G ` G ) ( A L B ) )
130	57, 2, 58, 60, 61, 119, 129	reu2eqd 3403	. . . . 5 |- ( ( ph /\ X = A ) -> A = B )
131	55, 130	mtand 691	. . . 4 |- ( ph -> -. X = A )
132	131	neqned 2801	. . 3 |- ( ph -> X =/= A )
133		mideulem2.7	. . 3 |- ( ph -> M e. P )
134	132	necomd 2849	. . . 4 |- ( ph -> A =/= X )
135		eqid 2622	. . . . 5 |- ( S ` A ) = ( S ` A )
136		eqid 2622	. . . . 5 |- ( S ` M ) = ( S ` M )
137	5, 6, 7, 8, 27, 9, 10, 135, 17	mircl 25556	. . . . 5 |- ( ph -> ( ( S ` A ) ` O ) e. P )
138		mideulem2.4	. . . . 5 |- ( ph -> Z e. P )
139		mideulem2.5	. . . . 5 |- ( ph -> X e. ( ( ( S ` A ) ` O ) I Z ) )
140	83	orcd 407	. . . . . . . . 9 |- ( ph -> ( X e. ( A L B ) \/ A = B ) )
141	5, 8, 7, 9, 10, 11, 54, 140	colcom 25453	. . . . . . . 8 |- ( ph -> ( X e. ( B L A ) \/ B = A ) )
142	5, 8, 7, 9, 11, 10, 54, 141	colrot1 25454	. . . . . . 7 |- ( ph -> ( B e. ( A L X ) \/ A = X ) )
143	5, 6, 7, 8, 27, 9, 11, 10, 17, 54, 96, 84, 142	ragcol 25594	. . . . . 6 |- ( ph -> <" X A O "> e. (∟G ` G ) )
144	5, 6, 7, 8, 27, 9, 54, 10, 17	israg 25592	. . . . . 6 |- ( ph -> ( <" X A O "> e. (∟G ` G ) <-> ( X .- O ) = ( X .- ( ( S ` A ) ` O ) ) ) )
145	143, 144	mpbid 222	. . . . 5 |- ( ph -> ( X .- O ) = ( X .- ( ( S ` A ) ` O ) ) )
146		mideulem2.6	. . . . . 6 |- ( ph -> ( X .- Z ) = ( X .- R ) )
147	146	eqcomd 2628	. . . . 5 |- ( ph -> ( X .- R ) = ( X .- Z ) )
148		eqidd 2623	. . . . 5 |- ( ph -> ( ( S ` A ) ` O ) = ( ( S ` A ) ` O ) )
149		mideulem2.8	. . . . . . . 8 |- ( ph -> R = ( ( S ` M ) ` Z ) )
150	149	eqcomd 2628	. . . . . . 7 |- ( ph -> ( ( S ` M ) ` Z ) = R )
151	5, 6, 7, 8, 27, 9, 133, 136, 138, 150	mircom 25558	. . . . . 6 |- ( ph -> ( ( S ` M ) ` R ) = Z )
152	151	eqcomd 2628	. . . . 5 |- ( ph -> Z = ( ( S ` M ) ` R ) )
153	5, 6, 7, 8, 27, 9, 135, 136, 17, 137, 54, 14, 138, 10, 133, 79, 139, 145, 147, 148, 152	krippen 25586	. . . 4 |- ( ph -> X e. ( A I M ) )
154	5, 7, 8, 9, 10, 54, 133, 134, 153	btwnlng3 25516	. . 3 |- ( ph -> M e. ( A L X ) )
155	5, 7, 8, 9, 10, 11, 12, 54, 132, 83, 133, 154	tglineeltr 25526	. 2 |- ( ph -> M e. ( A L B ) )
156	5, 6, 7, 8, 9, 13, 122, 127	perpcom 25608	. 2 |- ( ph -> ( R L B ) (⟂G ` G ) ( A L B ) )
157		nelne2 2891	. . . . . 6 |- ( ( M e. ( A L B ) /\ -. R e. ( A L B ) ) -> M =/= R )
158	155, 51, 157	syl2anc 693	. . . . 5 |- ( ph -> M =/= R )
159	158	necomd 2849	. . . 4 |- ( ph -> R =/= M )
160	5, 7, 8, 9, 14, 133, 159	tgelrnln 25525	. . 3 |- ( ph -> ( R L M ) e. ran L )
161	5, 7, 8, 9, 14, 133, 159	tglinerflx2 25529	. . . . 5 |- ( ph -> M e. ( R L M ) )
162	155, 161	elind 3798	. . . 4 |- ( ph -> M e. ( ( A L B ) i^i ( R L M ) ) )
163	5, 7, 8, 9, 14, 133, 159	tglinerflx1 25528	. . . 4 |- ( ph -> R e. ( R L M ) )
164		simpr 477	. . . . . . . 8 |- ( ( ph /\ M = X ) -> M = X )
165	9	adantr 481	. . . . . . . . 9 |- ( ( ph /\ M = X ) -> G e. TarskiG )
166	133	adantr 481	. . . . . . . . 9 |- ( ( ph /\ M = X ) -> M e. P )
167	10	adantr 481	. . . . . . . . 9 |- ( ( ph /\ M = X ) -> A e. P )
168	17	adantr 481	. . . . . . . . 9 |- ( ( ph /\ M = X ) -> O e. P )
169	137	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ M = X ) -> ( ( S ` A ) ` O ) e. P )
170	145	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ M = X ) -> ( X .- O ) = ( X .- ( ( S ` A ) ` O ) ) )
171	164	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ph /\ M = X ) -> ( M .- O ) = ( X .- O ) )
172	164	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ph /\ M = X ) -> ( M .- ( ( S ` A ) ` O ) ) = ( X .- ( ( S ` A ) ` O ) ) )
173	170, 171, 172	3eqtr4rd 2667	. . . . . . . . . . 11 |- ( ( ph /\ M = X ) -> ( M .- ( ( S ` A ) ` O ) ) = ( M .- O ) )
174	138	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ M = X ) -> Z e. P )
175	14	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ M = X ) -> R e. P )
176	149	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ M = X ) -> R = ( ( S ` M ) ` Z ) )
177	176	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( ( ph /\ M = X ) -> ( M .- R ) = ( M .- ( ( S ` M ) ` Z ) ) )
178	5, 6, 7, 8, 27, 165, 166, 136, 174	mircgr 25552	. . . . . . . . . . . . . . 15 |- ( ( ph /\ M = X ) -> ( M .- ( ( S ` M ) ` Z ) ) = ( M .- Z ) )
179	177, 178	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ph /\ M = X ) -> ( M .- R ) = ( M .- Z ) )
180	5, 6, 7, 165, 166, 175, 166, 174, 179	tgcgrcomlr 25375	. . . . . . . . . . . . 13 |- ( ( ph /\ M = X ) -> ( R .- M ) = ( Z .- M ) )
181	83	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ M = X ) -> X e. ( A L B ) )
182	164, 181	eqeltrd 2701	. . . . . . . . . . . . . . 15 |- ( ( ph /\ M = X ) -> M e. ( A L B ) )
183	51	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ M = X ) -> -. R e. ( A L B ) )
184	182, 183, 157	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ph /\ M = X ) -> M =/= R )
185	184	necomd 2849	. . . . . . . . . . . . 13 |- ( ( ph /\ M = X ) -> R =/= M )
186	5, 6, 7, 165, 175, 166, 174, 166, 180, 185	tgcgrneq 25378	. . . . . . . . . . . 12 |- ( ( ph /\ M = X ) -> Z =/= M )
187	5, 6, 7, 8, 27, 9, 133, 136, 138	mirbtwn 25553	. . . . . . . . . . . . . . 15 |- ( ph -> M e. ( ( ( S ` M ) ` Z ) I Z ) )
188	149	oveq1d 6665	. . . . . . . . . . . . . . 15 |- ( ph -> ( R I Z ) = ( ( ( S ` M ) ` Z ) I Z ) )
189	187, 188	eleqtrrd 2704	. . . . . . . . . . . . . 14 |- ( ph -> M e. ( R I Z ) )
190	189	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ M = X ) -> M e. ( R I Z ) )
191	5, 6, 7, 165, 175, 166, 174, 190	tgbtwncom 25383	. . . . . . . . . . . 12 |- ( ( ph /\ M = X ) -> M e. ( Z I R ) )
192	139	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ M = X ) -> X e. ( ( ( S ` A ) ` O ) I Z ) )
193	164, 192	eqeltrd 2701	. . . . . . . . . . . . 13 |- ( ( ph /\ M = X ) -> M e. ( ( ( S ` A ) ` O ) I Z ) )
194	5, 6, 7, 165, 169, 166, 174, 193	tgbtwncom 25383	. . . . . . . . . . . 12 |- ( ( ph /\ M = X ) -> M e. ( Z I ( ( S ` A ) ` O ) ) )
195	78	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ M = X ) -> X e. ( R I O ) )
196	164, 195	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ( ph /\ M = X ) -> M e. ( R I O ) )
197	5, 7, 165, 174, 166, 175, 169, 168, 186, 185, 191, 194, 196	tgbtwnconn22 25474	. . . . . . . . . . 11 |- ( ( ph /\ M = X ) -> M e. ( ( ( S ` A ) ` O ) I O ) )
198	5, 6, 7, 8, 27, 165, 166, 136, 168, 169, 173, 197	ismir 25554	. . . . . . . . . 10 |- ( ( ph /\ M = X ) -> ( ( S ` A ) ` O ) = ( ( S ` M ) ` O ) )
199	198	eqcomd 2628	. . . . . . . . 9 |- ( ( ph /\ M = X ) -> ( ( S ` M ) ` O ) = ( ( S ` A ) ` O ) )
200	5, 6, 7, 8, 27, 165, 166, 167, 168, 199	miduniq1 25581	. . . . . . . 8 |- ( ( ph /\ M = X ) -> M = A )
201	164, 200	eqtr3d 2658	. . . . . . 7 |- ( ( ph /\ M = X ) -> X = A )
202	131, 201	mtand 691	. . . . . 6 |- ( ph -> -. M = X )
203	202	neqned 2801	. . . . 5 |- ( ph -> M =/= X )
204	203	necomd 2849	. . . 4 |- ( ph -> X =/= M )
205	151	oveq2d 6666	. . . . . 6 |- ( ph -> ( X .- ( ( S ` M ) ` R ) ) = ( X .- Z ) )
206	205, 146	eqtr2d 2657	. . . . 5 |- ( ph -> ( X .- R ) = ( X .- ( ( S ` M ) ` R ) ) )
207	5, 6, 7, 8, 27, 9, 54, 133, 14	israg 25592	. . . . 5 |- ( ph -> ( <" X M R "> e. (∟G ` G ) <-> ( X .- R ) = ( X .- ( ( S ` M ) ` R ) ) ) )
208	206, 207	mpbird 247	. . . 4 |- ( ph -> <" X M R "> e. (∟G ` G ) )
209	5, 6, 7, 8, 9, 13, 160, 162, 83, 163, 204, 159, 208	ragperp 25612	. . 3 |- ( ph -> ( A L B ) (⟂G ` G ) ( R L M ) )
210	5, 6, 7, 8, 9, 13, 160, 209	perpcom 25608	. 2 |- ( ph -> ( R L M ) (⟂G ` G ) ( A L B ) )
211	2, 4, 52, 53, 155, 156, 210	reu2eqd 3403	1 |- ( ph -> B = M )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E!wreu 2914   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-leg 25478  df-mir 25548  df-rag 25589  df-perpg 25591

This theorem is referenced by:  opphllem  25627