Theorem lmiisolem 25688

Description: Lemma for lmiiso 25689. (Contributed by Thierry Arnoux, 14-Dec-2019.)

Hypotheses

Ref	Expression
ismid.p	|- P = ( Base ` G )
ismid.d	|- .- = ( dist ` G )
ismid.i	|- I = (Itv ` G )
ismid.g	|- ( ph -> G e. TarskiG )
ismid.1	|- ( ph -> GDimTarskiG≥ 2 )
lmif.m	|- M = ( (lInvG ` G ) ` D )
lmif.l	|- L = (LineG ` G )
lmif.d	|- ( ph -> D e. ran L )
lmiiso.1	|- ( ph -> A e. P )
lmiiso.2	|- ( ph -> B e. P )
lmiisolem.s	|- S = ( (pInvG ` G ) ` Z )
lmiisolem.z	|- Z = ( ( A (midG ` G ) ( M ` A ) ) (midG ` G ) ( B (midG ` G ) ( M ` B ) ) )

Assertion

Ref	Expression
lmiisolem	|- ( ph -> ( ( M ` A ) .- ( M ` B ) ) = ( A .- B ) )

Proof of Theorem lmiisolem

Step	Hyp	Ref	Expression
1		ismid.p	. . . . . . . 8 |- P = ( Base ` G )
2		ismid.d	. . . . . . . 8 |- .- = ( dist ` G )
3		ismid.i	. . . . . . . 8 |- I = (Itv ` G )
4		ismid.g	. . . . . . . . 9 |- ( ph -> G e. TarskiG )
5	4	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( S ` A ) = Z ) -> G e. TarskiG )
6		lmiisolem.z	. . . . . . . . . 10 |- Z = ( ( A (midG ` G ) ( M ` A ) ) (midG ` G ) ( B (midG ` G ) ( M ` B ) ) )
7		ismid.1	. . . . . . . . . . 11 |- ( ph -> GDimTarskiG≥ 2 )
8		lmiiso.1	. . . . . . . . . . . 12 |- ( ph -> A e. P )
9		lmif.m	. . . . . . . . . . . . 13 |- M = ( (lInvG ` G ) ` D )
10		lmif.l	. . . . . . . . . . . . 13 |- L = (LineG ` G )
11		lmif.d	. . . . . . . . . . . . 13 |- ( ph -> D e. ran L )
12	1, 2, 3, 4, 7, 9, 10, 11, 8	lmicl 25678	. . . . . . . . . . . 12 |- ( ph -> ( M ` A ) e. P )
13	1, 2, 3, 4, 7, 8, 12	midcl 25669	. . . . . . . . . . 11 |- ( ph -> ( A (midG ` G ) ( M ` A ) ) e. P )
14		lmiiso.2	. . . . . . . . . . . 12 |- ( ph -> B e. P )
15	1, 2, 3, 4, 7, 9, 10, 11, 14	lmicl 25678	. . . . . . . . . . . 12 |- ( ph -> ( M ` B ) e. P )
16	1, 2, 3, 4, 7, 14, 15	midcl 25669	. . . . . . . . . . 11 |- ( ph -> ( B (midG ` G ) ( M ` B ) ) e. P )
17	1, 2, 3, 4, 7, 13, 16	midcl 25669	. . . . . . . . . 10 |- ( ph -> ( ( A (midG ` G ) ( M ` A ) ) (midG ` G ) ( B (midG ` G ) ( M ` B ) ) ) e. P )
18	6, 17	syl5eqel 2705	. . . . . . . . 9 |- ( ph -> Z e. P )
19	18	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( S ` A ) = Z ) -> Z e. P )
20		eqid 2622	. . . . . . . . . 10 |- (pInvG ` G ) = (pInvG ` G )
21		lmiisolem.s	. . . . . . . . . 10 |- S = ( (pInvG ` G ) ` Z )
22	1, 2, 3, 10, 20, 4, 18, 21, 8	mircl 25556	. . . . . . . . 9 |- ( ph -> ( S ` A ) e. P )
23	22	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( S ` A ) = Z ) -> ( S ` A ) e. P )
24	8	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( S ` A ) = Z ) -> A e. P )
25	1, 2, 3, 10, 20, 5, 19, 21, 24	mircgr 25552	. . . . . . . 8 |- ( ( ph /\ ( S ` A ) = Z ) -> ( Z .- ( S ` A ) ) = ( Z .- A ) )
26		simpr 477	. . . . . . . . 9 |- ( ( ph /\ ( S ` A ) = Z ) -> ( S ` A ) = Z )
27	26	eqcomd 2628	. . . . . . . 8 |- ( ( ph /\ ( S ` A ) = Z ) -> Z = ( S ` A ) )
28	1, 2, 3, 5, 19, 23, 19, 24, 25, 27	tgcgreq 25377	. . . . . . 7 |- ( ( ph /\ ( S ` A ) = Z ) -> Z = A )
29		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) )
30	29	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> ( ( A (midG ` G ) ( M ` A ) ) (midG ` G ) ( A (midG ` G ) ( M ` A ) ) ) = ( ( A (midG ` G ) ( M ` A ) ) (midG ` G ) ( B (midG ` G ) ( M ` B ) ) ) )
31	30, 6	syl6reqr 2675	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> Z = ( ( A (midG ` G ) ( M ` A ) ) (midG ` G ) ( A (midG ` G ) ( M ` A ) ) ) )
32	4	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> G e. TarskiG )
33	7	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> GDimTarskiG≥ 2 )
34	13	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> ( A (midG ` G ) ( M ` A ) ) e. P )
35	1, 2, 3, 32, 33, 34, 34	midid 25673	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> ( ( A (midG ` G ) ( M ` A ) ) (midG ` G ) ( A (midG ` G ) ( M ` A ) ) ) = ( A (midG ` G ) ( M ` A ) ) )
36	31, 35	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> Z = ( A (midG ` G ) ( M ` A ) ) )
37		eqidd 2623	. . . . . . . . . . . . 13 |- ( ph -> ( M ` A ) = ( M ` A ) )
38	1, 2, 3, 4, 7, 9, 10, 11, 8, 12	islmib 25679	. . . . . . . . . . . . 13 |- ( ph -> ( ( M ` A ) = ( M ` A ) <-> ( ( A (midG ` G ) ( M ` A ) ) e. D /\ ( D (⟂G ` G ) ( A L ( M ` A ) ) \/ A = ( M ` A ) ) ) ) )
39	37, 38	mpbid 222	. . . . . . . . . . . 12 |- ( ph -> ( ( A (midG ` G ) ( M ` A ) ) e. D /\ ( D (⟂G ` G ) ( A L ( M ` A ) ) \/ A = ( M ` A ) ) ) )
40	39	simpld 475	. . . . . . . . . . 11 |- ( ph -> ( A (midG ` G ) ( M ` A ) ) e. D )
41	40	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> ( A (midG ` G ) ( M ` A ) ) e. D )
42	36, 41	eqeltrd 2701	. . . . . . . . 9 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> Z e. D )
43	4	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> G e. TarskiG )
44	13	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> ( A (midG ` G ) ( M ` A ) ) e. P )
45	16	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> ( B (midG ` G ) ( M ` B ) ) e. P )
46	18	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> Z e. P )
47		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) )
48	1, 2, 3, 4, 7, 13, 16	midbtwn 25671	. . . . . . . . . . . . 13 |- ( ph -> ( ( A (midG ` G ) ( M ` A ) ) (midG ` G ) ( B (midG ` G ) ( M ` B ) ) ) e. ( ( A (midG ` G ) ( M ` A ) ) I ( B (midG ` G ) ( M ` B ) ) ) )
49	6, 48	syl5eqel 2705	. . . . . . . . . . . 12 |- ( ph -> Z e. ( ( A (midG ` G ) ( M ` A ) ) I ( B (midG ` G ) ( M ` B ) ) ) )
50	49	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> Z e. ( ( A (midG ` G ) ( M ` A ) ) I ( B (midG ` G ) ( M ` B ) ) ) )
51	1, 3, 10, 43, 44, 45, 46, 47, 50	btwnlng1 25514	. . . . . . . . . 10 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> Z e. ( ( A (midG ` G ) ( M ` A ) ) L ( B (midG ` G ) ( M ` B ) ) ) )
52	11	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> D e. ran L )
53	40	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> ( A (midG ` G ) ( M ` A ) ) e. D )
54		eqidd 2623	. . . . . . . . . . . . . 14 |- ( ph -> ( M ` B ) = ( M ` B ) )
55	1, 2, 3, 4, 7, 9, 10, 11, 14, 15	islmib 25679	. . . . . . . . . . . . . 14 |- ( ph -> ( ( M ` B ) = ( M ` B ) <-> ( ( B (midG ` G ) ( M ` B ) ) e. D /\ ( D (⟂G ` G ) ( B L ( M ` B ) ) \/ B = ( M ` B ) ) ) ) )
56	54, 55	mpbid 222	. . . . . . . . . . . . 13 |- ( ph -> ( ( B (midG ` G ) ( M ` B ) ) e. D /\ ( D (⟂G ` G ) ( B L ( M ` B ) ) \/ B = ( M ` B ) ) ) )
57	56	simpld 475	. . . . . . . . . . . 12 |- ( ph -> ( B (midG ` G ) ( M ` B ) ) e. D )
58	57	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> ( B (midG ` G ) ( M ` B ) ) e. D )
59	1, 3, 10, 43, 44, 45, 47, 47, 52, 53, 58	tglinethru 25531	. . . . . . . . . 10 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> D = ( ( A (midG ` G ) ( M ` A ) ) L ( B (midG ` G ) ( M ` B ) ) ) )
60	51, 59	eleqtrrd 2704	. . . . . . . . 9 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> Z e. D )
61	42, 60	pm2.61dane 2881	. . . . . . . 8 |- ( ph -> Z e. D )
62	61	adantr 481	. . . . . . 7 |- ( ( ph /\ ( S ` A ) = Z ) -> Z e. D )
63	28, 62	eqeltrrd 2702	. . . . . 6 |- ( ( ph /\ ( S ` A ) = Z ) -> A e. D )
64	1, 2, 3, 4, 7, 9, 10, 11, 8	lmiinv 25684	. . . . . . 7 |- ( ph -> ( ( M ` A ) = A <-> A e. D ) )
65	64	biimpar 502	. . . . . 6 |- ( ( ph /\ A e. D ) -> ( M ` A ) = A )
66	63, 65	syldan 487	. . . . 5 |- ( ( ph /\ ( S ` A ) = Z ) -> ( M ` A ) = A )
67	66, 28	eqtr4d 2659	. . . 4 |- ( ( ph /\ ( S ` A ) = Z ) -> ( M ` A ) = Z )
68	67	oveq1d 6665	. . 3 |- ( ( ph /\ ( S ` A ) = Z ) -> ( ( M ` A ) .- ( M ` B ) ) = ( Z .- ( M ` B ) ) )
69		eqidd 2623	. . . . . . . . 9 |- ( ( ph /\ B = ( M ` B ) ) -> Z = Z )
70	4	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ B = ( M ` B ) ) -> G e. TarskiG )
71	14	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ B = ( M ` B ) ) -> B e. P )
72	16	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ B = ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) e. P )
73	1, 2, 3, 4, 7, 14, 15	midbtwn 25671	. . . . . . . . . . . 12 |- ( ph -> ( B (midG ` G ) ( M ` B ) ) e. ( B I ( M ` B ) ) )
74	73	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ B = ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) e. ( B I ( M ` B ) ) )
75		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ B = ( M ` B ) ) -> B = ( M ` B ) )
76	75	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ph /\ B = ( M ` B ) ) -> ( B I B ) = ( B I ( M ` B ) ) )
77	74, 76	eleqtrrd 2704	. . . . . . . . . 10 |- ( ( ph /\ B = ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) e. ( B I B ) )
78	1, 2, 3, 70, 71, 72, 77	axtgbtwnid 25365	. . . . . . . . 9 |- ( ( ph /\ B = ( M ` B ) ) -> B = ( B (midG ` G ) ( M ` B ) ) )
79		eqidd 2623	. . . . . . . . 9 |- ( ( ph /\ B = ( M ` B ) ) -> B = B )
80	69, 78, 79	s3eqd 13609	. . . . . . . 8 |- ( ( ph /\ B = ( M ` B ) ) -> <" Z B B "> = <" Z ( B (midG ` G ) ( M ` B ) ) B "> )
81	1, 2, 3, 10, 20, 4, 18, 14, 14	ragtrivb 25597	. . . . . . . . 9 |- ( ph -> <" Z B B "> e. (∟G ` G ) )
82	81	adantr 481	. . . . . . . 8 |- ( ( ph /\ B = ( M ` B ) ) -> <" Z B B "> e. (∟G ` G ) )
83	80, 82	eqeltrrd 2702	. . . . . . 7 |- ( ( ph /\ B = ( M ` B ) ) -> <" Z ( B (midG ` G ) ( M ` B ) ) B "> e. (∟G ` G ) )
84	4	adantr 481	. . . . . . . 8 |- ( ( ph /\ B =/= ( M ` B ) ) -> G e. TarskiG )
85	61	adantr 481	. . . . . . . 8 |- ( ( ph /\ B =/= ( M ` B ) ) -> Z e. D )
86	57	adantr 481	. . . . . . . 8 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) e. D )
87	14	adantr 481	. . . . . . . 8 |- ( ( ph /\ B =/= ( M ` B ) ) -> B e. P )
88		df-ne 2795	. . . . . . . . . 10 |- ( B =/= ( M ` B ) <-> -. B = ( M ` B ) )
89	56	simprd 479	. . . . . . . . . . . 12 |- ( ph -> ( D (⟂G ` G ) ( B L ( M ` B ) ) \/ B = ( M ` B ) ) )
90	89	orcomd 403	. . . . . . . . . . 11 |- ( ph -> ( B = ( M ` B ) \/ D (⟂G ` G ) ( B L ( M ` B ) ) ) )
91	90	orcanai 952	. . . . . . . . . 10 |- ( ( ph /\ -. B = ( M ` B ) ) -> D (⟂G ` G ) ( B L ( M ` B ) ) )
92	88, 91	sylan2b 492	. . . . . . . . 9 |- ( ( ph /\ B =/= ( M ` B ) ) -> D (⟂G ` G ) ( B L ( M ` B ) ) )
93	15	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( M ` B ) e. P )
94		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ B =/= ( M ` B ) ) -> B =/= ( M ` B ) )
95	16	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) e. P )
96	4	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> G e. TarskiG )
97	14	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> B e. P )
98	15	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> ( M ` B ) e. P )
99	7	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> GDimTarskiG≥ 2 )
100		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> ( B (midG ` G ) ( M ` B ) ) = B )
101	1, 2, 3, 96, 99, 97, 98, 100	midcgr 25672	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> ( B .- B ) = ( B .- ( M ` B ) ) )
102	101	eqcomd 2628	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> ( B .- ( M ` B ) ) = ( B .- B ) )
103	1, 2, 3, 96, 97, 98, 97, 102	axtgcgrid 25362	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> B = ( M ` B ) )
104	103	ex 450	. . . . . . . . . . . . 13 |- ( ph -> ( ( B (midG ` G ) ( M ` B ) ) = B -> B = ( M ` B ) ) )
105	104	necon3d 2815	. . . . . . . . . . . 12 |- ( ph -> ( B =/= ( M ` B ) -> ( B (midG ` G ) ( M ` B ) ) =/= B ) )
106	105	imp 445	. . . . . . . . . . 11 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) =/= B )
107	73	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) e. ( B I ( M ` B ) ) )
108	1, 3, 10, 84, 87, 93, 95, 94, 107	btwnlng1 25514	. . . . . . . . . . 11 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) e. ( B L ( M ` B ) ) )
109	1, 3, 10, 84, 87, 93, 94, 95, 106, 108	tglineelsb2 25527	. . . . . . . . . 10 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( B L ( M ` B ) ) = ( B L ( B (midG ` G ) ( M ` B ) ) ) )
110	1, 3, 10, 84, 95, 87, 106	tglinecom 25530	. . . . . . . . . 10 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( ( B (midG ` G ) ( M ` B ) ) L B ) = ( B L ( B (midG ` G ) ( M ` B ) ) ) )
111	109, 110	eqtr4d 2659	. . . . . . . . 9 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( B L ( M ` B ) ) = ( ( B (midG ` G ) ( M ` B ) ) L B ) )
112	92, 111	breqtrd 4679	. . . . . . . 8 |- ( ( ph /\ B =/= ( M ` B ) ) -> D (⟂G ` G ) ( ( B (midG ` G ) ( M ` B ) ) L B ) )
113	1, 2, 3, 10, 84, 85, 86, 87, 112	perpdrag 25620	. . . . . . 7 |- ( ( ph /\ B =/= ( M ` B ) ) -> <" Z ( B (midG ` G ) ( M ` B ) ) B "> e. (∟G ` G ) )
114	83, 113	pm2.61dane 2881	. . . . . 6 |- ( ph -> <" Z ( B (midG ` G ) ( M ` B ) ) B "> e. (∟G ` G ) )
115	1, 2, 3, 10, 20, 4, 18, 16, 14	israg 25592	. . . . . 6 |- ( ph -> ( <" Z ( B (midG ` G ) ( M ` B ) ) B "> e. (∟G ` G ) <-> ( Z .- B ) = ( Z .- ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` B ) ) ) )
116	114, 115	mpbid 222	. . . . 5 |- ( ph -> ( Z .- B ) = ( Z .- ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` B ) ) )
117		eqidd 2623	. . . . . . 7 |- ( ph -> ( B (midG ` G ) ( M ` B ) ) = ( B (midG ` G ) ( M ` B ) ) )
118	1, 2, 3, 4, 7, 14, 15, 20, 16	ismidb 25670	. . . . . . 7 |- ( ph -> ( ( M ` B ) = ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` B ) <-> ( B (midG ` G ) ( M ` B ) ) = ( B (midG ` G ) ( M ` B ) ) ) )
119	117, 118	mpbird 247	. . . . . 6 |- ( ph -> ( M ` B ) = ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` B ) )
120	119	oveq2d 6666	. . . . 5 |- ( ph -> ( Z .- ( M ` B ) ) = ( Z .- ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` B ) ) )
121	116, 120	eqtr4d 2659	. . . 4 |- ( ph -> ( Z .- B ) = ( Z .- ( M ` B ) ) )
122	121	adantr 481	. . 3 |- ( ( ph /\ ( S ` A ) = Z ) -> ( Z .- B ) = ( Z .- ( M ` B ) ) )
123	28	oveq1d 6665	. . 3 |- ( ( ph /\ ( S ` A ) = Z ) -> ( Z .- B ) = ( A .- B ) )
124	68, 122, 123	3eqtr2d 2662	. 2 |- ( ( ph /\ ( S ` A ) = Z ) -> ( ( M ` A ) .- ( M ` B ) ) = ( A .- B ) )
125	4	adantr 481	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> G e. TarskiG )
126	22	adantr 481	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( S ` A ) e. P )
127	18	adantr 481	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> Z e. P )
128	8	adantr 481	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> A e. P )
129	1, 2, 3, 10, 20, 4, 18, 21, 12	mircl 25556	. . . . 5 |- ( ph -> ( S ` ( M ` A ) ) e. P )
130	129	adantr 481	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( S ` ( M ` A ) ) e. P )
131	12	adantr 481	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( M ` A ) e. P )
132	14	adantr 481	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> B e. P )
133	15	adantr 481	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( M ` B ) e. P )
134		simpr 477	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( S ` A ) =/= Z )
135	1, 2, 3, 10, 20, 125, 127, 21, 128	mirbtwn 25553	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> Z e. ( ( S ` A ) I A ) )
136	1, 2, 3, 10, 20, 125, 127, 21, 131	mirbtwn 25553	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> Z e. ( ( S ` ( M ` A ) ) I ( M ` A ) ) )
137		eqidd 2623	. . . . . . . . . . . 12 |- ( ( ph /\ A = ( M ` A ) ) -> Z = Z )
138	4	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ A = ( M ` A ) ) -> G e. TarskiG )
139	8	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ A = ( M ` A ) ) -> A e. P )
140	13	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ A = ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) e. P )
141	1, 2, 3, 4, 7, 8, 12	midbtwn 25671	. . . . . . . . . . . . . . 15 |- ( ph -> ( A (midG ` G ) ( M ` A ) ) e. ( A I ( M ` A ) ) )
142	141	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ A = ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) e. ( A I ( M ` A ) ) )
143		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ph /\ A = ( M ` A ) ) -> A = ( M ` A ) )
144	143	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( ph /\ A = ( M ` A ) ) -> ( A I A ) = ( A I ( M ` A ) ) )
145	142, 144	eleqtrrd 2704	. . . . . . . . . . . . 13 |- ( ( ph /\ A = ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) e. ( A I A ) )
146	1, 2, 3, 138, 139, 140, 145	axtgbtwnid 25365	. . . . . . . . . . . 12 |- ( ( ph /\ A = ( M ` A ) ) -> A = ( A (midG ` G ) ( M ` A ) ) )
147		eqidd 2623	. . . . . . . . . . . 12 |- ( ( ph /\ A = ( M ` A ) ) -> A = A )
148	137, 146, 147	s3eqd 13609	. . . . . . . . . . 11 |- ( ( ph /\ A = ( M ` A ) ) -> <" Z A A "> = <" Z ( A (midG ` G ) ( M ` A ) ) A "> )
149	1, 2, 3, 10, 20, 4, 18, 8, 8	ragtrivb 25597	. . . . . . . . . . . 12 |- ( ph -> <" Z A A "> e. (∟G ` G ) )
150	149	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ A = ( M ` A ) ) -> <" Z A A "> e. (∟G ` G ) )
151	148, 150	eqeltrrd 2702	. . . . . . . . . 10 |- ( ( ph /\ A = ( M ` A ) ) -> <" Z ( A (midG ` G ) ( M ` A ) ) A "> e. (∟G ` G ) )
152	4	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ A =/= ( M ` A ) ) -> G e. TarskiG )
153	61	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ A =/= ( M ` A ) ) -> Z e. D )
154	40	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) e. D )
155	8	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ A =/= ( M ` A ) ) -> A e. P )
156		df-ne 2795	. . . . . . . . . . . . 13 |- ( A =/= ( M ` A ) <-> -. A = ( M ` A ) )
157	39	simprd 479	. . . . . . . . . . . . . . 15 |- ( ph -> ( D (⟂G ` G ) ( A L ( M ` A ) ) \/ A = ( M ` A ) ) )
158	157	orcomd 403	. . . . . . . . . . . . . 14 |- ( ph -> ( A = ( M ` A ) \/ D (⟂G ` G ) ( A L ( M ` A ) ) ) )
159	158	orcanai 952	. . . . . . . . . . . . 13 |- ( ( ph /\ -. A = ( M ` A ) ) -> D (⟂G ` G ) ( A L ( M ` A ) ) )
160	156, 159	sylan2b 492	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= ( M ` A ) ) -> D (⟂G ` G ) ( A L ( M ` A ) ) )
161	12	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( M ` A ) e. P )
162		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= ( M ` A ) ) -> A =/= ( M ` A ) )
163	13	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) e. P )
164	4	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> G e. TarskiG )
165	8	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> A e. P )
166	12	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> ( M ` A ) e. P )
167	7	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> GDimTarskiG≥ 2 )
168		simpr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> ( A (midG ` G ) ( M ` A ) ) = A )
169	1, 2, 3, 164, 167, 165, 166, 168	midcgr 25672	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> ( A .- A ) = ( A .- ( M ` A ) ) )
170	169	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> ( A .- ( M ` A ) ) = ( A .- A ) )
171	1, 2, 3, 164, 165, 166, 165, 170	axtgcgrid 25362	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> A = ( M ` A ) )
172	171	ex 450	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( A (midG ` G ) ( M ` A ) ) = A -> A = ( M ` A ) ) )
173	172	necon3d 2815	. . . . . . . . . . . . . . 15 |- ( ph -> ( A =/= ( M ` A ) -> ( A (midG ` G ) ( M ` A ) ) =/= A ) )
174	173	imp 445	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) =/= A )
175	141	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) e. ( A I ( M ` A ) ) )
176	1, 3, 10, 152, 155, 161, 163, 162, 175	btwnlng1 25514	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) e. ( A L ( M ` A ) ) )
177	1, 3, 10, 152, 155, 161, 162, 163, 174, 176	tglineelsb2 25527	. . . . . . . . . . . . 13 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( A L ( M ` A ) ) = ( A L ( A (midG ` G ) ( M ` A ) ) ) )
178	1, 3, 10, 152, 163, 155, 174	tglinecom 25530	. . . . . . . . . . . . 13 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( ( A (midG ` G ) ( M ` A ) ) L A ) = ( A L ( A (midG ` G ) ( M ` A ) ) ) )
179	177, 178	eqtr4d 2659	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( A L ( M ` A ) ) = ( ( A (midG ` G ) ( M ` A ) ) L A ) )
180	160, 179	breqtrd 4679	. . . . . . . . . . 11 |- ( ( ph /\ A =/= ( M ` A ) ) -> D (⟂G ` G ) ( ( A (midG ` G ) ( M ` A ) ) L A ) )
181	1, 2, 3, 10, 152, 153, 154, 155, 180	perpdrag 25620	. . . . . . . . . 10 |- ( ( ph /\ A =/= ( M ` A ) ) -> <" Z ( A (midG ` G ) ( M ` A ) ) A "> e. (∟G ` G ) )
182	151, 181	pm2.61dane 2881	. . . . . . . . 9 |- ( ph -> <" Z ( A (midG ` G ) ( M ` A ) ) A "> e. (∟G ` G ) )
183	1, 2, 3, 10, 20, 4, 18, 13, 8	israg 25592	. . . . . . . . 9 |- ( ph -> ( <" Z ( A (midG ` G ) ( M ` A ) ) A "> e. (∟G ` G ) <-> ( Z .- A ) = ( Z .- ( ( (pInvG ` G ) ` ( A (midG ` G ) ( M ` A ) ) ) ` A ) ) ) )
184	182, 183	mpbid 222	. . . . . . . 8 |- ( ph -> ( Z .- A ) = ( Z .- ( ( (pInvG ` G ) ` ( A (midG ` G ) ( M ` A ) ) ) ` A ) ) )
185		eqidd 2623	. . . . . . . . . 10 |- ( ph -> ( A (midG ` G ) ( M ` A ) ) = ( A (midG ` G ) ( M ` A ) ) )
186	1, 2, 3, 4, 7, 8, 12, 20, 13	ismidb 25670	. . . . . . . . . 10 |- ( ph -> ( ( M ` A ) = ( ( (pInvG ` G ) ` ( A (midG ` G ) ( M ` A ) ) ) ` A ) <-> ( A (midG ` G ) ( M ` A ) ) = ( A (midG ` G ) ( M ` A ) ) ) )
187	185, 186	mpbird 247	. . . . . . . . 9 |- ( ph -> ( M ` A ) = ( ( (pInvG ` G ) ` ( A (midG ` G ) ( M ` A ) ) ) ` A ) )
188	187	oveq2d 6666	. . . . . . . 8 |- ( ph -> ( Z .- ( M ` A ) ) = ( Z .- ( ( (pInvG ` G ) ` ( A (midG ` G ) ( M ` A ) ) ) ` A ) ) )
189	184, 188	eqtr4d 2659	. . . . . . 7 |- ( ph -> ( Z .- A ) = ( Z .- ( M ` A ) ) )
190	1, 2, 3, 10, 20, 4, 18, 21, 8	mircgr 25552	. . . . . . 7 |- ( ph -> ( Z .- ( S ` A ) ) = ( Z .- A ) )
191	1, 2, 3, 10, 20, 4, 18, 21, 12	mircgr 25552	. . . . . . 7 |- ( ph -> ( Z .- ( S ` ( M ` A ) ) ) = ( Z .- ( M ` A ) ) )
192	189, 190, 191	3eqtr4d 2666	. . . . . 6 |- ( ph -> ( Z .- ( S ` A ) ) = ( Z .- ( S ` ( M ` A ) ) ) )
193	192	adantr 481	. . . . 5 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( Z .- ( S ` A ) ) = ( Z .- ( S ` ( M ` A ) ) ) )
194	1, 2, 3, 125, 127, 126, 127, 130, 193	tgcgrcomlr 25375	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( ( S ` A ) .- Z ) = ( ( S ` ( M ` A ) ) .- Z ) )
195	189	adantr 481	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( Z .- A ) = ( Z .- ( M ` A ) ) )
196	21	fveq1i 6192	. . . . . . . . . 10 |- ( S ` ( A (midG ` G ) ( M ` A ) ) ) = ( ( (pInvG ` G ) ` Z ) ` ( A (midG ` G ) ( M ` A ) ) )
197	1, 2, 3, 4, 7, 8, 12, 21, 18	mirmid 25675	. . . . . . . . . 10 |- ( ph -> ( ( S ` A ) (midG ` G ) ( S ` ( M ` A ) ) ) = ( S ` ( A (midG ` G ) ( M ` A ) ) ) )
198	6	eqcomi 2631	. . . . . . . . . . 11 |- ( ( A (midG ` G ) ( M ` A ) ) (midG ` G ) ( B (midG ` G ) ( M ` B ) ) ) = Z
199	1, 2, 3, 4, 7, 13, 16, 20, 18	ismidb 25670	. . . . . . . . . . 11 |- ( ph -> ( ( B (midG ` G ) ( M ` B ) ) = ( ( (pInvG ` G ) ` Z ) ` ( A (midG ` G ) ( M ` A ) ) ) <-> ( ( A (midG ` G ) ( M ` A ) ) (midG ` G ) ( B (midG ` G ) ( M ` B ) ) ) = Z ) )
200	198, 199	mpbiri 248	. . . . . . . . . 10 |- ( ph -> ( B (midG ` G ) ( M ` B ) ) = ( ( (pInvG ` G ) ` Z ) ` ( A (midG ` G ) ( M ` A ) ) ) )
201	196, 197, 200	3eqtr4a 2682	. . . . . . . . 9 |- ( ph -> ( ( S ` A ) (midG ` G ) ( S ` ( M ` A ) ) ) = ( B (midG ` G ) ( M ` B ) ) )
202	1, 2, 3, 4, 7, 22, 129, 20, 16	ismidb 25670	. . . . . . . . 9 |- ( ph -> ( ( S ` ( M ` A ) ) = ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` ( S ` A ) ) <-> ( ( S ` A ) (midG ` G ) ( S ` ( M ` A ) ) ) = ( B (midG ` G ) ( M ` B ) ) ) )
203	201, 202	mpbird 247	. . . . . . . 8 |- ( ph -> ( S ` ( M ` A ) ) = ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` ( S ` A ) ) )
204	119, 203	oveq12d 6668	. . . . . . 7 |- ( ph -> ( ( M ` B ) .- ( S ` ( M ` A ) ) ) = ( ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` B ) .- ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` ( S ` A ) ) ) )
205		eqid 2622	. . . . . . . 8 |- ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) = ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) )
206	1, 2, 3, 10, 20, 4, 16, 205, 14, 22	miriso 25565	. . . . . . 7 |- ( ph -> ( ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` B ) .- ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` ( S ` A ) ) ) = ( B .- ( S ` A ) ) )
207	204, 206	eqtr2d 2657	. . . . . 6 |- ( ph -> ( B .- ( S ` A ) ) = ( ( M ` B ) .- ( S ` ( M ` A ) ) ) )
208	207	adantr 481	. . . . 5 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( B .- ( S ` A ) ) = ( ( M ` B ) .- ( S ` ( M ` A ) ) ) )
209	1, 2, 3, 125, 132, 126, 133, 130, 208	tgcgrcomlr 25375	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( ( S ` A ) .- B ) = ( ( S ` ( M ` A ) ) .- ( M ` B ) ) )
210	121	adantr 481	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( Z .- B ) = ( Z .- ( M ` B ) ) )
211	1, 2, 3, 125, 126, 127, 128, 130, 127, 131, 132, 133, 134, 135, 136, 194, 195, 209, 210	axtg5seg 25364	. . 3 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( A .- B ) = ( ( M ` A ) .- ( M ` B ) ) )
212	211	eqcomd 2628	. 2 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( ( M ` A ) .- ( M ` B ) ) = ( A .- B ) )
213	124, 212	pm2.61dane 2881	1 |- ( ph -> ( ( M ` A ) .- ( M ` B ) ) = ( A .- B ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ran crn 5115   ` cfv 5888  (class class class)co 6650   2c2 11070   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Dim_TarskiG≥cstrkgld 25333  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547  ∟Gcrag 25588  ⟂Gcperpg 25590  midGcmid 25664  lInvGclmi 25665

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-trkgld 25351  df-trkg 25352  df-cgrg 25406  df-leg 25478  df-mir 25548  df-rag 25589  df-perpg 25591  df-mid 25666  df-lmi 25667

This theorem is referenced by:  lmiiso  25689