Theorem lmiisolem 24378

Description: Lemma for lmiiso 24379 (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 465	. . . . . . . 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 24369	. . . . . . . . . . . 12 |- ( ph -> ( M ` A ) e. P )
13	1, 2, 3, 4, 7, 8, 12	midcl 24360	. . . . . . . . . . 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 24369	. . . . . . . . . . . 12 |- ( ph -> ( M ` B ) e. P )
16	1, 2, 3, 4, 7, 14, 15	midcl 24360	. . . . . . . . . . 11 |- ( ph -> ( B (midG ` G ) ( M ` B ) ) e. P )
17	1, 2, 3, 4, 7, 13, 16	midcl 24360	. . . . . . . . . 10 |- ( ph -> ( ( A (midG ` G ) ( M ` A ) ) (midG ` G ) ( B (midG ` G ) ( M ` B ) ) ) e. P )
18	6, 17	syl5eqel 2549	. . . . . . . . 9 |- ( ph -> Z e. P )
19	18	adantr 465	. . . . . . . 8 |- ( ( ph /\ ( S ` A ) = Z ) -> Z e. P )
20		eqid 2457	. . . . . . . . . 10 |- (pInvG ` G ) = (pInvG ` G )
21		lmiisolem.s	. . . . . . . . . 10 |- S = ( (pInvG ` G ) ` Z )
22	1, 2, 3, 10, 20, 4, 18, 21, 8	mircl 24259	. . . . . . . . 9 |- ( ph -> ( S ` A ) e. P )
23	22	adantr 465	. . . . . . . 8 |- ( ( ph /\ ( S ` A ) = Z ) -> ( S ` A ) e. P )
24	8	adantr 465	. . . . . . . 8 |- ( ( ph /\ ( S ` A ) = Z ) -> A e. P )
25	1, 2, 3, 10, 20, 5, 19, 21, 24	mircgr 24255	. . . . . . . 8 |- ( ( ph /\ ( S ` A ) = Z ) -> ( Z .- ( S ` A ) ) = ( Z .- A ) )
26		simpr 461	. . . . . . . . 9 |- ( ( ph /\ ( S ` A ) = Z ) -> ( S ` A ) = Z )
27	26	eqcomd 2465	. . . . . . . 8 |- ( ( ph /\ ( S ` A ) = Z ) -> Z = ( S ` A ) )
28	1, 2, 3, 5, 19, 23, 19, 24, 25, 27	tgcgreq 24090	. . . . . . 7 |- ( ( ph /\ ( S ` A ) = Z ) -> Z = A )
29		simpr 461	. . . . . . . . . . . . 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 6312	. . . . . . . . . . . 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 2517	. . . . . . . . . . 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 465	. . . . . . . . . . . 12 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> G e. TarskiG )
33	7	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> GDimTarskiG≥ 2 )
34	13	adantr 465	. . . . . . . . . . . 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 24364	. . . . . . . . . . 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 2498	. . . . . . . . . 10 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> Z = ( A (midG ` G ) ( M ` A ) ) )
37		eqidd 2458	. . . . . . . . . . . . 13 |- ( ph -> ( M ` A ) = ( M ` A ) )
38	1, 2, 3, 4, 7, 9, 10, 11, 8, 12	islmib 24370	. . . . . . . . . . . . 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 210	. . . . . . . . . . . 12 |- ( ph -> ( ( A (midG ` G ) ( M ` A ) ) e. D /\ ( D (⟂G ` G ) ( A L ( M ` A ) ) \/ A = ( M ` A ) ) ) )
40	39	simpld 459	. . . . . . . . . . 11 |- ( ph -> ( A (midG ` G ) ( M ` A ) ) e. D )
41	40	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> ( A (midG ` G ) ( M ` A ) ) e. D )
42	36, 41	eqeltrd 2545	. . . . . . . . 9 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = ( B (midG ` G ) ( M ` B ) ) ) -> Z e. D )
43	4	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> G e. TarskiG )
44	13	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> ( A (midG ` G ) ( M ` A ) ) e. P )
45	16	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> ( B (midG ` G ) ( M ` B ) ) e. P )
46	18	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> Z e. P )
47		simpr 461	. . . . . . . . . . 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 24362	. . . . . . . . . . . . 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 2549	. . . . . . . . . . . 12 |- ( ph -> Z e. ( ( A (midG ` G ) ( M ` A ) ) I ( B (midG ` G ) ( M ` B ) ) ) )
50	49	adantr 465	. . . . . . . . . . 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 24216	. . . . . . . . . 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 465	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> D e. ran L )
53	40	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> ( A (midG ` G ) ( M ` A ) ) e. D )
54		eqidd 2458	. . . . . . . . . . . . . 14 |- ( ph -> ( M ` B ) = ( M ` B ) )
55	1, 2, 3, 4, 7, 9, 10, 11, 14, 15	islmib 24370	. . . . . . . . . . . . . 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 210	. . . . . . . . . . . . 13 |- ( ph -> ( ( B (midG ` G ) ( M ` B ) ) e. D /\ ( D (⟂G ` G ) ( B L ( M ` B ) ) \/ B = ( M ` B ) ) ) )
57	56	simpld 459	. . . . . . . . . . . 12 |- ( ph -> ( B (midG ` G ) ( M ` B ) ) e. D )
58	57	adantr 465	. . . . . . . . . . 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 24233	. . . . . . . . . 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 2548	. . . . . . . . 9 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) =/= ( B (midG ` G ) ( M ` B ) ) ) -> Z e. D )
61	42, 60	pm2.61dane 2775	. . . . . . . 8 |- ( ph -> Z e. D )
62	61	adantr 465	. . . . . . 7 |- ( ( ph /\ ( S ` A ) = Z ) -> Z e. D )
63	28, 62	eqeltrrd 2546	. . . . . 6 |- ( ( ph /\ ( S ` A ) = Z ) -> A e. D )
64	1, 2, 3, 4, 7, 9, 10, 11, 8	lmiinv 24375	. . . . . . 7 |- ( ph -> ( ( M ` A ) = A <-> A e. D ) )
65	64	biimpar 485	. . . . . 6 |- ( ( ph /\ A e. D ) -> ( M ` A ) = A )
66	63, 65	syldan 470	. . . . 5 |- ( ( ph /\ ( S ` A ) = Z ) -> ( M ` A ) = A )
67	66, 28	eqtr4d 2501	. . . 4 |- ( ( ph /\ ( S ` A ) = Z ) -> ( M ` A ) = Z )
68	67	oveq1d 6311	. . 3 |- ( ( ph /\ ( S ` A ) = Z ) -> ( ( M ` A ) .- ( M ` B ) ) = ( Z .- ( M ` B ) ) )
69		eqidd 2458	. . . . . . . . 9 |- ( ( ph /\ B = ( M ` B ) ) -> Z = Z )
70	4	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ B = ( M ` B ) ) -> G e. TarskiG )
71	14	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ B = ( M ` B ) ) -> B e. P )
72	16	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ B = ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) e. P )
73	1, 2, 3, 4, 7, 14, 15	midbtwn 24362	. . . . . . . . . . . 12 |- ( ph -> ( B (midG ` G ) ( M ` B ) ) e. ( B I ( M ` B ) ) )
74	73	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ B = ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) e. ( B I ( M ` B ) ) )
75		simpr 461	. . . . . . . . . . . 12 |- ( ( ph /\ B = ( M ` B ) ) -> B = ( M ` B ) )
76	75	oveq2d 6312	. . . . . . . . . . 11 |- ( ( ph /\ B = ( M ` B ) ) -> ( B I B ) = ( B I ( M ` B ) ) )
77	74, 76	eleqtrrd 2548	. . . . . . . . . 10 |- ( ( ph /\ B = ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) e. ( B I B ) )
78	1, 2, 3, 70, 71, 72, 77	axtgbtwnid 24080	. . . . . . . . 9 |- ( ( ph /\ B = ( M ` B ) ) -> B = ( B (midG ` G ) ( M ` B ) ) )
79		eqidd 2458	. . . . . . . . 9 |- ( ( ph /\ B = ( M ` B ) ) -> B = B )
80	69, 78, 79	s3eqd 12840	. . . . . . . 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 24296	. . . . . . . . 9 |- ( ph -> <" Z B B "> e. (∟G ` G ) )
82	81	adantr 465	. . . . . . . 8 |- ( ( ph /\ B = ( M ` B ) ) -> <" Z B B "> e. (∟G ` G ) )
83	80, 82	eqeltrrd 2546	. . . . . . 7 |- ( ( ph /\ B = ( M ` B ) ) -> <" Z ( B (midG ` G ) ( M ` B ) ) B "> e. (∟G ` G ) )
84	4	adantr 465	. . . . . . . 8 |- ( ( ph /\ B =/= ( M ` B ) ) -> G e. TarskiG )
85	61	adantr 465	. . . . . . . 8 |- ( ( ph /\ B =/= ( M ` B ) ) -> Z e. D )
86	57	adantr 465	. . . . . . . 8 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) e. D )
87	14	adantr 465	. . . . . . . 8 |- ( ( ph /\ B =/= ( M ` B ) ) -> B e. P )
88		df-ne 2654	. . . . . . . . . 10 |- ( B =/= ( M ` B ) <-> -. B = ( M ` B ) )
89	56	simprd 463	. . . . . . . . . . . 12 |- ( ph -> ( D (⟂G ` G ) ( B L ( M ` B ) ) \/ B = ( M ` B ) ) )
90	89	orcomd 388	. . . . . . . . . . 11 |- ( ph -> ( B = ( M ` B ) \/ D (⟂G ` G ) ( B L ( M ` B ) ) ) )
91	90	orcanai 913	. . . . . . . . . 10 |- ( ( ph /\ -. B = ( M ` B ) ) -> D (⟂G ` G ) ( B L ( M ` B ) ) )
92	88, 91	sylan2b 475	. . . . . . . . 9 |- ( ( ph /\ B =/= ( M ` B ) ) -> D (⟂G ` G ) ( B L ( M ` B ) ) )
93	15	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( M ` B ) e. P )
94		simpr 461	. . . . . . . . . . 11 |- ( ( ph /\ B =/= ( M ` B ) ) -> B =/= ( M ` B ) )
95	16	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) e. P )
96	4	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> G e. TarskiG )
97	14	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> B e. P )
98	15	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> ( M ` B ) e. P )
99	7	adantr 465	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> GDimTarskiG≥ 2 )
100		simpr 461	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> ( B (midG ` G ) ( M ` B ) ) = B )
101	1, 2, 3, 96, 99, 97, 98, 100	midcgr 24363	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> ( B .- B ) = ( B .- ( M ` B ) ) )
102	101	eqcomd 2465	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> ( B .- ( M ` B ) ) = ( B .- B ) )
103	1, 2, 3, 96, 97, 98, 97, 102	axtgcgrid 24077	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( B (midG ` G ) ( M ` B ) ) = B ) -> B = ( M ` B ) )
104	103	ex 434	. . . . . . . . . . . . 13 |- ( ph -> ( ( B (midG ` G ) ( M ` B ) ) = B -> B = ( M ` B ) ) )
105	104	necon3d 2681	. . . . . . . . . . . 12 |- ( ph -> ( B =/= ( M ` B ) -> ( B (midG ` G ) ( M ` B ) ) =/= B ) )
106	105	imp 429	. . . . . . . . . . 11 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( B (midG ` G ) ( M ` B ) ) =/= B )
107	73	adantr 465	. . . . . . . . . . . 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 24216	. . . . . . . . . . 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 24229	. . . . . . . . . 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 24232	. . . . . . . . . 10 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( ( B (midG ` G ) ( M ` B ) ) L B ) = ( B L ( B (midG ` G ) ( M ` B ) ) ) )
111	109, 110	eqtr4d 2501	. . . . . . . . 9 |- ( ( ph /\ B =/= ( M ` B ) ) -> ( B L ( M ` B ) ) = ( ( B (midG ` G ) ( M ` B ) ) L B ) )
112	92, 111	breqtrd 4480	. . . . . . . 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 24319	. . . . . . 7 |- ( ( ph /\ B =/= ( M ` B ) ) -> <" Z ( B (midG ` G ) ( M ` B ) ) B "> e. (∟G ` G ) )
114	83, 113	pm2.61dane 2775	. . . . . 6 |- ( ph -> <" Z ( B (midG ` G ) ( M ` B ) ) B "> e. (∟G ` G ) )
115	1, 2, 3, 10, 20, 4, 18, 16, 14	israg 24291	. . . . . 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 210	. . . . 5 |- ( ph -> ( Z .- B ) = ( Z .- ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` B ) ) )
117		eqidd 2458	. . . . . . 7 |- ( ph -> ( B (midG ` G ) ( M ` B ) ) = ( B (midG ` G ) ( M ` B ) ) )
118	1, 2, 3, 4, 7, 14, 15, 20, 16	ismidb 24361	. . . . . . 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 232	. . . . . 6 |- ( ph -> ( M ` B ) = ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` B ) )
120	119	oveq2d 6312	. . . . 5 |- ( ph -> ( Z .- ( M ` B ) ) = ( Z .- ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` B ) ) )
121	116, 120	eqtr4d 2501	. . . 4 |- ( ph -> ( Z .- B ) = ( Z .- ( M ` B ) ) )
122	121	adantr 465	. . 3 |- ( ( ph /\ ( S ` A ) = Z ) -> ( Z .- B ) = ( Z .- ( M ` B ) ) )
123	28	oveq1d 6311	. . 3 |- ( ( ph /\ ( S ` A ) = Z ) -> ( Z .- B ) = ( A .- B ) )
124	68, 122, 123	3eqtr2d 2504	. 2 |- ( ( ph /\ ( S ` A ) = Z ) -> ( ( M ` A ) .- ( M ` B ) ) = ( A .- B ) )
125	4	adantr 465	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> G e. TarskiG )
126	22	adantr 465	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( S ` A ) e. P )
127	18	adantr 465	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> Z e. P )
128	8	adantr 465	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> A e. P )
129	1, 2, 3, 10, 20, 4, 18, 21, 12	mircl 24259	. . . . 5 |- ( ph -> ( S ` ( M ` A ) ) e. P )
130	129	adantr 465	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( S ` ( M ` A ) ) e. P )
131	12	adantr 465	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( M ` A ) e. P )
132	14	adantr 465	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> B e. P )
133	15	adantr 465	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( M ` B ) e. P )
134		simpr 461	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( S ` A ) =/= Z )
135	1, 2, 3, 10, 20, 125, 127, 21, 128	mirbtwn 24256	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> Z e. ( ( S ` A ) I A ) )
136	1, 2, 3, 10, 20, 125, 127, 21, 131	mirbtwn 24256	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> Z e. ( ( S ` ( M ` A ) ) I ( M ` A ) ) )
137		eqidd 2458	. . . . . . . . . . . 12 |- ( ( ph /\ A = ( M ` A ) ) -> Z = Z )
138	4	adantr 465	. . . . . . . . . . . . 13 |- ( ( ph /\ A = ( M ` A ) ) -> G e. TarskiG )
139	8	adantr 465	. . . . . . . . . . . . 13 |- ( ( ph /\ A = ( M ` A ) ) -> A e. P )
140	13	adantr 465	. . . . . . . . . . . . 13 |- ( ( ph /\ A = ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) e. P )
141	1, 2, 3, 4, 7, 8, 12	midbtwn 24362	. . . . . . . . . . . . . . 15 |- ( ph -> ( A (midG ` G ) ( M ` A ) ) e. ( A I ( M ` A ) ) )
142	141	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ph /\ A = ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) e. ( A I ( M ` A ) ) )
143		simpr 461	. . . . . . . . . . . . . . 15 |- ( ( ph /\ A = ( M ` A ) ) -> A = ( M ` A ) )
144	143	oveq2d 6312	. . . . . . . . . . . . . 14 |- ( ( ph /\ A = ( M ` A ) ) -> ( A I A ) = ( A I ( M ` A ) ) )
145	142, 144	eleqtrrd 2548	. . . . . . . . . . . . 13 |- ( ( ph /\ A = ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) e. ( A I A ) )
146	1, 2, 3, 138, 139, 140, 145	axtgbtwnid 24080	. . . . . . . . . . . 12 |- ( ( ph /\ A = ( M ` A ) ) -> A = ( A (midG ` G ) ( M ` A ) ) )
147		eqidd 2458	. . . . . . . . . . . 12 |- ( ( ph /\ A = ( M ` A ) ) -> A = A )
148	137, 146, 147	s3eqd 12840	. . . . . . . . . . 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 24296	. . . . . . . . . . . 12 |- ( ph -> <" Z A A "> e. (∟G ` G ) )
150	149	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ A = ( M ` A ) ) -> <" Z A A "> e. (∟G ` G ) )
151	148, 150	eqeltrrd 2546	. . . . . . . . . 10 |- ( ( ph /\ A = ( M ` A ) ) -> <" Z ( A (midG ` G ) ( M ` A ) ) A "> e. (∟G ` G ) )
152	4	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ A =/= ( M ` A ) ) -> G e. TarskiG )
153	61	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ A =/= ( M ` A ) ) -> Z e. D )
154	40	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) e. D )
155	8	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ A =/= ( M ` A ) ) -> A e. P )
156		df-ne 2654	. . . . . . . . . . . . 13 |- ( A =/= ( M ` A ) <-> -. A = ( M ` A ) )
157	39	simprd 463	. . . . . . . . . . . . . . 15 |- ( ph -> ( D (⟂G ` G ) ( A L ( M ` A ) ) \/ A = ( M ` A ) ) )
158	157	orcomd 388	. . . . . . . . . . . . . 14 |- ( ph -> ( A = ( M ` A ) \/ D (⟂G ` G ) ( A L ( M ` A ) ) ) )
159	158	orcanai 913	. . . . . . . . . . . . 13 |- ( ( ph /\ -. A = ( M ` A ) ) -> D (⟂G ` G ) ( A L ( M ` A ) ) )
160	156, 159	sylan2b 475	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= ( M ` A ) ) -> D (⟂G ` G ) ( A L ( M ` A ) ) )
161	12	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( M ` A ) e. P )
162		simpr 461	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= ( M ` A ) ) -> A =/= ( M ` A ) )
163	13	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) e. P )
164	4	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> G e. TarskiG )
165	8	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> A e. P )
166	12	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> ( M ` A ) e. P )
167	7	adantr 465	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> GDimTarskiG≥ 2 )
168		simpr 461	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> ( A (midG ` G ) ( M ` A ) ) = A )
169	1, 2, 3, 164, 167, 165, 166, 168	midcgr 24363	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> ( A .- A ) = ( A .- ( M ` A ) ) )
170	169	eqcomd 2465	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> ( A .- ( M ` A ) ) = ( A .- A ) )
171	1, 2, 3, 164, 165, 166, 165, 170	axtgcgrid 24077	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( A (midG ` G ) ( M ` A ) ) = A ) -> A = ( M ` A ) )
172	171	ex 434	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( A (midG ` G ) ( M ` A ) ) = A -> A = ( M ` A ) ) )
173	172	necon3d 2681	. . . . . . . . . . . . . . 15 |- ( ph -> ( A =/= ( M ` A ) -> ( A (midG ` G ) ( M ` A ) ) =/= A ) )
174	173	imp 429	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( A (midG ` G ) ( M ` A ) ) =/= A )
175	141	adantr 465	. . . . . . . . . . . . . . 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 24216	. . . . . . . . . . . . . 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 24229	. . . . . . . . . . . . 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 24232	. . . . . . . . . . . . 13 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( ( A (midG ` G ) ( M ` A ) ) L A ) = ( A L ( A (midG ` G ) ( M ` A ) ) ) )
179	177, 178	eqtr4d 2501	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= ( M ` A ) ) -> ( A L ( M ` A ) ) = ( ( A (midG ` G ) ( M ` A ) ) L A ) )
180	160, 179	breqtrd 4480	. . . . . . . . . . 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 24319	. . . . . . . . . 10 |- ( ( ph /\ A =/= ( M ` A ) ) -> <" Z ( A (midG ` G ) ( M ` A ) ) A "> e. (∟G ` G ) )
182	151, 181	pm2.61dane 2775	. . . . . . . . 9 |- ( ph -> <" Z ( A (midG ` G ) ( M ` A ) ) A "> e. (∟G ` G ) )
183	1, 2, 3, 10, 20, 4, 18, 13, 8	israg 24291	. . . . . . . . 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 210	. . . . . . . 8 |- ( ph -> ( Z .- A ) = ( Z .- ( ( (pInvG ` G ) ` ( A (midG ` G ) ( M ` A ) ) ) ` A ) ) )
185		eqidd 2458	. . . . . . . . . 10 |- ( ph -> ( A (midG ` G ) ( M ` A ) ) = ( A (midG ` G ) ( M ` A ) ) )
186	1, 2, 3, 4, 7, 8, 12, 20, 13	ismidb 24361	. . . . . . . . . 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 232	. . . . . . . . 9 |- ( ph -> ( M ` A ) = ( ( (pInvG ` G ) ` ( A (midG ` G ) ( M ` A ) ) ) ` A ) )
188	187	oveq2d 6312	. . . . . . . 8 |- ( ph -> ( Z .- ( M ` A ) ) = ( Z .- ( ( (pInvG ` G ) ` ( A (midG ` G ) ( M ` A ) ) ) ` A ) ) )
189	184, 188	eqtr4d 2501	. . . . . . 7 |- ( ph -> ( Z .- A ) = ( Z .- ( M ` A ) ) )
190	1, 2, 3, 10, 20, 4, 18, 21, 8	mircgr 24255	. . . . . . 7 |- ( ph -> ( Z .- ( S ` A ) ) = ( Z .- A ) )
191	1, 2, 3, 10, 20, 4, 18, 21, 12	mircgr 24255	. . . . . . 7 |- ( ph -> ( Z .- ( S ` ( M ` A ) ) ) = ( Z .- ( M ` A ) ) )
192	189, 190, 191	3eqtr4d 2508	. . . . . 6 |- ( ph -> ( Z .- ( S ` A ) ) = ( Z .- ( S ` ( M ` A ) ) ) )
193	192	adantr 465	. . . . 5 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( Z .- ( S ` A ) ) = ( Z .- ( S ` ( M ` A ) ) ) )
194	1, 2, 3, 125, 127, 126, 127, 130, 193	tgcgrcomlr 24088	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( ( S ` A ) .- Z ) = ( ( S ` ( M ` A ) ) .- Z ) )
195	189	adantr 465	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( Z .- A ) = ( Z .- ( M ` A ) ) )
196	21	fveq1i 5873	. . . . . . . . . 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 24366	. . . . . . . . . 10 |- ( ph -> ( ( S ` A ) (midG ` G ) ( S ` ( M ` A ) ) ) = ( S ` ( A (midG ` G ) ( M ` A ) ) ) )
198	6	eqcomi 2470	. . . . . . . . . . 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 24361	. . . . . . . . . . 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 233	. . . . . . . . . 10 |- ( ph -> ( B (midG ` G ) ( M ` B ) ) = ( ( (pInvG ` G ) ` Z ) ` ( A (midG ` G ) ( M ` A ) ) ) )
201	196, 197, 200	3eqtr4a 2524	. . . . . . . . 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 24361	. . . . . . . . 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 232	. . . . . . . 8 |- ( ph -> ( S ` ( M ` A ) ) = ( ( (pInvG ` G ) ` ( B (midG ` G ) ( M ` B ) ) ) ` ( S ` A ) ) )
204	119, 203	oveq12d 6314	. . . . . . 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 2457	. . . . . . . 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 24267	. . . . . . 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 2499	. . . . . 6 |- ( ph -> ( B .- ( S ` A ) ) = ( ( M ` B ) .- ( S ` ( M ` A ) ) ) )
208	207	adantr 465	. . . . 5 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( B .- ( S ` A ) ) = ( ( M ` B ) .- ( S ` ( M ` A ) ) ) )
209	1, 2, 3, 125, 132, 126, 133, 130, 208	tgcgrcomlr 24088	. . . 4 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( ( S ` A ) .- B ) = ( ( S ` ( M ` A ) ) .- ( M ` B ) ) )
210	121	adantr 465	. . . 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 24079	. . 3 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( A .- B ) = ( ( M ` A ) .- ( M ` B ) ) )
212	211	eqcomd 2465	. 2 |- ( ( ph /\ ( S ` A ) =/= Z ) -> ( ( M ` A ) .- ( M ` B ) ) = ( A .- B ) )
213	124, 212	pm2.61dane 2775	1 |- ( ph -> ( ( M ` A ) .- ( M ` B ) ) = ( A .- B ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1395   e. wcel 1819   =/= wne 2652   class class class wbr 4456   ran crn 5009   ` cfv 5594  (class class class)co 6296   2c2 10606   <"cs3 12819   Basecbs 14735   distcds 14812  TarskiGcstrkg 24042  Dim_TarskiG≥cstrkgld 24046  Itvcitv 24049  LineGclng 24050  pInvGcmir 24250  ∟Gcrag 24287  ⟂Gcperpg 24289  midGcmid 24355  lInvGclmi 24356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-concat 12548  df-s1 12549  df-s2 12825  df-s3 12826  df-trkgc 24061  df-trkgb 24062  df-trkgcb 24063  df-trkgld 24065  df-trkg 24067  df-cgrg 24120  df-leg 24187  df-mir 24251  df-rag 24288  df-perpg 24290  df-mid 24357  df-lmi 24358

This theorem is referenced by:  lmiiso  24379