Theorem imsmetlem 9655

Description: Lemma for imsmet 9656.

Hypotheses

Ref	Expression
imsmetlem.1	|- X = (BaseSet` U)
imsmetlem.2	|- G = (+v` U)
imsmetlem.7	|- M = (inv` G)
imsmetlem.4	|- S = (.s` U)
imsmetlem.5	|- Z = (0v` U)
imsmetlem.6	|- N = (norm` U)
imsmetlem.8	|- D = (IndMet` U)
imsmetlem.9	|- U e. NrmCVec

Assertion

Ref	Expression
imsmetlem	|- D e. Met

Proof of Theorem imsmetlem

Step	Hyp	Ref	Expression
1		imsmetlem.1	. . 3 |- X = (BaseSet` U)
2		fvex 4689	. . 3 |- (BaseSet` U) e. _V
3	1, 2	eqeltri 1967	. 2 |- X e. _V
4		imsmetlem.9	. . 3 |- U e. NrmCVec
5		imsmetlem.8	. . . 4 |- D = (IndMet` U)
6	1, 5	imsdf 9652	. . 3 |- (U e. NrmCVec -> D:(X X. X)-->RR)
7	4, 6	ax-mp 7	. 2 |- D:(X X. X)-->RR
8		imsmetlem.2	. . . . . 6 |- G = (+v` U)
9		imsmetlem.4	. . . . . 6 |- S = (.s` U)
10		imsmetlem.6	. . . . . 6 |- N = (norm` U)
11	1, 8, 9, 10, 5	imsdval2 9650	. . . . 5 |- ((U e. NrmCVec /\ x e. X /\ y e. X) -> (xDy) = (N` (xG(-u1Sy))))
12	4, 11	mp3an1 1178	. . . 4 |- ((x e. X /\ y e. X) -> (xDy) = (N` (xG(-u1Sy))))
13	12	eqeq1d 1892	. . 3 |- ((x e. X /\ y e. X) -> ((xDy) = 0 <-> (N` (xG(-u1Sy))) = 0))
14	1, 8	nvgcl 9571	. . . . . 6 |- ((U e. NrmCVec /\ x e. X /\ (-u1Sy) e. X) -> (xG(-u1Sy)) e. X)
15	4, 14	mp3an1 1178	. . . . 5 |- ((x e. X /\ (-u1Sy) e. X) -> (xG(-u1Sy)) e. X)
16		ax1cn 6422	. . . . . . 7 |- 1 e. CC
17	16	negcli 6526	. . . . . 6 |- -u1 e. CC
18	1, 9	nvscl 9579	. . . . . 6 |- ((U e. NrmCVec /\ -u1 e. CC /\ y e. X) -> (-u1Sy) e. X)
19	4, 17, 18	mp3an12 1181	. . . . 5 |- (y e. X -> (-u1Sy) e. X)
20	15, 19	sylan2 500	. . . 4 |- ((x e. X /\ y e. X) -> (xG(-u1Sy)) e. X)
21		imsmetlem.5	. . . . . 6 |- Z = (0v` U)
22	1, 21, 10	nvz 9629	. . . . 5 |- ((U e. NrmCVec /\ (xG(-u1Sy)) e. X) -> ((N` (xG(-u1Sy))) = 0 <-> (xG(-u1Sy)) = Z))
23	4, 22	mpan 759	. . . 4 |- ((xG(-u1Sy)) e. X -> ((N` (xG(-u1Sy))) = 0 <-> (xG(-u1Sy)) = Z))
24	20, 23	syl 12	. . 3 |- ((x e. X /\ y e. X) -> ((N` (xG(-u1Sy))) = 0 <-> (xG(-u1Sy)) = Z))
25	1, 21	nvzcl 9587	. . . . . . 7 |- (U e. NrmCVec -> Z e. X)
26	4, 25	ax-mp 7	. . . . . 6 |- Z e. X
27	1, 8	nvrcan 9576	. . . . . . 7 |- ((U e. NrmCVec /\ ((xG(-u1Sy)) e. X /\ Z e. X /\ y e. X)) -> (((xG(-u1Sy))Gy) = (ZGy) <-> (xG(-u1Sy)) = Z))
28	4, 27	mpan 759	. . . . . 6 |- (((xG(-u1Sy)) e. X /\ Z e. X /\ y e. X) -> (((xG(-u1Sy))Gy) = (ZGy) <-> (xG(-u1Sy)) = Z))
29	26, 28	mp3an2 1179	. . . . 5 |- (((xG(-u1Sy)) e. X /\ y e. X) -> (((xG(-u1Sy))Gy) = (ZGy) <-> (xG(-u1Sy)) = Z))
30	20, 29	sylancom 531	. . . 4 |- ((x e. X /\ y e. X) -> (((xG(-u1Sy))Gy) = (ZGy) <-> (xG(-u1Sy)) = Z))
31		simpl 346	. . . . . . 7 |- ((x e. X /\ y e. X) -> x e. X)
32	19	adantl 424	. . . . . . 7 |- ((x e. X /\ y e. X) -> (-u1Sy) e. X)
33		simpr 350	. . . . . . 7 |- ((x e. X /\ y e. X) -> y e. X)
34	1, 8	nvass 9573	. . . . . . . 8 |- ((U e. NrmCVec /\ (x e. X /\ (-u1Sy) e. X /\ y e. X)) -> ((xG(-u1Sy))Gy) = (xG((-u1Sy)Gy)))
35	4, 34	mpan 759	. . . . . . 7 |- ((x e. X /\ (-u1Sy) e. X /\ y e. X) -> ((xG(-u1Sy))Gy) = (xG((-u1Sy)Gy)))
36	31, 32, 33, 35	syl111anc 1100	. . . . . 6 |- ((x e. X /\ y e. X) -> ((xG(-u1Sy))Gy) = (xG((-u1Sy)Gy)))
37	1, 8, 9, 21	nvlinv 9606	. . . . . . . . 9 |- ((U e. NrmCVec /\ y e. X) -> ((-u1Sy)Gy) = Z)
38	4, 37	mpan 759	. . . . . . . 8 |- (y e. X -> ((-u1Sy)Gy) = Z)
39	38	adantl 424	. . . . . . 7 |- ((x e. X /\ y e. X) -> ((-u1Sy)Gy) = Z)
40	39	opreq2d 4898	. . . . . 6 |- ((x e. X /\ y e. X) -> (xG((-u1Sy)Gy)) = (xGZ))
41	1, 8, 21	nv0rid 9588	. . . . . . . 8 |- ((U e. NrmCVec /\ x e. X) -> (xGZ) = x)
42	4, 41	mpan 759	. . . . . . 7 |- (x e. X -> (xGZ) = x)
43	42	adantr 425	. . . . . 6 |- ((x e. X /\ y e. X) -> (xGZ) = x)
44	36, 40, 43	3eqtrd 1929	. . . . 5 |- ((x e. X /\ y e. X) -> ((xG(-u1Sy))Gy) = x)
45	1, 8, 21	nv0lid 9589	. . . . . . 7 |- ((U e. NrmCVec /\ y e. X) -> (ZGy) = y)
46	4, 45	mpan 759	. . . . . 6 |- (y e. X -> (ZGy) = y)
47	46	adantl 424	. . . . 5 |- ((x e. X /\ y e. X) -> (ZGy) = y)
48	44, 47	eqeq12d 1899	. . . 4 |- ((x e. X /\ y e. X) -> (((xG(-u1Sy))Gy) = (ZGy) <-> x = y))
49	30, 48	bitr3d 589	. . 3 |- ((x e. X /\ y e. X) -> ((xG(-u1Sy)) = Z <-> x = y))
50	13, 24, 49	3bitrd 603	. 2 |- ((x e. X /\ y e. X) -> ((xDy) = 0 <-> x = y))
51		simpr 350	. . . . . . 7 |- ((z e. X /\ x e. X) -> x e. X)
52	1, 9	nvscl 9579	. . . . . . . . 9 |- ((U e. NrmCVec /\ -u1 e. CC /\ z e. X) -> (-u1Sz) e. X)
53	4, 17, 52	mp3an12 1181	. . . . . . . 8 |- (z e. X -> (-u1Sz) e. X)
54	53	adantr 425	. . . . . . 7 |- ((z e. X /\ x e. X) -> (-u1Sz) e. X)
55	1, 8	nvgcl 9571	. . . . . . . 8 |- ((U e. NrmCVec /\ x e. X /\ (-u1Sz) e. X) -> (xG(-u1Sz)) e. X)
56	4, 55	mp3an1 1178	. . . . . . 7 |- ((x e. X /\ (-u1Sz) e. X) -> (xG(-u1Sz)) e. X)
57	51, 54, 56	syl11anc 524	. . . . . 6 |- ((z e. X /\ x e. X) -> (xG(-u1Sz)) e. X)
58	57	3adant3 896	. . . . 5 |- ((z e. X /\ x e. X /\ y e. X) -> (xG(-u1Sz)) e. X)
59	1, 8	nvgcl 9571	. . . . . . . 8 |- ((U e. NrmCVec /\ z e. X /\ (-u1Sy) e. X) -> (zG(-u1Sy)) e. X)
60	4, 59	mp3an1 1178	. . . . . . 7 |- ((z e. X /\ (-u1Sy) e. X) -> (zG(-u1Sy)) e. X)
61	60, 19	sylan2 500	. . . . . 6 |- ((z e. X /\ y e. X) -> (zG(-u1Sy)) e. X)
62	61	3adant2 895	. . . . 5 |- ((z e. X /\ x e. X /\ y e. X) -> (zG(-u1Sy)) e. X)
63	1, 8, 10	nvtri 9630	. . . . . 6 |- ((U e. NrmCVec /\ (xG(-u1Sz)) e. X /\ (zG(-u1Sy)) e. X) -> (N` ((xG(-u1Sz))G(zG(-u1Sy)))) <_ ((N` (xG(-u1Sz))) + (N` (zG(-u1Sy)))))
64	4, 63	mp3an1 1178	. . . . 5 |- (((xG(-u1Sz)) e. X /\ (zG(-u1Sy)) e. X) -> (N` ((xG(-u1Sz))G(zG(-u1Sy)))) <_ ((N` (xG(-u1Sz))) + (N` (zG(-u1Sy)))))
65	58, 62, 64	syl11anc 524	. . . 4 |- ((z e. X /\ x e. X /\ y e. X) -> (N` ((xG(-u1Sz))G(zG(-u1Sy)))) <_ ((N` (xG(-u1Sz))) + (N` (zG(-u1Sy)))))
66	12	3adant1 894	. . . . 5 |- ((z e. X /\ x e. X /\ y e. X) -> (xDy) = (N` (xG(-u1Sy))))
67		simp1 876	. . . . . . . 8 |- ((z e. X /\ x e. X /\ y e. X) -> z e. X)
68	19	3ad2ant3 899	. . . . . . . 8 |- ((z e. X /\ x e. X /\ y e. X) -> (-u1Sy) e. X)
69	1, 8	nvass 9573	. . . . . . . . 9 |- ((U e. NrmCVec /\ ((xG(-u1Sz)) e. X /\ z e. X /\ (-u1Sy) e. X)) -> (((xG(-u1Sz))Gz)G(-u1Sy)) = ((xG(-u1Sz))G(zG(-u1Sy))))
70	4, 69	mpan 759	. . . . . . . 8 |- (((xG(-u1Sz)) e. X /\ z e. X /\ (-u1Sy) e. X) -> (((xG(-u1Sz))Gz)G(-u1Sy)) = ((xG(-u1Sz))G(zG(-u1Sy))))
71	58, 67, 68, 70	syl111anc 1100	. . . . . . 7 |- ((z e. X /\ x e. X /\ y e. X) -> (((xG(-u1Sz))Gz)G(-u1Sy)) = ((xG(-u1Sz))G(zG(-u1Sy))))
72		simpl 346	. . . . . . . . . . 11 |- ((z e. X /\ x e. X) -> z e. X)
73	1, 8	nvass 9573	. . . . . . . . . . . 12 |- ((U e. NrmCVec /\ (x e. X /\ (-u1Sz) e. X /\ z e. X)) -> ((xG(-u1Sz))Gz) = (xG((-u1Sz)Gz)))
74	4, 73	mpan 759	. . . . . . . . . . 11 |- ((x e. X /\ (-u1Sz) e. X /\ z e. X) -> ((xG(-u1Sz))Gz) = (xG((-u1Sz)Gz)))
75	51, 54, 72, 74	syl111anc 1100	. . . . . . . . . 10 |- ((z e. X /\ x e. X) -> ((xG(-u1Sz))Gz) = (xG((-u1Sz)Gz)))
76	1, 8, 9, 21	nvlinv 9606	. . . . . . . . . . . . 13 |- ((U e. NrmCVec /\ z e. X) -> ((-u1Sz)Gz) = Z)
77	4, 76	mpan 759	. . . . . . . . . . . 12 |- (z e. X -> ((-u1Sz)Gz) = Z)
78	77	adantr 425	. . . . . . . . . . 11 |- ((z e. X /\ x e. X) -> ((-u1Sz)Gz) = Z)
79	78	opreq2d 4898	. . . . . . . . . 10 |- ((z e. X /\ x e. X) -> (xG((-u1Sz)Gz)) = (xGZ))
80	42	adantl 424	. . . . . . . . . 10 |- ((z e. X /\ x e. X) -> (xGZ) = x)
81	75, 79, 80	3eqtrd 1929	. . . . . . . . 9 |- ((z e. X /\ x e. X) -> ((xG(-u1Sz))Gz) = x)
82	81	3adant3 896	. . . . . . . 8 |- ((z e. X /\ x e. X /\ y e. X) -> ((xG(-u1Sz))Gz) = x)
83	82	opreq1d 4897	. . . . . . 7 |- ((z e. X /\ x e. X /\ y e. X) -> (((xG(-u1Sz))Gz)G(-u1Sy)) = (xG(-u1Sy)))
84	71, 83	eqtr3d 1927	. . . . . 6 |- ((z e. X /\ x e. X /\ y e. X) -> ((xG(-u1Sz))G(zG(-u1Sy))) = (xG(-u1Sy)))
85	84	fveq2d 4685	. . . . 5 |- ((z e. X /\ x e. X /\ y e. X) -> (N` ((xG(-u1Sz))G(zG(-u1Sy)))) = (N` (xG(-u1Sy))))
86	66, 85	eqtr4d 1928	. . . 4 |- ((z e. X /\ x e. X /\ y e. X) -> (xDy) = (N` ((xG(-u1Sz))G(zG(-u1Sy)))))
87	1, 8, 9, 10, 5	imsdval2 9650	. . . . . . . 8 |- ((U e. NrmCVec /\ z e. X /\ x e. X) -> (zDx) = (N` (zG(-u1Sx))))
88	4, 87	mp3an1 1178	. . . . . . 7 |- ((z e. X /\ x e. X) -> (zDx) = (N` (zG(-u1Sx))))
89	1, 8, 9, 10	nvdif 9625	. . . . . . . 8 |- ((U e. NrmCVec /\ z e. X /\ x e. X) -> (N` (zG(-u1Sx))) = (N` (xG(-u1Sz))))
90	4, 89	mp3an1 1178	. . . . . . 7 |- ((z e. X /\ x e. X) -> (N` (zG(-u1Sx))) = (N` (xG(-u1Sz))))
91	88, 90	eqtrd 1925	. . . . . 6 |- ((z e. X /\ x e. X) -> (zDx) = (N` (xG(-u1Sz))))
92	91	3adant3 896	. . . . 5 |- ((z e. X /\ x e. X /\ y e. X) -> (zDx) = (N` (xG(-u1Sz))))
93	1, 8, 9, 10, 5	imsdval2 9650	. . . . . . 7 |- ((U e. NrmCVec /\ z e. X /\ y e. X) -> (zDy) = (N` (zG(-u1Sy))))
94	4, 93	mp3an1 1178	. . . . . 6 |- ((z e. X /\ y e. X) -> (zDy) = (N` (zG(-u1Sy))))
95	94	3adant2 895	. . . . 5 |- ((z e. X /\ x e. X /\ y e. X) -> (zDy) = (N` (zG(-u1Sy))))
96	92, 95	opreq12d 4900	. . . 4 |- ((z e. X /\ x e. X /\ y e. X) -> ((zDx) + (zDy)) = ((N` (xG(-u1Sz))) + (N` (zG(-u1Sy)))))
97	65, 86, 96	3brtr4d 3367	. . 3 |- ((z e. X /\ x e. X /\ y e. X) -> (xDy) <_ ((zDx) + (zDy)))
98	97	3coml 1075	. 2 |- ((x e. X /\ y e. X /\ z e. X) -> (xDy) <_ ((zDx) + (zDy)))
99	3, 7, 50, 98	ismeti 9079	1 |- D e. Met

Syntax hints:   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292   class class class wbr 3338   X. cxp 3984  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389  -ucneg 6446   <_ cle 6448  Metcme 9066  invcgn 9313  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  0vcn0v 9539  normcnm 9541  IndMetcims 9542

This theorem is referenced by:  imsmet 9656

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-met 9070  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-vs 9550  df-nm 9551  df-ims 9552

Copyright terms: Public domain