Theorem nvtri 27525

Description: Triangle inequality for the norm of a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvtri.1	|- X = ( BaseSet ` U )
nvtri.2	|- G = ( +v ` U )
nvtri.6	|- N = ( normCV ` U )

Assertion

Ref	Expression
nvtri	|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( A G B ) ) <_ ( ( N ` A ) + ( N ` B ) ) )

Proof of Theorem nvtri

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nvtri.1	. . . . . . 7 |- X = ( BaseSet ` U )
2		nvtri.2	. . . . . . 7 |- G = ( +v ` U )
3		eqid 2622	. . . . . . . . 9 |- ( .sOLD ` U ) = ( .sOLD ` U )
4	3	smfval 27460	. . . . . . . 8 |- ( .sOLD ` U ) = ( 2nd ` ( 1st ` U ) )
5	4	eqcomi 2631	. . . . . . 7 |- ( 2nd ` ( 1st ` U ) ) = ( .sOLD ` U )
6		eqid 2622	. . . . . . 7 |- ( 0vec ` U ) = ( 0vec ` U )
7		nvtri.6	. . . . . . 7 |- N = ( normCV ` U )
8	1, 2, 5, 6, 7	nvi 27469	. . . . . 6 |- ( U e. NrmCVec -> ( <. G , ( 2nd ` ( 1st ` U ) ) >. e. CVecOLD /\ N : X --> RR /\ A. x e. X ( ( ( N ` x ) = 0 -> x = ( 0vec ` U ) ) /\ A. y e. CC ( N ` ( y ( 2nd ` ( 1st ` U ) ) x ) ) = ( ( abs ` y ) x. ( N ` x ) ) /\ A. y e. X ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) ) ) )
9	8	simp3d 1075	. . . . 5 |- ( U e. NrmCVec -> A. x e. X ( ( ( N ` x ) = 0 -> x = ( 0vec ` U ) ) /\ A. y e. CC ( N ` ( y ( 2nd ` ( 1st ` U ) ) x ) ) = ( ( abs ` y ) x. ( N ` x ) ) /\ A. y e. X ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) ) )
10		simp3 1063	. . . . . 6 |- ( ( ( ( N ` x ) = 0 -> x = ( 0vec ` U ) ) /\ A. y e. CC ( N ` ( y ( 2nd ` ( 1st ` U ) ) x ) ) = ( ( abs ` y ) x. ( N ` x ) ) /\ A. y e. X ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) ) -> A. y e. X ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) )
11	10	ralimi 2952	. . . . 5 |- ( A. x e. X ( ( ( N ` x ) = 0 -> x = ( 0vec ` U ) ) /\ A. y e. CC ( N ` ( y ( 2nd ` ( 1st ` U ) ) x ) ) = ( ( abs ` y ) x. ( N ` x ) ) /\ A. y e. X ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) ) -> A. x e. X A. y e. X ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) )
12	9, 11	syl 17	. . . 4 |- ( U e. NrmCVec -> A. x e. X A. y e. X ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) )
13		oveq1 6657	. . . . . . 7 |- ( x = A -> ( x G y ) = ( A G y ) )
14	13	fveq2d 6195	. . . . . 6 |- ( x = A -> ( N ` ( x G y ) ) = ( N ` ( A G y ) ) )
15		fveq2 6191	. . . . . . 7 |- ( x = A -> ( N ` x ) = ( N ` A ) )
16	15	oveq1d 6665	. . . . . 6 |- ( x = A -> ( ( N ` x ) + ( N ` y ) ) = ( ( N ` A ) + ( N ` y ) ) )
17	14, 16	breq12d 4666	. . . . 5 |- ( x = A -> ( ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) <-> ( N ` ( A G y ) ) <_ ( ( N ` A ) + ( N ` y ) ) ) )
18		oveq2 6658	. . . . . . 7 |- ( y = B -> ( A G y ) = ( A G B ) )
19	18	fveq2d 6195	. . . . . 6 |- ( y = B -> ( N ` ( A G y ) ) = ( N ` ( A G B ) ) )
20		fveq2 6191	. . . . . . 7 |- ( y = B -> ( N ` y ) = ( N ` B ) )
21	20	oveq2d 6666	. . . . . 6 |- ( y = B -> ( ( N ` A ) + ( N ` y ) ) = ( ( N ` A ) + ( N ` B ) ) )
22	19, 21	breq12d 4666	. . . . 5 |- ( y = B -> ( ( N ` ( A G y ) ) <_ ( ( N ` A ) + ( N ` y ) ) <-> ( N ` ( A G B ) ) <_ ( ( N ` A ) + ( N ` B ) ) ) )
23	17, 22	rspc2v 3322	. . . 4 |- ( ( A e. X /\ B e. X ) -> ( A. x e. X A. y e. X ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) -> ( N ` ( A G B ) ) <_ ( ( N ` A ) + ( N ` B ) ) ) )
24	12, 23	syl5 34	. . 3 |- ( ( A e. X /\ B e. X ) -> ( U e. NrmCVec -> ( N ` ( A G B ) ) <_ ( ( N ` A ) + ( N ` B ) ) ) )
25	24	3impia 1261	. 2 |- ( ( A e. X /\ B e. X /\ U e. NrmCVec ) -> ( N ` ( A G B ) ) <_ ( ( N ` A ) + ( N ` B ) ) )
26	25	3comr 1273	1 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( A G B ) ) <_ ( ( N ` A ) + ( N ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   <_ cle 10075   abscabs 13974   CVecOLDcvc 27413   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   0veccn0v 27443   norm_CVcnmcv 27445

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-1st 7168  df-2nd 7169  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455

This theorem is referenced by:  nvmtri  27526  nvabs  27527  nvge0  27528  imsmetlem  27545  vacn  27549