Theorem nmvs 22480

Description: Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
isnlm.v	|- V = ( Base ` W )
isnlm.n	|- N = ( norm ` W )
isnlm.s	|- .x. = ( .s ` W )
isnlm.f	|- F = (Scalar ` W )
isnlm.k	|- K = ( Base ` F )
isnlm.a	|- A = ( norm ` F )

Assertion

Ref	Expression
nmvs	|- ( ( W e. NrmMod /\ X e. K /\ Y e. V ) -> ( N ` ( X .x. Y ) ) = ( ( A ` X ) x. ( N ` Y ) ) )

Proof of Theorem nmvs

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

Step	Hyp	Ref	Expression
1		isnlm.v	. . . . 5 |- V = ( Base ` W )
2		isnlm.n	. . . . 5 |- N = ( norm ` W )
3		isnlm.s	. . . . 5 |- .x. = ( .s ` W )
4		isnlm.f	. . . . 5 |- F = (Scalar ` W )
5		isnlm.k	. . . . 5 |- K = ( Base ` F )
6		isnlm.a	. . . . 5 |- A = ( norm ` F )
7	1, 2, 3, 4, 5, 6	isnlm 22479	. . . 4 |- ( W e. NrmMod <-> ( ( W e. NrmGrp /\ W e. LMod /\ F e. NrmRing ) /\ A. x e. K A. y e. V ( N ` ( x .x. y ) ) = ( ( A ` x ) x. ( N ` y ) ) ) )
8	7	simprbi 480	. . 3 |- ( W e. NrmMod -> A. x e. K A. y e. V ( N ` ( x .x. y ) ) = ( ( A ` x ) x. ( N ` y ) ) )
9		oveq1 6657	. . . . . 6 |- ( x = X -> ( x .x. y ) = ( X .x. y ) )
10	9	fveq2d 6195	. . . . 5 |- ( x = X -> ( N ` ( x .x. y ) ) = ( N ` ( X .x. y ) ) )
11		fveq2 6191	. . . . . 6 |- ( x = X -> ( A ` x ) = ( A ` X ) )
12	11	oveq1d 6665	. . . . 5 |- ( x = X -> ( ( A ` x ) x. ( N ` y ) ) = ( ( A ` X ) x. ( N ` y ) ) )
13	10, 12	eqeq12d 2637	. . . 4 |- ( x = X -> ( ( N ` ( x .x. y ) ) = ( ( A ` x ) x. ( N ` y ) ) <-> ( N ` ( X .x. y ) ) = ( ( A ` X ) x. ( N ` y ) ) ) )
14		oveq2 6658	. . . . . 6 |- ( y = Y -> ( X .x. y ) = ( X .x. Y ) )
15	14	fveq2d 6195	. . . . 5 |- ( y = Y -> ( N ` ( X .x. y ) ) = ( N ` ( X .x. Y ) ) )
16		fveq2 6191	. . . . . 6 |- ( y = Y -> ( N ` y ) = ( N ` Y ) )
17	16	oveq2d 6666	. . . . 5 |- ( y = Y -> ( ( A ` X ) x. ( N ` y ) ) = ( ( A ` X ) x. ( N ` Y ) ) )
18	15, 17	eqeq12d 2637	. . . 4 |- ( y = Y -> ( ( N ` ( X .x. y ) ) = ( ( A ` X ) x. ( N ` y ) ) <-> ( N ` ( X .x. Y ) ) = ( ( A ` X ) x. ( N ` Y ) ) ) )
19	13, 18	rspc2v 3322	. . 3 |- ( ( X e. K /\ Y e. V ) -> ( A. x e. K A. y e. V ( N ` ( x .x. y ) ) = ( ( A ` x ) x. ( N ` y ) ) -> ( N ` ( X .x. Y ) ) = ( ( A ` X ) x. ( N ` Y ) ) ) )
20	8, 19	syl5com 31	. 2 |- ( W e. NrmMod -> ( ( X e. K /\ Y e. V ) -> ( N ` ( X .x. Y ) ) = ( ( A ` X ) x. ( N ` Y ) ) ) )
21	20	3impib 1262	1 |- ( ( W e. NrmMod /\ X e. K /\ Y e. V ) -> ( N ` ( X .x. Y ) ) = ( ( A ` X ) x. ( N ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   x. cmul 9941   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   LModclmod 18863   normcnm 22381  NrmGrpcngp 22382  NrmRingcnrg 22384  NrmModcnlm 22385

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-nlm 22391

This theorem is referenced by:  nlmdsdi  22485  nlmdsdir  22486  nlmmul0or  22487  lssnlm  22505  nmoleub2lem3  22915  nmoleub3  22919  ncvsprp  22952  cphnmvs  22990  nmmulg  30012