Theorem isnlm 22479

Description: A normed (left) module is a module which is also a normed group over a normed ring, such that the norm distributes over scalar multiplication. (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
isnlm	|- ( 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 ) ) ) )

Distinct variable groups:   x, y, A   x, N, y   x, V, y   x, K   x, W, y   x, .x. , y

Allowed substitution hints:   F( x, y)   K( y)

Proof of Theorem isnlm

Dummy variables w f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		anass 681	. 2 |- ( ( ( W e. (NrmGrp i^i LMod ) /\ F e. NrmRing ) /\ A. x e. K A. y e. V ( N ` ( x .x. y ) ) = ( ( A ` x ) x. ( N ` y ) ) ) <-> ( W e. (NrmGrp i^i LMod ) /\ ( F e. NrmRing /\ A. x e. K A. y e. V ( N ` ( x .x. y ) ) = ( ( A ` x ) x. ( N ` y ) ) ) ) )
2		df-3an 1039	. . . 4 |- ( ( W e. NrmGrp /\ W e. LMod /\ F e. NrmRing ) <-> ( ( W e. NrmGrp /\ W e. LMod ) /\ F e. NrmRing ) )
3		elin 3796	. . . . 5 |- ( W e. (NrmGrp i^i LMod ) <-> ( W e. NrmGrp /\ W e. LMod ) )
4	3	anbi1i 731	. . . 4 |- ( ( W e. (NrmGrp i^i LMod ) /\ F e. NrmRing ) <-> ( ( W e. NrmGrp /\ W e. LMod ) /\ F e. NrmRing ) )
5	2, 4	bitr4i 267	. . 3 |- ( ( W e. NrmGrp /\ W e. LMod /\ F e. NrmRing ) <-> ( W e. (NrmGrp i^i LMod ) /\ F e. NrmRing ) )
6	5	anbi1i 731	. 2 |- ( ( ( 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 ) ) ) <-> ( ( W e. (NrmGrp i^i LMod ) /\ F e. NrmRing ) /\ A. x e. K A. y e. V ( N ` ( x .x. y ) ) = ( ( A ` x ) x. ( N ` y ) ) ) )
7		fvexd 6203	. . . 4 |- ( w = W -> (Scalar ` w ) e. _V )
8		id 22	. . . . . . 7 |- ( f = (Scalar ` w ) -> f = (Scalar ` w ) )
9		fveq2 6191	. . . . . . . 8 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
10		isnlm.f	. . . . . . . 8 |- F = (Scalar ` W )
11	9, 10	syl6eqr 2674	. . . . . . 7 |- ( w = W -> (Scalar ` w ) = F )
12	8, 11	sylan9eqr 2678	. . . . . 6 |- ( ( w = W /\ f = (Scalar ` w ) ) -> f = F )
13	12	eleq1d 2686	. . . . 5 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( f e. NrmRing <-> F e. NrmRing ) )
14	12	fveq2d 6195	. . . . . . 7 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( Base ` f ) = ( Base ` F ) )
15		isnlm.k	. . . . . . 7 |- K = ( Base ` F )
16	14, 15	syl6eqr 2674	. . . . . 6 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( Base ` f ) = K )
17		simpl 473	. . . . . . . . 9 |- ( ( w = W /\ f = (Scalar ` w ) ) -> w = W )
18	17	fveq2d 6195	. . . . . . . 8 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( Base ` w ) = ( Base ` W ) )
19		isnlm.v	. . . . . . . 8 |- V = ( Base ` W )
20	18, 19	syl6eqr 2674	. . . . . . 7 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( Base ` w ) = V )
21	17	fveq2d 6195	. . . . . . . . . 10 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( norm ` w ) = ( norm ` W ) )
22		isnlm.n	. . . . . . . . . 10 |- N = ( norm ` W )
23	21, 22	syl6eqr 2674	. . . . . . . . 9 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( norm ` w ) = N )
24	17	fveq2d 6195	. . . . . . . . . . 11 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( .s ` w ) = ( .s ` W ) )
25		isnlm.s	. . . . . . . . . . 11 |- .x. = ( .s ` W )
26	24, 25	syl6eqr 2674	. . . . . . . . . 10 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( .s ` w ) = .x. )
27	26	oveqd 6667	. . . . . . . . 9 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( x ( .s ` w ) y ) = ( x .x. y ) )
28	23, 27	fveq12d 6197	. . . . . . . 8 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( ( norm ` w ) ` ( x ( .s ` w ) y ) ) = ( N ` ( x .x. y ) ) )
29	12	fveq2d 6195	. . . . . . . . . . 11 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( norm ` f ) = ( norm ` F ) )
30		isnlm.a	. . . . . . . . . . 11 |- A = ( norm ` F )
31	29, 30	syl6eqr 2674	. . . . . . . . . 10 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( norm ` f ) = A )
32	31	fveq1d 6193	. . . . . . . . 9 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( ( norm ` f ) ` x ) = ( A ` x ) )
33	23	fveq1d 6193	. . . . . . . . 9 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( ( norm ` w ) ` y ) = ( N ` y ) )
34	32, 33	oveq12d 6668	. . . . . . . 8 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( ( ( norm ` f ) ` x ) x. ( ( norm ` w ) ` y ) ) = ( ( A ` x ) x. ( N ` y ) ) )
35	28, 34	eqeq12d 2637	. . . . . . 7 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( ( ( norm ` w ) ` ( x ( .s ` w ) y ) ) = ( ( ( norm ` f ) ` x ) x. ( ( norm ` w ) ` y ) ) <-> ( N ` ( x .x. y ) ) = ( ( A ` x ) x. ( N ` y ) ) ) )
36	20, 35	raleqbidv 3152	. . . . . 6 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( A. y e. ( Base ` w ) ( ( norm ` w ) ` ( x ( .s ` w ) y ) ) = ( ( ( norm ` f ) ` x ) x. ( ( norm ` w ) ` y ) ) <-> A. y e. V ( N ` ( x .x. y ) ) = ( ( A ` x ) x. ( N ` y ) ) ) )
37	16, 36	raleqbidv 3152	. . . . 5 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( A. x e. ( Base ` f ) A. y e. ( Base ` w ) ( ( norm ` w ) ` ( x ( .s ` w ) y ) ) = ( ( ( norm ` f ) ` x ) x. ( ( norm ` w ) ` y ) ) <-> A. x e. K A. y e. V ( N ` ( x .x. y ) ) = ( ( A ` x ) x. ( N ` y ) ) ) )
38	13, 37	anbi12d 747	. . . 4 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( ( f e. NrmRing /\ A. x e. ( Base ` f ) A. y e. ( Base ` w ) ( ( norm ` w ) ` ( x ( .s ` w ) y ) ) = ( ( ( norm ` f ) ` x ) x. ( ( norm ` w ) ` y ) ) ) <-> ( F e. NrmRing /\ A. x e. K A. y e. V ( N ` ( x .x. y ) ) = ( ( A ` x ) x. ( N ` y ) ) ) ) )
39	7, 38	sbcied 3472	. . 3 |- ( w = W -> ( [. (Scalar ` w ) / f ]. ( f e. NrmRing /\ A. x e. ( Base ` f ) A. y e. ( Base ` w ) ( ( norm ` w ) ` ( x ( .s ` w ) y ) ) = ( ( ( norm ` f ) ` x ) x. ( ( norm ` w ) ` y ) ) ) <-> ( F e. NrmRing /\ A. x e. K A. y e. V ( N ` ( x .x. y ) ) = ( ( A ` x ) x. ( N ` y ) ) ) ) )
40		df-nlm 22391	. . 3 |- NrmMod = { w e. (NrmGrp i^i LMod ) | [. (Scalar ` w ) / f ]. ( f e. NrmRing /\ A. x e. ( Base ` f ) A. y e. ( Base ` w ) ( ( norm ` w ) ` ( x ( .s ` w ) y ) ) = ( ( ( norm ` f ) ` x ) x. ( ( norm ` w ) ` y ) ) ) }
41	39, 40	elrab2 3366	. 2 |- ( W e. NrmMod <-> ( W e. (NrmGrp i^i LMod ) /\ ( F e. NrmRing /\ A. x e. K A. y e. V ( N ` ( x .x. y ) ) = ( ( A ` x ) x. ( N ` y ) ) ) ) )
42	1, 6, 41	3bitr4ri 293	1 |- ( 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 ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   i^i cin 3573   ` 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:  nmvs  22480  nlmngp  22481  nlmlmod  22482  nlmnrg  22483  sranlm  22488  lssnlm  22505  isncvsngp  22949  tchcph  23036  cnzh  30014  rezh  30015