Theorem isnlm 24183

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	⊢ · = ( ·𝑠 ‘𝑊)
isnlm.f	⊢ 𝐹 = (Scalar‘𝑊)
isnlm.k	⊢ 𝐾 = (Base‘𝐹)
isnlm.a	⊢ 𝐴 = (norm‘𝐹)

Assertion

Ref	Expression
isnlm	⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑁,𝑦   𝑥,𝑉,𝑦   𝑥,𝐾   𝑥,𝑊,𝑦   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 469	. 2 ⊢ (((𝑊 ∈ (NrmGrp ∩ LMod) ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))) ↔ (𝑊 ∈ (NrmGrp ∩ LMod) ∧ (𝐹 ∈ NrmRing ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))))
2		df-3an 1089	. . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ NrmRing))
3		elin 3963	. . . . 5 ⊢ (𝑊 ∈ (NrmGrp ∩ LMod) ↔ (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod))
4	3	anbi1i 624	. . . 4 ⊢ ((𝑊 ∈ (NrmGrp ∩ LMod) ∧ 𝐹 ∈ NrmRing) ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ NrmRing))
5	2, 4	bitr4i 277	. . 3 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ↔ (𝑊 ∈ (NrmGrp ∩ LMod) ∧ 𝐹 ∈ NrmRing))
6	5	anbi1i 624	. 2 ⊢ (((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))) ↔ ((𝑊 ∈ (NrmGrp ∩ LMod) ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))))
7		fvexd 6903	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
8		id 22	. . . . . . 7 ⊢ (𝑓 = (Scalar‘𝑤) → 𝑓 = (Scalar‘𝑤))
9		fveq2 6888	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
10		isnlm.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
11	9, 10	eqtr4di 2790	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
12	8, 11	sylan9eqr 2794	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → 𝑓 = 𝐹)
13	12	eleq1d 2818	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (𝑓 ∈ NrmRing ↔ 𝐹 ∈ NrmRing))
14	12	fveq2d 6892	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = (Base‘𝐹))
15		isnlm.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐹)
16	14, 15	eqtr4di 2790	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = 𝐾)
17		simpl 483	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → 𝑤 = 𝑊)
18	17	fveq2d 6892	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑤) = (Base‘𝑊))
19		isnlm.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
20	18, 19	eqtr4di 2790	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑤) = 𝑉)
21	17	fveq2d 6892	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (norm‘𝑤) = (norm‘𝑊))
22		isnlm.n	. . . . . . . . . 10 ⊢ 𝑁 = (norm‘𝑊)
23	21, 22	eqtr4di 2790	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (norm‘𝑤) = 𝑁)
24	17	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
25		isnlm.s	. . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝑊)
26	24, 25	eqtr4di 2790	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ( ·𝑠 ‘𝑤) = · )
27	26	oveqd 7422	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (𝑥( ·𝑠 ‘𝑤)𝑦) = (𝑥 · 𝑦))
28	23, 27	fveq12d 6895	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ((norm‘𝑤)‘(𝑥( ·𝑠 ‘𝑤)𝑦)) = (𝑁‘(𝑥 · 𝑦)))
29	12	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (norm‘𝑓) = (norm‘𝐹))
30		isnlm.a	. . . . . . . . . . 11 ⊢ 𝐴 = (norm‘𝐹)
31	29, 30	eqtr4di 2790	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (norm‘𝑓) = 𝐴)
32	31	fveq1d 6890	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ((norm‘𝑓)‘𝑥) = (𝐴‘𝑥))
33	23	fveq1d 6890	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ((norm‘𝑤)‘𝑦) = (𝑁‘𝑦))
34	32, 33	oveq12d 7423	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))
35	28, 34	eqeq12d 2748	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (((norm‘𝑤)‘(𝑥( ·𝑠 ‘𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)) ↔ (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))))
36	20, 35	raleqbidv 3342	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠 ‘𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)) ↔ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))))
37	16, 36	raleqbidv 3342	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (∀𝑥 ∈ (Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠 ‘𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)) ↔ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))))
38	13, 37	anbi12d 631	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ((𝑓 ∈ NrmRing ∧ ∀𝑥 ∈ (Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠 ‘𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦))) ↔ (𝐹 ∈ NrmRing ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))))
39	7, 38	sbcied 3821	. . 3 ⊢ (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 𝑓](𝑓 ∈ NrmRing ∧ ∀𝑥 ∈ (Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠 ‘𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦))) ↔ (𝐹 ∈ NrmRing ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))))
40		df-nlm 24086	. . 3 ⊢ NrmMod = {𝑤 ∈ (NrmGrp ∩ LMod) ∣ [(Scalar‘𝑤) / 𝑓](𝑓 ∈ NrmRing ∧ ∀𝑥 ∈ (Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠 ‘𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)))}
41	39, 40	elrab2 3685	. 2 ⊢ (𝑊 ∈ NrmMod ↔ (𝑊 ∈ (NrmGrp ∩ LMod) ∧ (𝐹 ∈ NrmRing ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))))
42	1, 6, 41	3bitr4ri 303	1 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474  [wsbc 3776   ∩ cin 3946  ‘cfv 6540  (class class class)co 7405   · cmul 11111  Basecbs 17140  Scalarcsca 17196   ·_𝑠 cvsca 17197  LModclmod 20463  normcnm 24076  NrmGrpcngp 24077  NrmRingcnrg 24079  NrmModcnlm 24080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7408  df-nlm 24086

This theorem is referenced by:  nmvs  24184  nlmngp  24185  nlmlmod  24186  nlmnrg  24187  sranlm  24192  lssnlm  24209  isncvsngp  24657  tcphcph  24745  cnzh  32938  rezh  32939