Theorem isnlm 23888

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 470	. 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 3908	. . . . 5 ⊢ (𝑊 ∈ (NrmGrp ∩ LMod) ↔ (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod))
4	3	anbi1i 625	. . . 4 ⊢ ((𝑊 ∈ (NrmGrp ∩ LMod) ∧ 𝐹 ∈ NrmRing) ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ NrmRing))
5	2, 4	bitr4i 278	. . 3 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ↔ (𝑊 ∈ (NrmGrp ∩ LMod) ∧ 𝐹 ∈ NrmRing))
6	5	anbi1i 625	. 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 6819	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
8		id 22	. . . . . . 7 ⊢ (𝑓 = (Scalar‘𝑤) → 𝑓 = (Scalar‘𝑤))
9		fveq2 6804	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
10		isnlm.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
11	9, 10	eqtr4di 2794	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
12	8, 11	sylan9eqr 2798	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → 𝑓 = 𝐹)
13	12	eleq1d 2821	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (𝑓 ∈ NrmRing ↔ 𝐹 ∈ NrmRing))
14	12	fveq2d 6808	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = (Base‘𝐹))
15		isnlm.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐹)
16	14, 15	eqtr4di 2794	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = 𝐾)
17		simpl 484	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → 𝑤 = 𝑊)
18	17	fveq2d 6808	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑤) = (Base‘𝑊))
19		isnlm.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
20	18, 19	eqtr4di 2794	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑤) = 𝑉)
21	17	fveq2d 6808	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (norm‘𝑤) = (norm‘𝑊))
22		isnlm.n	. . . . . . . . . 10 ⊢ 𝑁 = (norm‘𝑊)
23	21, 22	eqtr4di 2794	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (norm‘𝑤) = 𝑁)
24	17	fveq2d 6808	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
25		isnlm.s	. . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝑊)
26	24, 25	eqtr4di 2794	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ( ·𝑠 ‘𝑤) = · )
27	26	oveqd 7324	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (𝑥( ·𝑠 ‘𝑤)𝑦) = (𝑥 · 𝑦))
28	23, 27	fveq12d 6811	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ((norm‘𝑤)‘(𝑥( ·𝑠 ‘𝑤)𝑦)) = (𝑁‘(𝑥 · 𝑦)))
29	12	fveq2d 6808	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (norm‘𝑓) = (norm‘𝐹))
30		isnlm.a	. . . . . . . . . . 11 ⊢ 𝐴 = (norm‘𝐹)
31	29, 30	eqtr4di 2794	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (norm‘𝑓) = 𝐴)
32	31	fveq1d 6806	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ((norm‘𝑓)‘𝑥) = (𝐴‘𝑥))
33	23	fveq1d 6806	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ((norm‘𝑤)‘𝑦) = (𝑁‘𝑦))
34	32, 33	oveq12d 7325	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))
35	28, 34	eqeq12d 2752	. . . . . . 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 3355	. . . . . 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 3355	. . . . 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 632	. . . 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 3766	. . 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 23791	. . 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 3632	. 2 ⊢ (𝑊 ∈ NrmMod ↔ (𝑊 ∈ (NrmGrp ∩ LMod) ∧ (𝐹 ∈ NrmRing ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))))
42	1, 6, 41	3bitr4ri 304	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 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  ∀wral 3062  Vcvv 3437  [wsbc 3721   ∩ cin 3891  ‘cfv 6458  (class class class)co 7307   · cmul 10926  Basecbs 16961  Scalarcsca 17014   ·_𝑠 cvsca 17015  LModclmod 20172  normcnm 23781  NrmGrpcngp 23782  NrmRingcnrg 23784  NrmModcnlm 23785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-nul 5239

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rab 3306  df-v 3439  df-sbc 3722  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-iota 6410  df-fv 6466  df-ov 7310  df-nlm 23791

This theorem is referenced by:  nmvs  23889  nlmngp  23890  nlmlmod  23891  nlmnrg  23892  sranlm  23897  lssnlm  23914  isncvsngp  24362  tcphcph  24450  cnzh  31969  rezh  31970