Theorem isnlm 23278

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 471	. 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 1085	. . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ NrmRing))
3		elin 4168	. . . . 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 280	. . 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 6679	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
8		id 22	. . . . . . 7 ⊢ (𝑓 = (Scalar‘𝑤) → 𝑓 = (Scalar‘𝑤))
9		fveq2 6664	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
10		isnlm.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
11	9, 10	syl6eqr 2874	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
12	8, 11	sylan9eqr 2878	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → 𝑓 = 𝐹)
13	12	eleq1d 2897	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (𝑓 ∈ NrmRing ↔ 𝐹 ∈ NrmRing))
14	12	fveq2d 6668	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = (Base‘𝐹))
15		isnlm.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐹)
16	14, 15	syl6eqr 2874	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = 𝐾)
17		simpl 485	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → 𝑤 = 𝑊)
18	17	fveq2d 6668	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑤) = (Base‘𝑊))
19		isnlm.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
20	18, 19	syl6eqr 2874	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑤) = 𝑉)
21	17	fveq2d 6668	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (norm‘𝑤) = (norm‘𝑊))
22		isnlm.n	. . . . . . . . . 10 ⊢ 𝑁 = (norm‘𝑊)
23	21, 22	syl6eqr 2874	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (norm‘𝑤) = 𝑁)
24	17	fveq2d 6668	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
25		isnlm.s	. . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝑊)
26	24, 25	syl6eqr 2874	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ( ·𝑠 ‘𝑤) = · )
27	26	oveqd 7167	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (𝑥( ·𝑠 ‘𝑤)𝑦) = (𝑥 · 𝑦))
28	23, 27	fveq12d 6671	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ((norm‘𝑤)‘(𝑥( ·𝑠 ‘𝑤)𝑦)) = (𝑁‘(𝑥 · 𝑦)))
29	12	fveq2d 6668	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (norm‘𝑓) = (norm‘𝐹))
30		isnlm.a	. . . . . . . . . . 11 ⊢ 𝐴 = (norm‘𝐹)
31	29, 30	syl6eqr 2874	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (norm‘𝑓) = 𝐴)
32	31	fveq1d 6666	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ((norm‘𝑓)‘𝑥) = (𝐴‘𝑥))
33	23	fveq1d 6666	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ((norm‘𝑤)‘𝑦) = (𝑁‘𝑦))
34	32, 33	oveq12d 7168	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))
35	28, 34	eqeq12d 2837	. . . . . . 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 3401	. . . . . 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 3401	. . . . 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 3813	. . 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 23190	. . 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 3682	. 2 ⊢ (𝑊 ∈ NrmMod ↔ (𝑊 ∈ (NrmGrp ∩ LMod) ∧ (𝐹 ∈ NrmRing ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))))
42	1, 6, 41	3bitr4ri 306	1 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494  [wsbc 3771   ∩ cin 3934  ‘cfv 6349  (class class class)co 7150   · cmul 10536  Basecbs 16477  Scalarcsca 16562   ·_𝑠 cvsca 16563  LModclmod 19628  normcnm 23180  NrmGrpcngp 23181  NrmRingcnrg 23183  NrmModcnlm 23184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5202

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-iota 6308  df-fv 6357  df-ov 7153  df-nlm 23190

This theorem is referenced by:  nmvs  23279  nlmngp  23280  nlmlmod  23281  nlmnrg  23282  sranlm  23287  lssnlm  23304  isncvsngp  23747  tcphcph  23834  cnzh  31206  rezh  31207