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Theorem isnlm 22856
 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 · = ( ·𝑠𝑊)
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) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
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.
StepHypRef Expression
1 anass 462 . 2 (((𝑊 ∈ (NrmGrp ∩ LMod) ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))) ↔ (𝑊 ∈ (NrmGrp ∩ LMod) ∧ (𝐹 ∈ NrmRing ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)))))
2 df-3an 1113 . . . 4 ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ NrmRing))
3 elin 4025 . . . . 5 (𝑊 ∈ (NrmGrp ∩ LMod) ↔ (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod))
43anbi1i 617 . . . 4 ((𝑊 ∈ (NrmGrp ∩ LMod) ∧ 𝐹 ∈ NrmRing) ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ NrmRing))
52, 4bitr4i 270 . . 3 ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ↔ (𝑊 ∈ (NrmGrp ∩ LMod) ∧ 𝐹 ∈ NrmRing))
65anbi1i 617 . 2 (((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))) ↔ ((𝑊 ∈ (NrmGrp ∩ LMod) ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
7 fvexd 6452 . . . 4 (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
8 id 22 . . . . . . 7 (𝑓 = (Scalar‘𝑤) → 𝑓 = (Scalar‘𝑤))
9 fveq2 6437 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
10 isnlm.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
119, 10syl6eqr 2879 . . . . . . 7 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
128, 11sylan9eqr 2883 . . . . . 6 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → 𝑓 = 𝐹)
1312eleq1d 2891 . . . . 5 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (𝑓 ∈ NrmRing ↔ 𝐹 ∈ NrmRing))
1412fveq2d 6441 . . . . . . 7 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = (Base‘𝐹))
15 isnlm.k . . . . . . 7 𝐾 = (Base‘𝐹)
1614, 15syl6eqr 2879 . . . . . 6 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = 𝐾)
17 simpl 476 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → 𝑤 = 𝑊)
1817fveq2d 6441 . . . . . . . 8 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑤) = (Base‘𝑊))
19 isnlm.v . . . . . . . 8 𝑉 = (Base‘𝑊)
2018, 19syl6eqr 2879 . . . . . . 7 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑤) = 𝑉)
2117fveq2d 6441 . . . . . . . . . 10 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (norm‘𝑤) = (norm‘𝑊))
22 isnlm.n . . . . . . . . . 10 𝑁 = (norm‘𝑊)
2321, 22syl6eqr 2879 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (norm‘𝑤) = 𝑁)
2417fveq2d 6441 . . . . . . . . . . 11 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ( ·𝑠𝑤) = ( ·𝑠𝑊))
25 isnlm.s . . . . . . . . . . 11 · = ( ·𝑠𝑊)
2624, 25syl6eqr 2879 . . . . . . . . . 10 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ( ·𝑠𝑤) = · )
2726oveqd 6927 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (𝑥( ·𝑠𝑤)𝑦) = (𝑥 · 𝑦))
2823, 27fveq12d 6444 . . . . . . . 8 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ((norm‘𝑤)‘(𝑥( ·𝑠𝑤)𝑦)) = (𝑁‘(𝑥 · 𝑦)))
2912fveq2d 6441 . . . . . . . . . . 11 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (norm‘𝑓) = (norm‘𝐹))
30 isnlm.a . . . . . . . . . . 11 𝐴 = (norm‘𝐹)
3129, 30syl6eqr 2879 . . . . . . . . . 10 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (norm‘𝑓) = 𝐴)
3231fveq1d 6439 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ((norm‘𝑓)‘𝑥) = (𝐴𝑥))
3323fveq1d 6439 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ((norm‘𝑤)‘𝑦) = (𝑁𝑦))
3432, 33oveq12d 6928 . . . . . . . 8 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)) = ((𝐴𝑥) · (𝑁𝑦)))
3528, 34eqeq12d 2840 . . . . . . 7 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (((norm‘𝑤)‘(𝑥( ·𝑠𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)) ↔ (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
3620, 35raleqbidv 3364 . . . . . 6 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)) ↔ ∀𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
3716, 36raleqbidv 3364 . . . . 5 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (∀𝑥 ∈ (Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)) ↔ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
3813, 37anbi12d 624 . . . 4 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ((𝑓 ∈ NrmRing ∧ ∀𝑥 ∈ (Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦))) ↔ (𝐹 ∈ NrmRing ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)))))
397, 38sbcied 3699 . . 3 (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 𝑓](𝑓 ∈ NrmRing ∧ ∀𝑥 ∈ (Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦))) ↔ (𝐹 ∈ NrmRing ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)))))
40 df-nlm 22768 . . 3 NrmMod = {𝑤 ∈ (NrmGrp ∩ LMod) ∣ [(Scalar‘𝑤) / 𝑓](𝑓 ∈ NrmRing ∧ ∀𝑥 ∈ (Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)))}
4139, 40elrab2 3589 . 2 (𝑊 ∈ NrmMod ↔ (𝑊 ∈ (NrmGrp ∩ LMod) ∧ (𝐹 ∈ NrmRing ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)))))
421, 6, 413bitr4ri 296 1 (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∀wral 3117  Vcvv 3414  [wsbc 3662   ∩ cin 3797  ‘cfv 6127  (class class class)co 6910   · cmul 10264  Basecbs 16229  Scalarcsca 16315   ·𝑠 cvsca 16316  LModclmod 19226  normcnm 22758  NrmGrpcngp 22759  NrmRingcnrg 22761  NrmModcnlm 22762 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913  df-nlm 22768 This theorem is referenced by:  nmvs  22857  nlmngp  22858  nlmlmod  22859  nlmnrg  22860  sranlm  22865  lssnlm  22882  isncvsngp  23325  tcphcph  23412  cnzh  30555  rezh  30556
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