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Theorem isnlm 22698
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 459 . 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 1072 . . . 4 ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ NrmRing))
3 elin 3945 . . . . 5 (𝑊 ∈ (NrmGrp ∩ LMod) ↔ (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod))
43anbi1i 602 . . . 4 ((𝑊 ∈ (NrmGrp ∩ LMod) ∧ 𝐹 ∈ NrmRing) ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ NrmRing))
52, 4bitr4i 267 . . 3 ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ↔ (𝑊 ∈ (NrmGrp ∩ LMod) ∧ 𝐹 ∈ NrmRing))
65anbi1i 602 . 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 6344 . . . 4 (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
8 id 22 . . . . . . 7 (𝑓 = (Scalar‘𝑤) → 𝑓 = (Scalar‘𝑤))
9 fveq2 6332 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
10 isnlm.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
119, 10syl6eqr 2822 . . . . . . 7 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
128, 11sylan9eqr 2826 . . . . . 6 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → 𝑓 = 𝐹)
1312eleq1d 2834 . . . . 5 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (𝑓 ∈ NrmRing ↔ 𝐹 ∈ NrmRing))
1412fveq2d 6336 . . . . . . 7 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = (Base‘𝐹))
15 isnlm.k . . . . . . 7 𝐾 = (Base‘𝐹)
1614, 15syl6eqr 2822 . . . . . 6 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) = 𝐾)
17 simpl 468 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → 𝑤 = 𝑊)
1817fveq2d 6336 . . . . . . . 8 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑤) = (Base‘𝑊))
19 isnlm.v . . . . . . . 8 𝑉 = (Base‘𝑊)
2018, 19syl6eqr 2822 . . . . . . 7 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑤) = 𝑉)
2117fveq2d 6336 . . . . . . . . . 10 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (norm‘𝑤) = (norm‘𝑊))
22 isnlm.n . . . . . . . . . 10 𝑁 = (norm‘𝑊)
2321, 22syl6eqr 2822 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (norm‘𝑤) = 𝑁)
2417fveq2d 6336 . . . . . . . . . . 11 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ( ·𝑠𝑤) = ( ·𝑠𝑊))
25 isnlm.s . . . . . . . . . . 11 · = ( ·𝑠𝑊)
2624, 25syl6eqr 2822 . . . . . . . . . 10 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ( ·𝑠𝑤) = · )
2726oveqd 6809 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (𝑥( ·𝑠𝑤)𝑦) = (𝑥 · 𝑦))
2823, 27fveq12d 6338 . . . . . . . 8 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ((norm‘𝑤)‘(𝑥( ·𝑠𝑤)𝑦)) = (𝑁‘(𝑥 · 𝑦)))
2912fveq2d 6336 . . . . . . . . . . 11 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (norm‘𝑓) = (norm‘𝐹))
30 isnlm.a . . . . . . . . . . 11 𝐴 = (norm‘𝐹)
3129, 30syl6eqr 2822 . . . . . . . . . 10 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (norm‘𝑓) = 𝐴)
3231fveq1d 6334 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ((norm‘𝑓)‘𝑥) = (𝐴𝑥))
3323fveq1d 6334 . . . . . . . . 9 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ((norm‘𝑤)‘𝑦) = (𝑁𝑦))
3432, 33oveq12d 6810 . . . . . . . 8 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)) = ((𝐴𝑥) · (𝑁𝑦)))
3528, 34eqeq12d 2785 . . . . . . 7 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (((norm‘𝑤)‘(𝑥( ·𝑠𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)) ↔ (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
3620, 35raleqbidv 3300 . . . . . 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 3300 . . . . 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 608 . . . 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 3622 . . 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 22610 . . 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 3516 . 2 (𝑊 ∈ NrmMod ↔ (𝑊 ∈ (NrmGrp ∩ LMod) ∧ (𝐹 ∈ NrmRing ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)))))
421, 6, 413bitr4ri 293 1 (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  Vcvv 3349  [wsbc 3585  cin 3720  cfv 6031  (class class class)co 6792   · cmul 10142  Basecbs 16063  Scalarcsca 16151   ·𝑠 cvsca 16152  LModclmod 19072  normcnm 22600  NrmGrpcngp 22601  NrmRingcnrg 22603  NrmModcnlm 22604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-ov 6795  df-nlm 22610
This theorem is referenced by:  nmvs  22699  nlmngp  22700  nlmlmod  22701  nlmnrg  22702  sranlm  22707  lssnlm  22724  isncvsngp  23167  tchcph  23254  cnzh  30348  rezh  30349
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