Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sranlm Structured version   Visualization version   GIF version

Theorem sranlm 22298
 Description: The subring algebra over a normed ring is a normed left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypothesis
Ref Expression
sranlm.a 𝐴 = ((subringAlg ‘𝑊)‘𝑆)
Assertion
Ref Expression
sranlm ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmMod)

Proof of Theorem sranlm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nrgngp 22276 . . . . 5 (𝑊 ∈ NrmRing → 𝑊 ∈ NrmGrp)
21adantr 480 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ NrmGrp)
3 eqidd 2611 . . . . 5 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝑊))
4 sranlm.a . . . . . . 7 𝐴 = ((subringAlg ‘𝑊)‘𝑆)
54a1i 11 . . . . . 6 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
6 eqid 2610 . . . . . . . 8 (Base‘𝑊) = (Base‘𝑊)
76subrgss 18604 . . . . . . 7 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
87adantl 481 . . . . . 6 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
95, 8srabase 18999 . . . . 5 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
105, 8sraaddg 19000 . . . . . 6 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g𝑊) = (+g𝐴))
1110oveqdr 6573 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g𝑊)𝑦) = (𝑥(+g𝐴)𝑦))
125, 8srads 19007 . . . . . 6 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (dist‘𝑊) = (dist‘𝐴))
1312reseq1d 5316 . . . . 5 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝐴) ↾ ((Base‘𝑊) × (Base‘𝑊))))
145, 8sratopn 19006 . . . . 5 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (TopOpen‘𝑊) = (TopOpen‘𝐴))
153, 9, 11, 13, 14ngppropd 22251 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ∈ NrmGrp ↔ 𝐴 ∈ NrmGrp))
162, 15mpbid 221 . . 3 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmGrp)
174sralmod 19008 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
1817adantl 481 . . 3 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
195, 8srasca 19002 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊s 𝑆) = (Scalar‘𝐴))
20 eqid 2610 . . . . 5 (𝑊s 𝑆) = (𝑊s 𝑆)
2120subrgnrg 22287 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊s 𝑆) ∈ NrmRing)
2219, 21eqeltrrd 2689 . . 3 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Scalar‘𝐴) ∈ NrmRing)
2316, 18, 223jca 1235 . 2 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ NrmGrp ∧ 𝐴 ∈ LMod ∧ (Scalar‘𝐴) ∈ NrmRing))
24 eqid 2610 . . . . . . . 8 (norm‘𝑊) = (norm‘𝑊)
25 eqid 2610 . . . . . . . 8 (AbsVal‘𝑊) = (AbsVal‘𝑊)
2624, 25nrgabv 22275 . . . . . . 7 (𝑊 ∈ NrmRing → (norm‘𝑊) ∈ (AbsVal‘𝑊))
2726ad2antrr 758 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) ∈ (AbsVal‘𝑊))
288adantr 480 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ⊆ (Base‘𝑊))
29 simprl 790 . . . . . . . 8 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘(Scalar‘𝐴)))
3020subrgbas 18612 . . . . . . . . . . 11 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊s 𝑆)))
3130adantl 481 . . . . . . . . . 10 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊s 𝑆)))
3219fveq2d 6107 . . . . . . . . . 10 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘(𝑊s 𝑆)) = (Base‘(Scalar‘𝐴)))
3331, 32eqtrd 2644 . . . . . . . . 9 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(Scalar‘𝐴)))
3433adantr 480 . . . . . . . 8 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 = (Base‘(Scalar‘𝐴)))
3529, 34eleqtrrd 2691 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥𝑆)
3628, 35sseldd 3569 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝑊))
37 simprr 792 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
389adantr 480 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (Base‘𝑊) = (Base‘𝐴))
3937, 38eleqtrrd 2691 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝑊))
40 eqid 2610 . . . . . . 7 (.r𝑊) = (.r𝑊)
4125, 6, 40abvmul 18652 . . . . . 6 (((norm‘𝑊) ∈ (AbsVal‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → ((norm‘𝑊)‘(𝑥(.r𝑊)𝑦)) = (((norm‘𝑊)‘𝑥) · ((norm‘𝑊)‘𝑦)))
4227, 36, 39, 41syl3anc 1318 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘(𝑥(.r𝑊)𝑦)) = (((norm‘𝑊)‘𝑥) · ((norm‘𝑊)‘𝑦)))
439, 10, 12nmpropd 22208 . . . . . . 7 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (norm‘𝑊) = (norm‘𝐴))
4443adantr 480 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) = (norm‘𝐴))
455, 8sravsca 19003 . . . . . . 7 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r𝑊) = ( ·𝑠𝐴))
4645oveqdr 6573 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(.r𝑊)𝑦) = (𝑥( ·𝑠𝐴)𝑦))
4744, 46fveq12d 6109 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘(𝑥(.r𝑊)𝑦)) = ((norm‘𝐴)‘(𝑥( ·𝑠𝐴)𝑦)))
4842, 47eqtr3d 2646 . . . 4 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (((norm‘𝑊)‘𝑥) · ((norm‘𝑊)‘𝑦)) = ((norm‘𝐴)‘(𝑥( ·𝑠𝐴)𝑦)))
49 subrgsubg 18609 . . . . . . . 8 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
5049ad2antlr 759 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ∈ (SubGrp‘𝑊))
51 eqid 2610 . . . . . . . 8 (norm‘(𝑊s 𝑆)) = (norm‘(𝑊s 𝑆))
5220, 24, 51subgnm2 22248 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝑊) ∧ 𝑥𝑆) → ((norm‘(𝑊s 𝑆))‘𝑥) = ((norm‘𝑊)‘𝑥))
5350, 35, 52syl2anc 691 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘(𝑊s 𝑆))‘𝑥) = ((norm‘𝑊)‘𝑥))
5419adantr 480 . . . . . . . 8 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑊s 𝑆) = (Scalar‘𝐴))
5554fveq2d 6107 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘(𝑊s 𝑆)) = (norm‘(Scalar‘𝐴)))
5655fveq1d 6105 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘(𝑊s 𝑆))‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
5753, 56eqtr3d 2646 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
5844fveq1d 6105 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑦) = ((norm‘𝐴)‘𝑦))
5957, 58oveq12d 6567 . . . 4 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (((norm‘𝑊)‘𝑥) · ((norm‘𝑊)‘𝑦)) = (((norm‘(Scalar‘𝐴))‘𝑥) · ((norm‘𝐴)‘𝑦)))
6048, 59eqtr3d 2646 . . 3 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝐴)‘(𝑥( ·𝑠𝐴)𝑦)) = (((norm‘(Scalar‘𝐴))‘𝑥) · ((norm‘𝐴)‘𝑦)))
6160ralrimivva 2954 . 2 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦 ∈ (Base‘𝐴)((norm‘𝐴)‘(𝑥( ·𝑠𝐴)𝑦)) = (((norm‘(Scalar‘𝐴))‘𝑥) · ((norm‘𝐴)‘𝑦)))
62 eqid 2610 . . 3 (Base‘𝐴) = (Base‘𝐴)
63 eqid 2610 . . 3 (norm‘𝐴) = (norm‘𝐴)
64 eqid 2610 . . 3 ( ·𝑠𝐴) = ( ·𝑠𝐴)
65 eqid 2610 . . 3 (Scalar‘𝐴) = (Scalar‘𝐴)
66 eqid 2610 . . 3 (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
67 eqid 2610 . . 3 (norm‘(Scalar‘𝐴)) = (norm‘(Scalar‘𝐴))
6862, 63, 64, 65, 66, 67isnlm 22289 . 2 (𝐴 ∈ NrmMod ↔ ((𝐴 ∈ NrmGrp ∧ 𝐴 ∈ LMod ∧ (Scalar‘𝐴) ∈ NrmRing) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦 ∈ (Base‘𝐴)((norm‘𝐴)‘(𝑥( ·𝑠𝐴)𝑦)) = (((norm‘(Scalar‘𝐴))‘𝑥) · ((norm‘𝐴)‘𝑦))))
6923, 61, 68sylanbrc 695 1 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmMod)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540   × cxp 5036  ‘cfv 5804  (class class class)co 6549   · cmul 9820  Basecbs 15695   ↾s cress 15696  +gcplusg 15768  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  distcds 15777  SubGrpcsubg 17411  SubRingcsubrg 18599  AbsValcabv 18639  LModclmod 18686  subringAlg csra 18989  normcnm 22191  NrmGrpcngp 22192  NrmRingcnrg 22194  NrmModcnlm 22195 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ico 12052  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ds 15791  df-rest 15906  df-topn 15907  df-0g 15925  df-topgen 15927  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-mgp 18313  df-ur 18325  df-ring 18372  df-subrg 18601  df-abv 18640  df-lmod 18688  df-sra 18993  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-xms 21935  df-ms 21936  df-nm 22197  df-ngp 22198  df-nrg 22200  df-nlm 22201 This theorem is referenced by:  rlmnlm  22302  srabn  22964
 Copyright terms: Public domain W3C validator