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Theorem sranlm 22428
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 22406 . . . . 5 (𝑊 ∈ NrmRing → 𝑊 ∈ NrmGrp)
21adantr 481 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ NrmGrp)
3 eqidd 2622 . . . . 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 2621 . . . . . . . 8 (Base‘𝑊) = (Base‘𝑊)
76subrgss 18721 . . . . . . 7 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
87adantl 482 . . . . . 6 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
95, 8srabase 19118 . . . . 5 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
105, 8sraaddg 19119 . . . . . 6 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g𝑊) = (+g𝐴))
1110oveqdr 6639 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g𝑊)𝑦) = (𝑥(+g𝐴)𝑦))
125, 8srads 19126 . . . . . 6 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (dist‘𝑊) = (dist‘𝐴))
1312reseq1d 5365 . . . . 5 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝐴) ↾ ((Base‘𝑊) × (Base‘𝑊))))
145, 8sratopn 19125 . . . . 5 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (TopOpen‘𝑊) = (TopOpen‘𝐴))
153, 9, 11, 13, 14ngppropd 22381 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ∈ NrmGrp ↔ 𝐴 ∈ NrmGrp))
162, 15mpbid 222 . . 3 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmGrp)
174sralmod 19127 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
1817adantl 482 . . 3 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
195, 8srasca 19121 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊s 𝑆) = (Scalar‘𝐴))
20 eqid 2621 . . . . 5 (𝑊s 𝑆) = (𝑊s 𝑆)
2120subrgnrg 22417 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊s 𝑆) ∈ NrmRing)
2219, 21eqeltrrd 2699 . . 3 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Scalar‘𝐴) ∈ NrmRing)
2316, 18, 223jca 1240 . 2 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ NrmGrp ∧ 𝐴 ∈ LMod ∧ (Scalar‘𝐴) ∈ NrmRing))
24 eqid 2621 . . . . . . . 8 (norm‘𝑊) = (norm‘𝑊)
25 eqid 2621 . . . . . . . 8 (AbsVal‘𝑊) = (AbsVal‘𝑊)
2624, 25nrgabv 22405 . . . . . . 7 (𝑊 ∈ NrmRing → (norm‘𝑊) ∈ (AbsVal‘𝑊))
2726ad2antrr 761 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) ∈ (AbsVal‘𝑊))
288adantr 481 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ⊆ (Base‘𝑊))
29 simprl 793 . . . . . . . 8 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘(Scalar‘𝐴)))
3020subrgbas 18729 . . . . . . . . . . 11 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊s 𝑆)))
3130adantl 482 . . . . . . . . . 10 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊s 𝑆)))
3219fveq2d 6162 . . . . . . . . . 10 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘(𝑊s 𝑆)) = (Base‘(Scalar‘𝐴)))
3331, 32eqtrd 2655 . . . . . . . . 9 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(Scalar‘𝐴)))
3433adantr 481 . . . . . . . 8 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 = (Base‘(Scalar‘𝐴)))
3529, 34eleqtrrd 2701 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥𝑆)
3628, 35sseldd 3589 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝑊))
37 simprr 795 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
389adantr 481 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (Base‘𝑊) = (Base‘𝐴))
3937, 38eleqtrrd 2701 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝑊))
40 eqid 2621 . . . . . . 7 (.r𝑊) = (.r𝑊)
4125, 6, 40abvmul 18769 . . . . . 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 1323 . . . . 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 22338 . . . . . . 7 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (norm‘𝑊) = (norm‘𝐴))
4443adantr 481 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) = (norm‘𝐴))
455, 8sravsca 19122 . . . . . . 7 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r𝑊) = ( ·𝑠𝐴))
4645oveqdr 6639 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(.r𝑊)𝑦) = (𝑥( ·𝑠𝐴)𝑦))
4744, 46fveq12d 6164 . . . . 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 2657 . . . 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 18726 . . . . . . . 8 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
5049ad2antlr 762 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ∈ (SubGrp‘𝑊))
51 eqid 2621 . . . . . . . 8 (norm‘(𝑊s 𝑆)) = (norm‘(𝑊s 𝑆))
5220, 24, 51subgnm2 22378 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝑊) ∧ 𝑥𝑆) → ((norm‘(𝑊s 𝑆))‘𝑥) = ((norm‘𝑊)‘𝑥))
5350, 35, 52syl2anc 692 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘(𝑊s 𝑆))‘𝑥) = ((norm‘𝑊)‘𝑥))
5419adantr 481 . . . . . . . 8 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑊s 𝑆) = (Scalar‘𝐴))
5554fveq2d 6162 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘(𝑊s 𝑆)) = (norm‘(Scalar‘𝐴)))
5655fveq1d 6160 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘(𝑊s 𝑆))‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
5753, 56eqtr3d 2657 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
5844fveq1d 6160 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑦) = ((norm‘𝐴)‘𝑦))
5957, 58oveq12d 6633 . . . 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 2657 . . 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 2967 . 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 2621 . . 3 (Base‘𝐴) = (Base‘𝐴)
63 eqid 2621 . . 3 (norm‘𝐴) = (norm‘𝐴)
64 eqid 2621 . . 3 ( ·𝑠𝐴) = ( ·𝑠𝐴)
65 eqid 2621 . . 3 (Scalar‘𝐴) = (Scalar‘𝐴)
66 eqid 2621 . . 3 (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
67 eqid 2621 . . 3 (norm‘(Scalar‘𝐴)) = (norm‘(Scalar‘𝐴))
6862, 63, 64, 65, 66, 67isnlm 22419 . 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 697 1 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  wss 3560   × cxp 5082  cfv 5857  (class class class)co 6615   · cmul 9901  Basecbs 15800  s cress 15801  +gcplusg 15881  .rcmulr 15882  Scalarcsca 15884   ·𝑠 cvsca 15885  distcds 15890  SubGrpcsubg 17528  SubRingcsubrg 18716  AbsValcabv 18756  LModclmod 18803  subringAlg csra 19108  normcnm 22321  NrmGrpcngp 22322  NrmRingcnrg 22324  NrmModcnlm 22325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-sup 8308  df-inf 8309  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ico 12139  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-mulr 15895  df-sca 15897  df-vsca 15898  df-ip 15899  df-tset 15900  df-ds 15904  df-rest 16023  df-topn 16024  df-0g 16042  df-topgen 16044  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-minusg 17366  df-sbg 17367  df-subg 17531  df-mgp 18430  df-ur 18442  df-ring 18489  df-subrg 18718  df-abv 18757  df-lmod 18805  df-sra 19112  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-top 20639  df-topon 20656  df-topsp 20677  df-bases 20690  df-xms 22065  df-ms 22066  df-nm 22327  df-ngp 22328  df-nrg 22330  df-nlm 22331
This theorem is referenced by:  rlmnlm  22432  srabn  23096
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