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Theorem sranlm 23958
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 23936 . . . . 5 (𝑊 ∈ NrmRing → 𝑊 ∈ NrmGrp)
21adantr 482 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ NrmGrp)
3 eqidd 2738 . . . . 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 2737 . . . . . . . 8 (Base‘𝑊) = (Base‘𝑊)
76subrgss 20134 . . . . . . 7 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
87adantl 483 . . . . . 6 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
95, 8srabase 20551 . . . . 5 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
105, 8sraaddg 20553 . . . . . 6 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g𝑊) = (+g𝐴))
1110oveqdr 7374 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g𝑊)𝑦) = (𝑥(+g𝐴)𝑦))
125, 8srads 20565 . . . . . 6 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (dist‘𝑊) = (dist‘𝐴))
1312reseq1d 5929 . . . . 5 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝐴) ↾ ((Base‘𝑊) × (Base‘𝑊))))
145, 8sratopn 20564 . . . . 5 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (TopOpen‘𝑊) = (TopOpen‘𝐴))
153, 9, 11, 13, 14ngppropd 23903 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ∈ NrmGrp ↔ 𝐴 ∈ NrmGrp))
162, 15mpbid 231 . . 3 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmGrp)
174sralmod 20567 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
1817adantl 483 . . 3 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
195, 8srasca 20557 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊s 𝑆) = (Scalar‘𝐴))
20 eqid 2737 . . . . 5 (𝑊s 𝑆) = (𝑊s 𝑆)
2120subrgnrg 23947 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊s 𝑆) ∈ NrmRing)
2219, 21eqeltrrd 2839 . . 3 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Scalar‘𝐴) ∈ NrmRing)
2316, 18, 223jca 1128 . 2 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ NrmGrp ∧ 𝐴 ∈ LMod ∧ (Scalar‘𝐴) ∈ NrmRing))
24 eqid 2737 . . . . . . . 8 (norm‘𝑊) = (norm‘𝑊)
25 eqid 2737 . . . . . . . 8 (AbsVal‘𝑊) = (AbsVal‘𝑊)
2624, 25nrgabv 23935 . . . . . . 7 (𝑊 ∈ NrmRing → (norm‘𝑊) ∈ (AbsVal‘𝑊))
2726ad2antrr 724 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) ∈ (AbsVal‘𝑊))
288adantr 482 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ⊆ (Base‘𝑊))
29 simprl 769 . . . . . . . 8 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘(Scalar‘𝐴)))
3020subrgbas 20142 . . . . . . . . . . 11 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊s 𝑆)))
3130adantl 483 . . . . . . . . . 10 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊s 𝑆)))
3219fveq2d 6838 . . . . . . . . . 10 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘(𝑊s 𝑆)) = (Base‘(Scalar‘𝐴)))
3331, 32eqtrd 2777 . . . . . . . . 9 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(Scalar‘𝐴)))
3433adantr 482 . . . . . . . 8 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 = (Base‘(Scalar‘𝐴)))
3529, 34eleqtrrd 2841 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥𝑆)
3628, 35sseldd 3940 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝑊))
37 simprr 771 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
389adantr 482 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (Base‘𝑊) = (Base‘𝐴))
3937, 38eleqtrrd 2841 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝑊))
40 eqid 2737 . . . . . . 7 (.r𝑊) = (.r𝑊)
4125, 6, 40abvmul 20199 . . . . . 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 1371 . . . . 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 23860 . . . . . . 7 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (norm‘𝑊) = (norm‘𝐴))
4443adantr 482 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) = (norm‘𝐴))
455, 8sravsca 20559 . . . . . . 7 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r𝑊) = ( ·𝑠𝐴))
4645oveqdr 7374 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(.r𝑊)𝑦) = (𝑥( ·𝑠𝐴)𝑦))
4744, 46fveq12d 6841 . . . . 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 2779 . . . 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 20139 . . . . . . . 8 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
5049ad2antlr 725 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ∈ (SubGrp‘𝑊))
51 eqid 2737 . . . . . . . 8 (norm‘(𝑊s 𝑆)) = (norm‘(𝑊s 𝑆))
5220, 24, 51subgnm2 23900 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝑊) ∧ 𝑥𝑆) → ((norm‘(𝑊s 𝑆))‘𝑥) = ((norm‘𝑊)‘𝑥))
5350, 35, 52syl2anc 585 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘(𝑊s 𝑆))‘𝑥) = ((norm‘𝑊)‘𝑥))
5419adantr 482 . . . . . . . 8 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑊s 𝑆) = (Scalar‘𝐴))
5554fveq2d 6838 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘(𝑊s 𝑆)) = (norm‘(Scalar‘𝐴)))
5655fveq1d 6836 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘(𝑊s 𝑆))‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
5753, 56eqtr3d 2779 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
5844fveq1d 6836 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑦) = ((norm‘𝐴)‘𝑦))
5957, 58oveq12d 7364 . . . 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 2779 . . 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 3195 . 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 2737 . . 3 (Base‘𝐴) = (Base‘𝐴)
63 eqid 2737 . . 3 (norm‘𝐴) = (norm‘𝐴)
64 eqid 2737 . . 3 ( ·𝑠𝐴) = ( ·𝑠𝐴)
65 eqid 2737 . . 3 (Scalar‘𝐴) = (Scalar‘𝐴)
66 eqid 2737 . . 3 (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
67 eqid 2737 . . 3 (norm‘(Scalar‘𝐴)) = (norm‘(Scalar‘𝐴))
6862, 63, 64, 65, 66, 67isnlm 23949 . 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 584 1 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1087   = wceq 1541  wcel 2106  wral 3062  wss 3905   × cxp 5625  cfv 6488  (class class class)co 7346   · cmul 10986  Basecbs 17014  s cress 17043  +gcplusg 17064  .rcmulr 17065  Scalarcsca 17067   ·𝑠 cvsca 17068  distcds 17073  SubGrpcsubg 18850  SubRingcsubrg 20129  AbsValcabv 20186  LModclmod 20233  subringAlg csra 20540  normcnm 23842  NrmGrpcngp 23843  NrmRingcnrg 23845  NrmModcnlm 23846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5237  ax-sep 5251  ax-nul 5258  ax-pow 5315  ax-pr 5379  ax-un 7659  ax-cnex 11037  ax-resscn 11038  ax-1cn 11039  ax-icn 11040  ax-addcl 11041  ax-addrcl 11042  ax-mulcl 11043  ax-mulrcl 11044  ax-mulcom 11045  ax-addass 11046  ax-mulass 11047  ax-distr 11048  ax-i2m1 11049  ax-1ne0 11050  ax-1rid 11051  ax-rnegex 11052  ax-rrecex 11053  ax-cnre 11054  ax-pre-lttri 11055  ax-pre-lttrn 11056  ax-pre-ltadd 11057  ax-pre-mulgt0 11058  ax-pre-sup 11059
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3735  df-csb 3851  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3924  df-nul 4278  df-if 4482  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5184  df-tr 5218  df-id 5525  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5582  df-we 5584  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-rn 5638  df-res 5639  df-ima 5640  df-pred 6246  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6440  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7302  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7790  df-1st 7908  df-2nd 7909  df-frecs 8176  df-wrecs 8207  df-recs 8281  df-rdg 8320  df-er 8578  df-map 8697  df-en 8814  df-dom 8815  df-sdom 8816  df-sup 9308  df-inf 9309  df-pnf 11121  df-mnf 11122  df-xr 11123  df-ltxr 11124  df-le 11125  df-sub 11317  df-neg 11318  df-div 11743  df-nn 12084  df-2 12146  df-3 12147  df-4 12148  df-5 12149  df-6 12150  df-7 12151  df-8 12152  df-9 12153  df-n0 12344  df-z 12430  df-dec 12548  df-uz 12693  df-q 12799  df-rp 12841  df-xneg 12958  df-xadd 12959  df-xmul 12960  df-ico 13195  df-sets 16967  df-slot 16985  df-ndx 16997  df-base 17015  df-ress 17044  df-plusg 17077  df-mulr 17078  df-sca 17080  df-vsca 17081  df-ip 17082  df-tset 17083  df-ds 17086  df-rest 17235  df-topn 17236  df-0g 17254  df-topgen 17256  df-mgm 18428  df-sgrp 18477  df-mnd 18488  df-grp 18681  df-minusg 18682  df-sbg 18683  df-subg 18853  df-mgp 19820  df-ur 19837  df-ring 19884  df-subrg 20131  df-abv 20187  df-lmod 20235  df-sra 20544  df-psmet 20699  df-xmet 20700  df-met 20701  df-bl 20702  df-mopn 20703  df-top 22153  df-topon 22170  df-topsp 22192  df-bases 22206  df-xms 23583  df-ms 23584  df-nm 23848  df-ngp 23849  df-nrg 23851  df-nlm 23852
This theorem is referenced by:  rlmnlm  23962  srabn  24634
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