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Theorem sranlm 24071
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 24049 . . . . 5 (𝑊 ∈ NrmRing → 𝑊 ∈ NrmGrp)
21adantr 482 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ NrmGrp)
3 eqidd 2734 . . . . 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 2733 . . . . . . . 8 (Base‘𝑊) = (Base‘𝑊)
76subrgss 20265 . . . . . . 7 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
87adantl 483 . . . . . 6 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
95, 8srabase 20685 . . . . 5 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
105, 8sraaddg 20687 . . . . . 6 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g𝑊) = (+g𝐴))
1110oveqdr 7389 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g𝑊)𝑦) = (𝑥(+g𝐴)𝑦))
125, 8srads 20699 . . . . . 6 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (dist‘𝑊) = (dist‘𝐴))
1312reseq1d 5940 . . . . 5 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝐴) ↾ ((Base‘𝑊) × (Base‘𝑊))))
145, 8sratopn 20698 . . . . 5 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (TopOpen‘𝑊) = (TopOpen‘𝐴))
153, 9, 11, 13, 14ngppropd 24016 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ∈ NrmGrp ↔ 𝐴 ∈ NrmGrp))
162, 15mpbid 231 . . 3 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmGrp)
174sralmod 20701 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
1817adantl 483 . . 3 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
195, 8srasca 20691 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊s 𝑆) = (Scalar‘𝐴))
20 eqid 2733 . . . . 5 (𝑊s 𝑆) = (𝑊s 𝑆)
2120subrgnrg 24060 . . . 4 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊s 𝑆) ∈ NrmRing)
2219, 21eqeltrrd 2835 . . 3 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Scalar‘𝐴) ∈ NrmRing)
2316, 18, 223jca 1129 . 2 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ NrmGrp ∧ 𝐴 ∈ LMod ∧ (Scalar‘𝐴) ∈ NrmRing))
24 eqid 2733 . . . . . . . 8 (norm‘𝑊) = (norm‘𝑊)
25 eqid 2733 . . . . . . . 8 (AbsVal‘𝑊) = (AbsVal‘𝑊)
2624, 25nrgabv 24048 . . . . . . 7 (𝑊 ∈ NrmRing → (norm‘𝑊) ∈ (AbsVal‘𝑊))
2726ad2antrr 725 . . . . . 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 770 . . . . . . . 8 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘(Scalar‘𝐴)))
3020subrgbas 20273 . . . . . . . . . . 11 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊s 𝑆)))
3130adantl 483 . . . . . . . . . 10 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊s 𝑆)))
3219fveq2d 6850 . . . . . . . . . 10 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘(𝑊s 𝑆)) = (Base‘(Scalar‘𝐴)))
3331, 32eqtrd 2773 . . . . . . . . 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 2837 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥𝑆)
3628, 35sseldd 3949 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝑊))
37 simprr 772 . . . . . . 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 2837 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝑊))
40 eqid 2733 . . . . . . 7 (.r𝑊) = (.r𝑊)
4125, 6, 40abvmul 20331 . . . . . 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 1372 . . . . 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 23973 . . . . . . 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 20693 . . . . . . 7 ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r𝑊) = ( ·𝑠𝐴))
4645oveqdr 7389 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(.r𝑊)𝑦) = (𝑥( ·𝑠𝐴)𝑦))
4744, 46fveq12d 6853 . . . . 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 2775 . . . 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 20270 . . . . . . . 8 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
5049ad2antlr 726 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ∈ (SubGrp‘𝑊))
51 eqid 2733 . . . . . . . 8 (norm‘(𝑊s 𝑆)) = (norm‘(𝑊s 𝑆))
5220, 24, 51subgnm2 24013 . . . . . . 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 6850 . . . . . . 7 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘(𝑊s 𝑆)) = (norm‘(Scalar‘𝐴)))
5655fveq1d 6848 . . . . . 6 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘(𝑊s 𝑆))‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
5753, 56eqtr3d 2775 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
5844fveq1d 6848 . . . . 5 (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑦) = ((norm‘𝐴)‘𝑦))
5957, 58oveq12d 7379 . . . 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 2775 . . 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 3194 . 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 2733 . . 3 (Base‘𝐴) = (Base‘𝐴)
63 eqid 2733 . . 3 (norm‘𝐴) = (norm‘𝐴)
64 eqid 2733 . . 3 ( ·𝑠𝐴) = ( ·𝑠𝐴)
65 eqid 2733 . . 3 (Scalar‘𝐴) = (Scalar‘𝐴)
66 eqid 2733 . . 3 (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
67 eqid 2733 . . 3 (norm‘(Scalar‘𝐴)) = (norm‘(Scalar‘𝐴))
6862, 63, 64, 65, 66, 67isnlm 24062 . 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 1088   = wceq 1542  wcel 2107  wral 3061  wss 3914   × cxp 5635  cfv 6500  (class class class)co 7361   · cmul 11064  Basecbs 17091  s cress 17120  +gcplusg 17141  .rcmulr 17142  Scalarcsca 17144   ·𝑠 cvsca 17145  distcds 17150  SubGrpcsubg 18930  SubRingcsubrg 20260  AbsValcabv 20318  LModclmod 20365  subringAlg csra 20674  normcnm 23955  NrmGrpcngp 23956  NrmRingcnrg 23958  NrmModcnlm 23959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-sup 9386  df-inf 9387  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-q 12882  df-rp 12924  df-xneg 13041  df-xadd 13042  df-xmul 13043  df-ico 13279  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-mulr 17155  df-sca 17157  df-vsca 17158  df-ip 17159  df-tset 17160  df-ds 17163  df-rest 17312  df-topn 17313  df-0g 17331  df-topgen 17333  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-grp 18759  df-minusg 18760  df-sbg 18761  df-subg 18933  df-mgp 19905  df-ur 19922  df-ring 19974  df-subrg 20262  df-abv 20319  df-lmod 20367  df-sra 20678  df-psmet 20811  df-xmet 20812  df-met 20813  df-bl 20814  df-mopn 20815  df-top 22266  df-topon 22283  df-topsp 22305  df-bases 22319  df-xms 23696  df-ms 23697  df-nm 23961  df-ngp 23962  df-nrg 23964  df-nlm 23965
This theorem is referenced by:  rlmnlm  24075  srabn  24747
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