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Theorem subrgpsr 19467
Description: A subring of the base ring induces a subring of power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
Hypotheses
Ref Expression
subrgpsr.s 𝑆 = (𝐼 mPwSer 𝑅)
subrgpsr.h 𝐻 = (𝑅s 𝑇)
subrgpsr.u 𝑈 = (𝐼 mPwSer 𝐻)
subrgpsr.b 𝐵 = (Base‘𝑈)
Assertion
Ref Expression
subrgpsr ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))

Proof of Theorem subrgpsr
Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgpsr.s . . . 4 𝑆 = (𝐼 mPwSer 𝑅)
2 simpl 472 . . . 4 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐼𝑉)
3 subrgrcl 18833 . . . . 5 (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
43adantl 481 . . . 4 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝑅 ∈ Ring)
51, 2, 4psrring 19459 . . 3 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝑆 ∈ Ring)
6 subrgpsr.u . . . . 5 𝑈 = (𝐼 mPwSer 𝐻)
7 subrgpsr.h . . . . . . 7 𝐻 = (𝑅s 𝑇)
87subrgring 18831 . . . . . 6 (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring)
98adantl 481 . . . . 5 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐻 ∈ Ring)
106, 2, 9psrring 19459 . . . 4 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝑈 ∈ Ring)
11 subrgpsr.b . . . . . 6 𝐵 = (Base‘𝑈)
1211a1i 11 . . . . 5 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = (Base‘𝑈))
13 eqid 2651 . . . . . 6 (𝑆s 𝐵) = (𝑆s 𝐵)
14 simpr 476 . . . . . 6 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝑇 ∈ (SubRing‘𝑅))
151, 7, 6, 11, 13, 14resspsrbas 19463 . . . . 5 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = (Base‘(𝑆s 𝐵)))
161, 7, 6, 11, 13, 14resspsradd 19464 . . . . 5 (((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑈)𝑦) = (𝑥(+g‘(𝑆s 𝐵))𝑦))
171, 7, 6, 11, 13, 14resspsrmul 19465 . . . . 5 (((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝑈)𝑦) = (𝑥(.r‘(𝑆s 𝐵))𝑦))
1812, 15, 16, 17ringpropd 18628 . . . 4 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (𝑈 ∈ Ring ↔ (𝑆s 𝐵) ∈ Ring))
1910, 18mpbid 222 . . 3 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (𝑆s 𝐵) ∈ Ring)
205, 19jca 553 . 2 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (𝑆 ∈ Ring ∧ (𝑆s 𝐵) ∈ Ring))
21 eqid 2651 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
2213, 21ressbasss 15979 . . . 4 (Base‘(𝑆s 𝐵)) ⊆ (Base‘𝑆)
2315, 22syl6eqss 3688 . . 3 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ⊆ (Base‘𝑆))
24 eqid 2651 . . . . . . . . . . . 12 (1r𝑅) = (1r𝑅)
2524subrg1cl 18836 . . . . . . . . . . 11 (𝑇 ∈ (SubRing‘𝑅) → (1r𝑅) ∈ 𝑇)
26 subrgsubg 18834 . . . . . . . . . . . 12 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
27 eqid 2651 . . . . . . . . . . . . 13 (0g𝑅) = (0g𝑅)
2827subg0cl 17649 . . . . . . . . . . . 12 (𝑇 ∈ (SubGrp‘𝑅) → (0g𝑅) ∈ 𝑇)
2926, 28syl 17 . . . . . . . . . . 11 (𝑇 ∈ (SubRing‘𝑅) → (0g𝑅) ∈ 𝑇)
3025, 29ifcld 4164 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅)) ∈ 𝑇)
3130adantl 481 . . . . . . . . 9 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅)) ∈ 𝑇)
327subrgbas 18837 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
3332adantl 481 . . . . . . . . 9 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝑇 = (Base‘𝐻))
3431, 33eleqtrd 2732 . . . . . . . 8 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅)) ∈ (Base‘𝐻))
3534adantr 480 . . . . . . 7 (((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅)) ∈ (Base‘𝐻))
36 eqid 2651 . . . . . . 7 (𝑥 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅))) = (𝑥 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅)))
3735, 36fmptd 6425 . . . . . 6 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (𝑥 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅))):{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
38 eqid 2651 . . . . . . . 8 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
39 eqid 2651 . . . . . . . 8 (1r𝑆) = (1r𝑆)
401, 2, 4, 38, 27, 24, 39psr1 19460 . . . . . . 7 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (1r𝑆) = (𝑥 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅))))
4140feq1d 6068 . . . . . 6 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → ((1r𝑆):{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻) ↔ (𝑥 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅))):{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻)))
4237, 41mpbird 247 . . . . 5 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (1r𝑆):{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
43 fvex 6239 . . . . . 6 (Base‘𝐻) ∈ V
44 ovex 6718 . . . . . . 7 (ℕ0𝑚 𝐼) ∈ V
4544rabex 4845 . . . . . 6 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V
4643, 45elmap 7928 . . . . 5 ((1r𝑆) ∈ ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ↔ (1r𝑆):{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
4742, 46sylibr 224 . . . 4 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (1r𝑆) ∈ ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
48 eqid 2651 . . . . 5 (Base‘𝐻) = (Base‘𝐻)
496, 48, 38, 11, 2psrbas 19426 . . . 4 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
5047, 49eleqtrrd 2733 . . 3 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (1r𝑆) ∈ 𝐵)
5123, 50jca 553 . 2 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (𝐵 ⊆ (Base‘𝑆) ∧ (1r𝑆) ∈ 𝐵))
5221, 39issubrg 18828 . 2 (𝐵 ∈ (SubRing‘𝑆) ↔ ((𝑆 ∈ Ring ∧ (𝑆s 𝐵) ∈ Ring) ∧ (𝐵 ⊆ (Base‘𝑆) ∧ (1r𝑆) ∈ 𝐵)))
5320, 51, 52sylanbrc 699 1 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {crab 2945  wss 3607  ifcif 4119  {csn 4210  cmpt 4762   × cxp 5141  ccnv 5142  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  Fincfn 7997  0cc0 9974  cn 11058  0cn0 11330  Basecbs 15904  s cress 15905  0gc0g 16147  SubGrpcsubg 17635  1rcur 18547  Ringcrg 18593  SubRingcsubrg 18824   mPwSer cmps 19399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-tset 16007  df-0g 16149  df-gsum 16150  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-subrg 18826  df-psr 19404
This theorem is referenced by:  ressmplbas2  19503  subrgmpl  19508
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