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Theorem subrgpsr 19186
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 471 . . . 4 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐼𝑉)
3 subrgrcl 18554 . . . . 5 (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
43adantl 480 . . . 4 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝑅 ∈ Ring)
51, 2, 4psrring 19178 . . 3 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝑆 ∈ Ring)
6 subrgpsr.u . . . . 5 𝑈 = (𝐼 mPwSer 𝐻)
7 subrgpsr.h . . . . . . 7 𝐻 = (𝑅s 𝑇)
87subrgring 18552 . . . . . 6 (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring)
98adantl 480 . . . . 5 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐻 ∈ Ring)
106, 2, 9psrring 19178 . . . 4 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝑈 ∈ Ring)
11 subrgpsr.b . . . . . 6 𝐵 = (Base‘𝑈)
1211a1i 11 . . . . 5 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = (Base‘𝑈))
13 eqid 2609 . . . . . 6 (𝑆s 𝐵) = (𝑆s 𝐵)
14 simpr 475 . . . . . 6 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝑇 ∈ (SubRing‘𝑅))
151, 7, 6, 11, 13, 14resspsrbas 19182 . . . . 5 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = (Base‘(𝑆s 𝐵)))
161, 7, 6, 11, 13, 14resspsradd 19183 . . . . 5 (((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑈)𝑦) = (𝑥(+g‘(𝑆s 𝐵))𝑦))
171, 7, 6, 11, 13, 14resspsrmul 19184 . . . . 5 (((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝑈)𝑦) = (𝑥(.r‘(𝑆s 𝐵))𝑦))
1812, 15, 16, 17ringpropd 18351 . . . 4 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (𝑈 ∈ Ring ↔ (𝑆s 𝐵) ∈ Ring))
1910, 18mpbid 220 . . 3 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (𝑆s 𝐵) ∈ Ring)
205, 19jca 552 . 2 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (𝑆 ∈ Ring ∧ (𝑆s 𝐵) ∈ Ring))
21 eqid 2609 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
2213, 21ressbasss 15705 . . . 4 (Base‘(𝑆s 𝐵)) ⊆ (Base‘𝑆)
2315, 22syl6eqss 3617 . . 3 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ⊆ (Base‘𝑆))
24 eqid 2609 . . . . . . . . . . . 12 (1r𝑅) = (1r𝑅)
2524subrg1cl 18557 . . . . . . . . . . 11 (𝑇 ∈ (SubRing‘𝑅) → (1r𝑅) ∈ 𝑇)
26 subrgsubg 18555 . . . . . . . . . . . 12 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
27 eqid 2609 . . . . . . . . . . . . 13 (0g𝑅) = (0g𝑅)
2827subg0cl 17371 . . . . . . . . . . . 12 (𝑇 ∈ (SubGrp‘𝑅) → (0g𝑅) ∈ 𝑇)
2926, 28syl 17 . . . . . . . . . . 11 (𝑇 ∈ (SubRing‘𝑅) → (0g𝑅) ∈ 𝑇)
3025, 29ifcld 4080 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅)) ∈ 𝑇)
3130adantl 480 . . . . . . . . 9 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅)) ∈ 𝑇)
327subrgbas 18558 . . . . . . . . . 10 (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
3332adantl 480 . . . . . . . . 9 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝑇 = (Base‘𝐻))
3431, 33eleqtrd 2689 . . . . . . . 8 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅)) ∈ (Base‘𝐻))
3534adantr 479 . . . . . . 7 (((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅)) ∈ (Base‘𝐻))
36 eqid 2609 . . . . . . 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 6277 . . . . . 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 2609 . . . . . . . 8 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
39 eqid 2609 . . . . . . . 8 (1r𝑆) = (1r𝑆)
401, 2, 4, 38, 27, 24, 39psr1 19179 . . . . . . 7 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (1r𝑆) = (𝑥 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (1r𝑅), (0g𝑅))))
4140feq1d 5929 . . . . . 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 245 . . . . 5 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (1r𝑆):{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
43 fvex 6098 . . . . . 6 (Base‘𝐻) ∈ V
44 ovex 6555 . . . . . . 7 (ℕ0𝑚 𝐼) ∈ V
4544rabex 4735 . . . . . 6 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V
4643, 45elmap 7749 . . . . 5 ((1r𝑆) ∈ ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ↔ (1r𝑆):{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
4742, 46sylibr 222 . . . 4 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (1r𝑆) ∈ ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
48 eqid 2609 . . . . 5 (Base‘𝐻) = (Base‘𝐻)
496, 48, 38, 11, 2psrbas 19145 . . . 4 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}))
5047, 49eleqtrrd 2690 . . 3 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (1r𝑆) ∈ 𝐵)
5123, 50jca 552 . 2 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → (𝐵 ⊆ (Base‘𝑆) ∧ (1r𝑆) ∈ 𝐵))
5221, 39issubrg 18549 . 2 (𝐵 ∈ (SubRing‘𝑆) ↔ ((𝑆 ∈ Ring ∧ (𝑆s 𝐵) ∈ Ring) ∧ (𝐵 ⊆ (Base‘𝑆) ∧ (1r𝑆) ∈ 𝐵)))
5320, 51, 52sylanbrc 694 1 ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  {crab 2899  wss 3539  ifcif 4035  {csn 4124  cmpt 4637   × cxp 5026  ccnv 5027  cima 5031  wf 5786  cfv 5790  (class class class)co 6527  𝑚 cmap 7721  Fincfn 7818  0cc0 9792  cn 10867  0cn0 11139  Basecbs 15641  s cress 15642  0gc0g 15869  SubGrpcsubg 17357  1rcur 18270  Ringcrg 18316  SubRingcsubrg 18545   mPwSer cmps 19118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-ofr 6773  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-oi 8275  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-seq 12619  df-hash 12935  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-mulr 15728  df-sca 15730  df-vsca 15731  df-tset 15733  df-0g 15871  df-gsum 15872  df-mre 16015  df-mrc 16016  df-acs 16018  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-mhm 17104  df-submnd 17105  df-grp 17194  df-minusg 17195  df-mulg 17310  df-subg 17360  df-ghm 17427  df-cntz 17519  df-cmn 17964  df-abl 17965  df-mgp 18259  df-ur 18271  df-ring 18318  df-subrg 18547  df-psr 19123
This theorem is referenced by:  ressmplbas2  19222  subrgmpl  19227
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