Theorem subrgpsr 19419

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 = (𝑅 ↾s 𝑇)
subrgpsr.u	⊢ 𝑈 = (𝐼 mPwSer 𝐻)
subrgpsr.b	⊢ 𝐵 = (Base‘𝑈)

Assertion

Ref	Expression
subrgpsr	⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))

Proof of Theorem subrgpsr

Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgpsr.s	. . . 4 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		simpl 473	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐼 ∈ 𝑉)
3		subrgrcl 18785	. . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
4	3	adantl 482	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑅 ∈ Ring)
5	1, 2, 4	psrring 19411	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑆 ∈ Ring)
6		subrgpsr.u	. . . . 5 ⊢ 𝑈 = (𝐼 mPwSer 𝐻)
7		subrgpsr.h	. . . . . . 7 ⊢ 𝐻 = (𝑅 ↾s 𝑇)
8	7	subrgring 18783	. . . . . 6 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring)
9	8	adantl 482	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐻 ∈ Ring)
10	6, 2, 9	psrring 19411	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑈 ∈ Ring)
11		subrgpsr.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑈)
12	11	a1i 11	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = (Base‘𝑈))
13		eqid 2622	. . . . . 6 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵)
14		simpr 477	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑇 ∈ (SubRing‘𝑅))
15	1, 7, 6, 11, 13, 14	resspsrbas 19415	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = (Base‘(𝑆 ↾s 𝐵)))
16	1, 7, 6, 11, 13, 14	resspsradd 19416	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑈)𝑦) = (𝑥(+g‘(𝑆 ↾s 𝐵))𝑦))
17	1, 7, 6, 11, 13, 14	resspsrmul 19417	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝑈)𝑦) = (𝑥(.r‘(𝑆 ↾s 𝐵))𝑦))
18	12, 15, 16, 17	ringpropd 18582	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (𝑈 ∈ Ring ↔ (𝑆 ↾s 𝐵) ∈ Ring))
19	10, 18	mpbid 222	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (𝑆 ↾s 𝐵) ∈ Ring)
20	5, 19	jca 554	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (𝑆 ∈ Ring ∧ (𝑆 ↾s 𝐵) ∈ Ring))
21		eqid 2622	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
22	13, 21	ressbasss 15932	. . . 4 ⊢ (Base‘(𝑆 ↾s 𝐵)) ⊆ (Base‘𝑆)
23	15, 22	syl6eqss 3655	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ⊆ (Base‘𝑆))
24		eqid 2622	. . . . . . . . . . . 12 ⊢ (1r‘𝑅) = (1r‘𝑅)
25	24	subrg1cl 18788	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (1r‘𝑅) ∈ 𝑇)
26		subrgsubg 18786	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
27		eqid 2622	. . . . . . . . . . . . 13 ⊢ (0g‘𝑅) = (0g‘𝑅)
28	27	subg0cl 17602	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ (SubGrp‘𝑅) → (0g‘𝑅) ∈ 𝑇)
29	26, 28	syl 17	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0g‘𝑅) ∈ 𝑇)
30	25, 29	ifcld 4131	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubRing‘𝑅) → if(𝑥 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)) ∈ 𝑇)
31	30	adantl 482	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → if(𝑥 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)) ∈ 𝑇)
32	7	subrgbas 18789	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
33	32	adantl 482	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑇 = (Base‘𝐻))
34	31, 33	eleqtrd 2703	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → if(𝑥 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝐻))
35	34	adantr 481	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → if(𝑥 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝐻))
36		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) = (𝑥 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)))
37	35, 36	fmptd 6385	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (𝑥 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
38		eqid 2622	. . . . . . . 8 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
39		eqid 2622	. . . . . . . 8 ⊢ (1r‘𝑆) = (1r‘𝑆)
40	1, 2, 4, 38, 27, 24, 39	psr1 19412	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1r‘𝑆) = (𝑥 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))))
41	40	feq1d 6030	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → ((1r‘𝑆):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻) ↔ (𝑥 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻)))
42	37, 41	mpbird 247	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1r‘𝑆):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
43		fvex 6201	. . . . . 6 ⊢ (Base‘𝐻) ∈ V
44		ovex 6678	. . . . . . 7 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V
45	44	rabex 4813	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V
46	43, 45	elmap 7886	. . . . 5 ⊢ ((1r‘𝑆) ∈ ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ↔ (1r‘𝑆):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
47	42, 46	sylibr 224	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1r‘𝑆) ∈ ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
48		eqid 2622	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
49	6, 48, 38, 11, 2	psrbas 19378	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
50	47, 49	eleqtrrd 2704	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1r‘𝑆) ∈ 𝐵)
51	23, 50	jca 554	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (𝐵 ⊆ (Base‘𝑆) ∧ (1r‘𝑆) ∈ 𝐵))
52	21, 39	issubrg 18780	. 2 ⊢ (𝐵 ∈ (SubRing‘𝑆) ↔ ((𝑆 ∈ Ring ∧ (𝑆 ↾s 𝐵) ∈ Ring) ∧ (𝐵 ⊆ (Base‘𝑆) ∧ (1r‘𝑆) ∈ 𝐵)))
53	20, 51, 52	sylanbrc 698	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916   ⊆ wss 3574  ifcif 4086  {csn 4177   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Fincfn 7955  0cc0 9936  ℕcn 11020  ℕ₀cn0 11292  Basecbs 15857   ↾_s cress 15858  0_gc0g 16100  SubGrpcsubg 17588  1_rcur 18501  Ringcrg 18547  SubRingcsubrg 18776   mPwSer cmps 19351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-psr 19356

This theorem is referenced by:  ressmplbas2  19455  subrgmpl  19460