subrgpsr - Metamath Proof Explorer

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Theorem subrgpsr 21916

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	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		simpl 482	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐼 ∈ 𝑉)
3		subrgrcl 20493	. . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
4	3	adantl 481	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑅 ∈ Ring)
5	1, 2, 4	psrring 21908	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑆 ∈ Ring)
6		subrgpsr.u	. . . 4 ⊢ 𝑈 = (𝐼 mPwSer 𝐻)
7		subrgpsr.h	. . . . . 6 ⊢ 𝐻 = (𝑅 ↾_s 𝑇)
8	7	subrgring 20491	. . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring)
9	8	adantl 481	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐻 ∈ Ring)
10	6, 2, 9	psrring 21908	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑈 ∈ Ring)
11		subrgpsr.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑈)
12	11	a1i 11	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = (Base‘𝑈))
13		eqid 2733	. . . . 5 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s 𝐵)
14		simpr 484	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑇 ∈ (SubRing‘𝑅))
15	1, 7, 6, 11, 13, 14	resspsrbas 21912	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = (Base‘(𝑆 ↾_s 𝐵)))
16	1, 7, 6, 11, 13, 14	resspsradd 21913	. . . 4 ⊢ (((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 21914	. . . 4 ⊢ (((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 20208	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (𝑈 ∈ Ring ↔ (𝑆 ↾_s 𝐵) ∈ Ring))
19	10, 18	mpbid 232	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (𝑆 ↾_s 𝐵) ∈ Ring)
20		eqid 2733	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
21	13, 20	ressbasss 17152	. . . 4 ⊢ (Base‘(𝑆 ↾_s 𝐵)) ⊆ (Base‘𝑆)
22	15, 21	eqsstrdi 3975	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ⊆ (Base‘𝑆))
23		eqid 2733	. . . . . . 7 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
24		eqid 2733	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
25		eqid 2733	. . . . . . 7 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
26		eqid 2733	. . . . . . 7 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
27	1, 2, 4, 23, 24, 25, 26	psr1 21909	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1_r‘𝑆) = (𝑥 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑥 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅))))
28	25	subrg1cl 20497	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (1_r‘𝑅) ∈ 𝑇)
29		subrgsubg 20494	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
30	24	subg0cl 19049	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (SubGrp‘𝑅) → (0_g‘𝑅) ∈ 𝑇)
31	29, 30	syl 17	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0_g‘𝑅) ∈ 𝑇)
32	28, 31	ifcld 4521	. . . . . . . . 9 ⊢ (𝑇 ∈ (SubRing‘𝑅) → if(𝑥 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅)) ∈ 𝑇)
33	32	adantl 481	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → if(𝑥 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅)) ∈ 𝑇)
34	7	subrgbas 20498	. . . . . . . . 9 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
35	34	adantl 481	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑇 = (Base‘𝐻))
36	33, 35	eleqtrd 2835	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → if(𝑥 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅)) ∈ (Base‘𝐻))
37	36	adantr 480	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → if(𝑥 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅)) ∈ (Base‘𝐻))
38	27, 37	fmpt3d 7055	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1_r‘𝑆):{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
39		fvex 6841	. . . . . 6 ⊢ (Base‘𝐻) ∈ V
40		ovex 7385	. . . . . . 7 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
41	40	rabex 5279	. . . . . 6 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∈ V
42	39, 41	elmap 8801	. . . . 5 ⊢ ((1_r‘𝑆) ∈ ((Base‘𝐻) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) ↔ (1_r‘𝑆):{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
43	38, 42	sylibr 234	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1_r‘𝑆) ∈ ((Base‘𝐻) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
44		eqid 2733	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
45	6, 44, 23, 11, 2	psrbas 21872	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = ((Base‘𝐻) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
46	43, 45	eleqtrrd 2836	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1_r‘𝑆) ∈ 𝐵)
47	22, 46	jca 511	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (𝐵 ⊆ (Base‘𝑆) ∧ (1_r‘𝑆) ∈ 𝐵))
48	20, 26	issubrg 20488	. 2 ⊢ (𝐵 ∈ (SubRing‘𝑆) ↔ ((𝑆 ∈ Ring ∧ (𝑆 ↾_s 𝐵) ∈ Ring) ∧ (𝐵 ⊆ (Base‘𝑆) ∧ (1_r‘𝑆) ∈ 𝐵)))
49	5, 19, 47, 48	syl21anbrc 1345	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {crab 3396 ⊆ wss 3898 ifcif 4474 {csn 4575 × cxp 5617 ^◡ccnv 5618 “ cima 5622 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 ↑_m cmap 8756 Fincfn 8875 0cc0 11013 ℕcn 12132 ℕ₀cn0 12388 Basecbs 17122 ↾_s cress 17143 0_gc0g 17345 SubGrpcsubg 19035 1_rcur 20101 Ringcrg 20153 SubRingcsubrg 20486 mPwSer cmps 21843

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-ofr 7617 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-sup 9333 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-fz 13410 df-fzo 13557 df-seq 13911 df-hash 14240 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-hom 17187 df-cco 17188 df-0g 17347 df-gsum 17348 df-prds 17353 df-pws 17355 df-mre 17490 df-mrc 17491 df-acs 17493 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-mhm 18693 df-submnd 18694 df-grp 18851 df-minusg 18852 df-mulg 18983 df-subg 19038 df-ghm 19127 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-subrng 20463 df-subrg 20487 df-psr 21848

This theorem is referenced by: ressmplbas2 21963 subrgmpl 21968

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