subrgpsr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > subrgpsr		Structured version Visualization version GIF version

Theorem subrgpsr 21887

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 20485	. . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
4	3	adantl 481	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑅 ∈ Ring)
5	1, 2, 4	psrring 21879	. 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 20483	. . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring)
9	8	adantl 481	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐻 ∈ Ring)
10	6, 2, 9	psrring 21879	. . 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 2729	. . . . 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 21883	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = (Base‘(𝑆 ↾_s 𝐵)))
16	1, 7, 6, 11, 13, 14	resspsradd 21884	. . . 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 21885	. . . 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 20197	. . 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 2729	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
21	13, 20	ressbasss 17209	. . . 4 ⊢ (Base‘(𝑆 ↾_s 𝐵)) ⊆ (Base‘𝑆)
22	15, 21	eqsstrdi 3991	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ⊆ (Base‘𝑆))
23		eqid 2729	. . . . . . 7 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
24		eqid 2729	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
25		eqid 2729	. . . . . . 7 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
26		eqid 2729	. . . . . . 7 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
27	1, 2, 4, 23, 24, 25, 26	psr1 21880	. . . . . 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 20489	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (1_r‘𝑅) ∈ 𝑇)
29		subrgsubg 20486	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
30	24	subg0cl 19066	. . . . . . . . . . 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 4535	. . . . . . . . 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 20490	. . . . . . . . 9 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
35	34	adantl 481	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑇 = (Base‘𝐻))
36	33, 35	eleqtrd 2830	. . . . . . 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 7088	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1_r‘𝑆):{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
39		fvex 6871	. . . . . 6 ⊢ (Base‘𝐻) ∈ V
40		ovex 7420	. . . . . . 7 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
41	40	rabex 5294	. . . . . 6 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∈ V
42	39, 41	elmap 8844	. . . . 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 2729	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
45	6, 44, 23, 11, 2	psrbas 21842	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = ((Base‘𝐻) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
46	43, 45	eleqtrrd 2831	. . 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 20480	. 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 1540 ∈ wcel 2109 {crab 3405 ⊆ wss 3914 ifcif 4488 {csn 4589 × cxp 5636 ^◡ccnv 5637 “ cima 5641 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ↑_m cmap 8799 Fincfn 8918 0cc0 11068 ℕcn 12186 ℕ₀cn0 12442 Basecbs 17179 ↾_s cress 17200 0_gc0g 17402 SubGrpcsubg 19052 1_rcur 20090 Ringcrg 20142 SubRingcsubrg 20478 mPwSer cmps 21813

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-ofr 7654 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-fzo 13616 df-seq 13967 df-hash 14296 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-mulg 19000 df-subg 19055 df-ghm 19145 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-subrng 20455 df-subrg 20479 df-psr 21818

This theorem is referenced by: ressmplbas2 21934 subrgmpl 21939

W3C validator