subrgpsr - Metamath Proof Explorer

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Theorem subrgpsr 22009

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 486	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐼 ∈ 𝑉)
3		subrgrcl 20605	. . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
4	3	adantl 485	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑅 ∈ Ring)
5	1, 2, 4	psrring 22001	. 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 20603	. . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring)
9	8	adantl 485	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐻 ∈ Ring)
10	6, 2, 9	psrring 22001	. . 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 2761	. . . . 5 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s 𝐵)
14		simpr 488	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑇 ∈ (SubRing‘𝑅))
15	1, 7, 6, 11, 13, 14	resspsrbas 22005	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = (Base‘(𝑆 ↾_s 𝐵)))
16	1, 7, 6, 11, 13, 14	resspsradd 22006	. . . 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 22007	. . . 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 20317	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (𝑈 ∈ Ring ↔ (𝑆 ↾_s 𝐵) ∈ Ring))
19	10, 18	mpbid 234	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (𝑆 ↾_s 𝐵) ∈ Ring)
20		eqid 2761	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
21	13, 20	ressbasss 17258	. . . 4 ⊢ (Base‘(𝑆 ↾_s 𝐵)) ⊆ (Base‘𝑆)
22	15, 21	eqsstrdi 3980	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ⊆ (Base‘𝑆))
23		eqid 2761	. . . . . . 7 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
24		eqid 2761	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
25		eqid 2761	. . . . . . 7 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
26		eqid 2761	. . . . . . 7 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
27	1, 2, 4, 23, 24, 25, 26	psr1 22002	. . . . . 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 20609	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (1_r‘𝑅) ∈ 𝑇)
29		subrgsubg 20606	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
30	24	subg0cl 19159	. . . . . . . . . . 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 4526	. . . . . . . . 9 ⊢ (𝑇 ∈ (SubRing‘𝑅) → if(𝑥 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅)) ∈ 𝑇)
33	32	adantl 485	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → if(𝑥 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅)) ∈ 𝑇)
34	7	subrgbas 20610	. . . . . . . . 9 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
35	34	adantl 485	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑇 = (Base‘𝐻))
36	33, 35	eleqtrd 2863	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → if(𝑥 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅)) ∈ (Base‘𝐻))
37	36	adantr 484	. . . . . 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 7093	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1_r‘𝑆):{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
39		fvex 6876	. . . . . 6 ⊢ (Base‘𝐻) ∈ V
40		ovex 7425	. . . . . . 7 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
41	40	rabex 5294	. . . . . 6 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∈ V
42	39, 41	elmap 8849	. . . . 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 236	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1_r‘𝑆) ∈ ((Base‘𝐻) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
44		eqid 2761	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
45	6, 44, 23, 11, 2	psrbas 21966	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = ((Base‘𝐻) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
46	43, 45	eleqtrrd 2864	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1_r‘𝑆) ∈ 𝐵)
47	22, 46	jca 519	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (𝐵 ⊆ (Base‘𝑆) ∧ (1_r‘𝑆) ∈ 𝐵))
48	20, 26	issubrg 20600	. 2 ⊢ (𝐵 ∈ (SubRing‘𝑆) ↔ ((𝑆 ∈ Ring ∧ (𝑆 ↾_s 𝐵) ∈ Ring) ∧ (𝐵 ⊆ (Base‘𝑆) ∧ (1_r‘𝑆) ∈ 𝐵)))
49	5, 19, 47, 48	syl21anbrc 1357	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 {crab 3413 ⊆ wss 3904 ifcif 4479 {csn 4581 × cxp 5643 ^◡ccnv 5644 “ cima 5648 ⟶wf 6513 ‘cfv 6517 (class class class)co 7392 ↑_m cmap 8803 Fincfn 8923 0cc0 11070 ℕcn 12207 ℕ₀cn0 12478 Basecbs 17228 ↾_s cress 17249 0_gc0g 17451 SubGrpcsubg 19145 1_rcur 20210 Ringcrg 20262 SubRingcsubrg 20598 mPwSer cmps 21936

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-ofr 7657 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-map 8805 df-pm 8806 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-sup 9385 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-fz 13510 df-fzo 13657 df-seq 14012 df-hash 14341 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-hom 17293 df-cco 17294 df-0g 17453 df-gsum 17454 df-prds 17459 df-pws 17461 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-mhm 18800 df-submnd 18801 df-grp 18961 df-minusg 18962 df-mulg 19093 df-subg 19148 df-ghm 19237 df-cntz 19340 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-subrng 20575 df-subrg 20599 df-psr 21941

This theorem is referenced by: ressmplbas2 22059 subrgmpl 22064

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