subrgpsr - Metamath Proof Explorer

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Theorem subrgpsr 21850

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 20474	. . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
4	3	adantl 481	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑅 ∈ Ring)
5	1, 2, 4	psrring 21842	. 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 20472	. . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring)
9	8	adantl 481	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐻 ∈ Ring)
10	6, 2, 9	psrring 21842	. . 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 2731	. . . . 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 21846	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = (Base‘(𝑆 ↾_s 𝐵)))
16	1, 7, 6, 11, 13, 14	resspsradd 21847	. . . 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 21848	. . . 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 20183	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (𝑈 ∈ Ring ↔ (𝑆 ↾_s 𝐵) ∈ Ring))
19	10, 18	mpbid 231	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (𝑆 ↾_s 𝐵) ∈ Ring)
20		eqid 2731	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
21	13, 20	ressbasss 17190	. . . 4 ⊢ (Base‘(𝑆 ↾_s 𝐵)) ⊆ (Base‘𝑆)
22	15, 21	eqsstrdi 4036	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ⊆ (Base‘𝑆))
23		eqid 2731	. . . . . . 7 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
24		eqid 2731	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
25		eqid 2731	. . . . . . 7 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
26		eqid 2731	. . . . . . 7 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
27	1, 2, 4, 23, 24, 25, 26	psr1 21843	. . . . . 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 20478	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (1_r‘𝑅) ∈ 𝑇)
29		subrgsubg 20475	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
30	24	subg0cl 19057	. . . . . . . . . . 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 4574	. . . . . . . . 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 20479	. . . . . . . . 9 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
35	34	adantl 481	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝑇 = (Base‘𝐻))
36	33, 35	eleqtrd 2834	. . . . . . 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 7117	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1_r‘𝑆):{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
39		fvex 6904	. . . . . 6 ⊢ (Base‘𝐻) ∈ V
40		ovex 7445	. . . . . . 7 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
41	40	rabex 5332	. . . . . 6 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∈ V
42	39, 41	elmap 8871	. . . . 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 233	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → (1_r‘𝑆) ∈ ((Base‘𝐻) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
44		eqid 2731	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
45	6, 44, 23, 11, 2	psrbas 21807	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 = ((Base‘𝐻) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
46	43, 45	eleqtrrd 2835	. . 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 20469	. 2 ⊢ (𝐵 ∈ (SubRing‘𝑆) ↔ ((𝑆 ∈ Ring ∧ (𝑆 ↾_s 𝐵) ∈ Ring) ∧ (𝐵 ⊆ (Base‘𝑆) ∧ (1_r‘𝑆) ∈ 𝐵)))
49	5, 19, 47, 48	syl21anbrc 1343	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {crab 3431 ⊆ wss 3948 ifcif 4528 {csn 4628 × cxp 5674 ^◡ccnv 5675 “ cima 5679 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ↑_m cmap 8826 Fincfn 8945 0cc0 11116 ℕcn 12219 ℕ₀cn0 12479 Basecbs 17151 ↾_s cress 17180 0_gc0g 17392 SubGrpcsubg 19043 1_rcur 20082 Ringcrg 20134 SubRingcsubrg 20465 mPwSer cmps 21767

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-mulg 18994 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-subrng 20442 df-subrg 20467 df-psr 21772

This theorem is referenced by: ressmplbas2 21893 subrgmpl 21898

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