Theorem subrgsubm 14480

Description: A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)

Hypothesis

Ref	Expression
subrgsubm.1	⊢ 𝑀 = (mulGrp‘𝑅)

Assertion

Ref	Expression
subrgsubm	⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubMnd‘𝑀))

Proof of Theorem subrgsubm

Step	Hyp	Ref	Expression
1		eqid 2234	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2	1	subrgss 14468	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
3		eqid 2234	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
4	3	subrg1cl 14475	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) ∈ 𝐴)
5		subrgrcl 14472	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
6		eqid 2234	. . . . 5 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴)
7		subrgsubm.1	. . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅)
8	6, 7	mgpress 14170	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑀 ↾s 𝐴) = (mulGrp‘(𝑅 ↾s 𝐴)))
9	5, 8	mpancom 422	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑀 ↾s 𝐴) = (mulGrp‘(𝑅 ↾s 𝐴)))
10	6	subrgring 14470	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ Ring)
11		eqid 2234	. . . . 5 ⊢ (mulGrp‘(𝑅 ↾s 𝐴)) = (mulGrp‘(𝑅 ↾s 𝐴))
12	11	ringmgp 14245	. . . 4 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (mulGrp‘(𝑅 ↾s 𝐴)) ∈ Mnd)
13	10, 12	syl 14	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (mulGrp‘(𝑅 ↾s 𝐴)) ∈ Mnd)
14	9, 13	eqeltrd 2311	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑀 ↾s 𝐴) ∈ Mnd)
15	7	ringmgp 14245	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
16		eqid 2234	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
17		eqid 2234	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
18		eqid 2234	. . . . . 6 ⊢ (𝑀 ↾s 𝐴) = (𝑀 ↾s 𝐴)
19	16, 17, 18	issubm2 13728	. . . . 5 ⊢ (𝑀 ∈ Mnd → (𝐴 ∈ (SubMnd‘𝑀) ↔ (𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ (𝑀 ↾s 𝐴) ∈ Mnd)))
20	15, 19	syl 14	. . . 4 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubMnd‘𝑀) ↔ (𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ (𝑀 ↾s 𝐴) ∈ Mnd)))
21	5, 20	syl 14	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ∈ (SubMnd‘𝑀) ↔ (𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ (𝑀 ↾s 𝐴) ∈ Mnd)))
22	7, 1	mgpbasg 14165	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑀))
23	22	sseq2d 3272	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝐴 ⊆ (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑀)))
24	7, 3	ringidvalg 14204	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) = (0g‘𝑀))
25	24	eleq1d 2303	. . . . . 6 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ∈ 𝐴 ↔ (0g‘𝑀) ∈ 𝐴))
26	23, 25	3anbi12d 1350	. . . . 5 ⊢ (𝑅 ∈ Ring → ((𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴 ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ↔ (𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ (𝑀 ↾s 𝐴) ∈ Mnd)))
27	26	bibi2d 232	. . . 4 ⊢ (𝑅 ∈ Ring → ((𝐴 ∈ (SubMnd‘𝑀) ↔ (𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴 ∧ (𝑀 ↾s 𝐴) ∈ Mnd)) ↔ (𝐴 ∈ (SubMnd‘𝑀) ↔ (𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ (𝑀 ↾s 𝐴) ∈ Mnd))))
28	5, 27	syl 14	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝐴 ∈ (SubMnd‘𝑀) ↔ (𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴 ∧ (𝑀 ↾s 𝐴) ∈ Mnd)) ↔ (𝐴 ∈ (SubMnd‘𝑀) ↔ (𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ (𝑀 ↾s 𝐴) ∈ Mnd))))
29	21, 28	mpbird 167	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ∈ (SubMnd‘𝑀) ↔ (𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴 ∧ (𝑀 ↾s 𝐴) ∈ Mnd)))
30	2, 4, 14, 29	mpbir3and 1207	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubMnd‘𝑀))

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205   ⊆ wss 3214  ‘cfv 5357  (class class class)co 6058  Basecbs 13296   ↾_s cress 13297  0_gc0g 13553  Mndcmnd 13677  SubMndcsubmnd 13713  mulGrpcmgp 14159  1_rcur 14202  Ringcrg 14239  SubRingcsubrg 14463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-submnd 13715  df-mgp 14160  df-ur 14203  df-ring 14241  df-subrg 14465

This theorem is referenced by:  resrhm  14494  resrhm2b  14495  rhmima  14497  lgseisenlem4  16072