Theorem primefld 20646

Description: The smallest sub division ring of a division ring, here named 𝑃, is a field, called the Prime Field of 𝑅. (Suggested by GL, 4-Aug-2023.) (Contributed by Thierry Arnoux, 21-Aug-2023.)

Hypothesis

Ref	Expression
primefld.1	⊢ 𝑃 = (𝑅 ↾s ∩ (SubDRing‘𝑅))

Assertion

Ref	Expression
primefld	⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Field)

Proof of Theorem primefld

Dummy variables 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		primefld.1	. . 3 ⊢ 𝑃 = (𝑅 ↾s ∩ (SubDRing‘𝑅))
2		id 22	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ DivRing)
3		issdrg 20629	. . . . . 6 ⊢ (𝑠 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑠 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑠) ∈ DivRing))
4	3	simp2bi 1143	. . . . 5 ⊢ (𝑠 ∈ (SubDRing‘𝑅) → 𝑠 ∈ (SubRing‘𝑅))
5	4	ssriv 3978	. . . 4 ⊢ (SubDRing‘𝑅) ⊆ (SubRing‘𝑅)
6	5	a1i 11	. . 3 ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) ⊆ (SubRing‘𝑅))
7		eqid 2724	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	7	sdrgid 20633	. . . 4 ⊢ (𝑅 ∈ DivRing → (Base‘𝑅) ∈ (SubDRing‘𝑅))
9	8	ne0d 4327	. . 3 ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) ≠ ∅)
10	3	simp3bi 1144	. . . 4 ⊢ (𝑠 ∈ (SubDRing‘𝑅) → (𝑅 ↾s 𝑠) ∈ DivRing)
11	10	adantl 481	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝑠 ∈ (SubDRing‘𝑅)) → (𝑅 ↾s 𝑠) ∈ DivRing)
12	1, 2, 6, 9, 11	subdrgint 20644	. 2 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ DivRing)
13		drngring 20584	. . . 4 ⊢ (𝑃 ∈ DivRing → 𝑃 ∈ Ring)
14	12, 13	syl 17	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Ring)
15		ssidd 3997	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ DivRing → (Base‘𝑅) ⊆ (Base‘𝑅))
16		eqid 2724	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
17		eqid 2724	. . . . . . . . . . . . . . 15 ⊢ (Cntz‘(mulGrp‘𝑅)) = (Cntz‘(mulGrp‘𝑅))
18	7, 16, 17	cntzsdrg 20643	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ DivRing ∧ (Base‘𝑅) ⊆ (Base‘𝑅)) → ((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)) ∈ (SubDRing‘𝑅))
19	2, 15, 18	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ DivRing → ((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)) ∈ (SubDRing‘𝑅))
20		intss1 4957	. . . . . . . . . . . . 13 ⊢ (((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)) ∈ (SubDRing‘𝑅) → ∩ (SubDRing‘𝑅) ⊆ ((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)))
21	19, 20	syl 17	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ DivRing → ∩ (SubDRing‘𝑅) ⊆ ((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)))
22	16, 7	mgpbas 20035	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
23	22, 17	cntrval 19225	. . . . . . . . . . . 12 ⊢ ((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)) = (Cntr‘(mulGrp‘𝑅))
24	21, 23	sseqtrdi 4024	. . . . . . . . . . 11 ⊢ (𝑅 ∈ DivRing → ∩ (SubDRing‘𝑅) ⊆ (Cntr‘(mulGrp‘𝑅)))
25	22	cntrss 19237	. . . . . . . . . . 11 ⊢ (Cntr‘(mulGrp‘𝑅)) ⊆ (Base‘𝑅)
26	24, 25	sstrdi 3986	. . . . . . . . . 10 ⊢ (𝑅 ∈ DivRing → ∩ (SubDRing‘𝑅) ⊆ (Base‘𝑅))
27	1, 7	ressbas2 17181	. . . . . . . . . 10 ⊢ (∩ (SubDRing‘𝑅) ⊆ (Base‘𝑅) → ∩ (SubDRing‘𝑅) = (Base‘𝑃))
28	26, 27	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ DivRing → ∩ (SubDRing‘𝑅) = (Base‘𝑃))
29	28, 24	eqsstrrd 4013	. . . . . . . 8 ⊢ (𝑅 ∈ DivRing → (Base‘𝑃) ⊆ (Cntr‘(mulGrp‘𝑅)))
30	29	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (Base‘𝑃) ⊆ (Cntr‘(mulGrp‘𝑅)))
31		simprl 768	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → 𝑥 ∈ (Base‘𝑃))
32	30, 31	sseldd 3975	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → 𝑥 ∈ (Cntr‘(mulGrp‘𝑅)))
33	28, 26	eqsstrrd 4013	. . . . . . . 8 ⊢ (𝑅 ∈ DivRing → (Base‘𝑃) ⊆ (Base‘𝑅))
34	33	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (Base‘𝑃) ⊆ (Base‘𝑅))
35		simprr 770	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → 𝑦 ∈ (Base‘𝑃))
36	34, 35	sseldd 3975	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → 𝑦 ∈ (Base‘𝑅))
37		eqid 2724	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
38	16, 37	mgpplusg 20033	. . . . . . 7 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅))
39		eqid 2724	. . . . . . 7 ⊢ (Cntr‘(mulGrp‘𝑅)) = (Cntr‘(mulGrp‘𝑅))
40	22, 38, 39	cntri 19238	. . . . . 6 ⊢ ((𝑥 ∈ (Cntr‘(mulGrp‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (𝑦(.r‘𝑅)𝑥))
41	32, 36, 40	syl2anc 583	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(.r‘𝑅)𝑦) = (𝑦(.r‘𝑅)𝑥))
42	8, 26	ssexd 5314	. . . . . . 7 ⊢ (𝑅 ∈ DivRing → ∩ (SubDRing‘𝑅) ∈ V)
43	1, 37	ressmulr 17251	. . . . . . 7 ⊢ (∩ (SubDRing‘𝑅) ∈ V → (.r‘𝑅) = (.r‘𝑃))
44	42, 43	syl 17	. . . . . 6 ⊢ (𝑅 ∈ DivRing → (.r‘𝑅) = (.r‘𝑃))
45	44	oveqdr 7429	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑃)𝑦))
46	44	oveqdr 7429	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑦(.r‘𝑅)𝑥) = (𝑦(.r‘𝑃)𝑥))
47	41, 45, 46	3eqtr3d 2772	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(.r‘𝑃)𝑦) = (𝑦(.r‘𝑃)𝑥))
48	47	ralrimivva 3192	. . 3 ⊢ (𝑅 ∈ DivRing → ∀𝑥 ∈ (Base‘𝑃)∀𝑦 ∈ (Base‘𝑃)(𝑥(.r‘𝑃)𝑦) = (𝑦(.r‘𝑃)𝑥))
49		eqid 2724	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
50		eqid 2724	. . . 4 ⊢ (.r‘𝑃) = (.r‘𝑃)
51	49, 50	iscrng2 20147	. . 3 ⊢ (𝑃 ∈ CRing ↔ (𝑃 ∈ Ring ∧ ∀𝑥 ∈ (Base‘𝑃)∀𝑦 ∈ (Base‘𝑃)(𝑥(.r‘𝑃)𝑦) = (𝑦(.r‘𝑃)𝑥)))
52	14, 48, 51	sylanbrc 582	. 2 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ CRing)
53		isfld 20588	. 2 ⊢ (𝑃 ∈ Field ↔ (𝑃 ∈ DivRing ∧ 𝑃 ∈ CRing))
54	12, 52, 53	sylanbrc 582	1 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Field)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3053  Vcvv 3466   ⊆ wss 3940  ∩ cint 4940  ‘cfv 6533  (class class class)co 7401  Basecbs 17143   ↾_s cress 17172  ._rcmulr 17197  Cntzccntz 19221  Cntrccntr 19222  mulGrpcmgp 20029  Ringcrg 20128  CRingccrg 20129  SubRingcsubrg 20459  DivRingcdr 20577  Fieldcfield 20578  SubDRingcsdrg 20627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-tpos 8206  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-0g 17386  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-submnd 18704  df-grp 18856  df-minusg 18857  df-subg 19040  df-cntz 19223  df-cntr 19224  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-ring 20130  df-cring 20131  df-oppr 20226  df-dvdsr 20249  df-unit 20250  df-invr 20280  df-dvr 20293  df-subrng 20436  df-subrg 20461  df-drng 20579  df-field 20580  df-sdrg 20628

This theorem is referenced by: (None)