primefld - Metamath Proof Explorer

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

Theorem primefld 20413

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 20396	. . . . . 6 ⊢ (𝑠 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑠 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾_s 𝑠) ∈ DivRing))
4	3	simp2bi 1146	. . . . 5 ⊢ (𝑠 ∈ (SubDRing‘𝑅) → 𝑠 ∈ (SubRing‘𝑅))
5	4	ssriv 3985	. . . 4 ⊢ (SubDRing‘𝑅) ⊆ (SubRing‘𝑅)
6	5	a1i 11	. . 3 ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) ⊆ (SubRing‘𝑅))
7		eqid 2732	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	7	sdrgid 20400	. . . 4 ⊢ (𝑅 ∈ DivRing → (Base‘𝑅) ∈ (SubDRing‘𝑅))
9	8	ne0d 4334	. . 3 ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) ≠ ∅)
10	3	simp3bi 1147	. . . 4 ⊢ (𝑠 ∈ (SubDRing‘𝑅) → (𝑅 ↾_s 𝑠) ∈ DivRing)
11	10	adantl 482	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝑠 ∈ (SubDRing‘𝑅)) → (𝑅 ↾_s 𝑠) ∈ DivRing)
12	1, 2, 6, 9, 11	subdrgint 20411	. 2 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ DivRing)
13		drngring 20314	. . . 4 ⊢ (𝑃 ∈ DivRing → 𝑃 ∈ Ring)
14	12, 13	syl 17	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Ring)
15		ssidd 4004	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ DivRing → (Base‘𝑅) ⊆ (Base‘𝑅))
16		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
17		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (Cntz‘(mulGrp‘𝑅)) = (Cntz‘(mulGrp‘𝑅))
18	7, 16, 17	cntzsdrg 20410	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ DivRing ∧ (Base‘𝑅) ⊆ (Base‘𝑅)) → ((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)) ∈ (SubDRing‘𝑅))
19	2, 15, 18	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ DivRing → ((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)) ∈ (SubDRing‘𝑅))
20		intss1 4966	. . . . . . . . . . . . 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 19987	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
23	22, 17	cntrval 19177	. . . . . . . . . . . 12 ⊢ ((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)) = (Cntr‘(mulGrp‘𝑅))
24	21, 23	sseqtrdi 4031	. . . . . . . . . . 11 ⊢ (𝑅 ∈ DivRing → ∩ (SubDRing‘𝑅) ⊆ (Cntr‘(mulGrp‘𝑅)))
25	22	cntrss 19189	. . . . . . . . . . 11 ⊢ (Cntr‘(mulGrp‘𝑅)) ⊆ (Base‘𝑅)
26	24, 25	sstrdi 3993	. . . . . . . . . 10 ⊢ (𝑅 ∈ DivRing → ∩ (SubDRing‘𝑅) ⊆ (Base‘𝑅))
27	1, 7	ressbas2 17178	. . . . . . . . . 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 4020	. . . . . . . 8 ⊢ (𝑅 ∈ DivRing → (Base‘𝑃) ⊆ (Cntr‘(mulGrp‘𝑅)))
30	29	adantr 481	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (Base‘𝑃) ⊆ (Cntr‘(mulGrp‘𝑅)))
31		simprl 769	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → 𝑥 ∈ (Base‘𝑃))
32	30, 31	sseldd 3982	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → 𝑥 ∈ (Cntr‘(mulGrp‘𝑅)))
33	28, 26	eqsstrrd 4020	. . . . . . . 8 ⊢ (𝑅 ∈ DivRing → (Base‘𝑃) ⊆ (Base‘𝑅))
34	33	adantr 481	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (Base‘𝑃) ⊆ (Base‘𝑅))
35		simprr 771	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → 𝑦 ∈ (Base‘𝑃))
36	34, 35	sseldd 3982	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → 𝑦 ∈ (Base‘𝑅))
37		eqid 2732	. . . . . . . 8 ⊢ (._r‘𝑅) = (._r‘𝑅)
38	16, 37	mgpplusg 19985	. . . . . . 7 ⊢ (._r‘𝑅) = (+_g‘(mulGrp‘𝑅))
39		eqid 2732	. . . . . . 7 ⊢ (Cntr‘(mulGrp‘𝑅)) = (Cntr‘(mulGrp‘𝑅))
40	22, 38, 39	cntri 19190	. . . . . 6 ⊢ ((𝑥 ∈ (Cntr‘(mulGrp‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(._r‘𝑅)𝑦) = (𝑦(._r‘𝑅)𝑥))
41	32, 36, 40	syl2anc 584	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(._r‘𝑅)𝑦) = (𝑦(._r‘𝑅)𝑥))
42	8, 26	ssexd 5323	. . . . . . 7 ⊢ (𝑅 ∈ DivRing → ∩ (SubDRing‘𝑅) ∈ V)
43	1, 37	ressmulr 17248	. . . . . . 7 ⊢ (∩ (SubDRing‘𝑅) ∈ V → (._r‘𝑅) = (._r‘𝑃))
44	42, 43	syl 17	. . . . . 6 ⊢ (𝑅 ∈ DivRing → (._r‘𝑅) = (._r‘𝑃))
45	44	oveqdr 7433	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(._r‘𝑅)𝑦) = (𝑥(._r‘𝑃)𝑦))
46	44	oveqdr 7433	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑦(._r‘𝑅)𝑥) = (𝑦(._r‘𝑃)𝑥))
47	41, 45, 46	3eqtr3d 2780	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(._r‘𝑃)𝑦) = (𝑦(._r‘𝑃)𝑥))
48	47	ralrimivva 3200	. . 3 ⊢ (𝑅 ∈ DivRing → ∀𝑥 ∈ (Base‘𝑃)∀𝑦 ∈ (Base‘𝑃)(𝑥(._r‘𝑃)𝑦) = (𝑦(._r‘𝑃)𝑥))
49		eqid 2732	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
50		eqid 2732	. . . 4 ⊢ (._r‘𝑃) = (._r‘𝑃)
51	49, 50	iscrng2 20068	. . 3 ⊢ (𝑃 ∈ CRing ↔ (𝑃 ∈ Ring ∧ ∀𝑥 ∈ (Base‘𝑃)∀𝑦 ∈ (Base‘𝑃)(𝑥(._r‘𝑃)𝑦) = (𝑦(._r‘𝑃)𝑥)))
52	14, 48, 51	sylanbrc 583	. 2 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ CRing)
53		isfld 20318	. 2 ⊢ (𝑃 ∈ Field ↔ (𝑃 ∈ DivRing ∧ 𝑃 ∈ CRing))
54	12, 52, 53	sylanbrc 583	1 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Field)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ⊆ wss 3947 ∩ cint 4949 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 ↾_s cress 17169 ._rcmulr 17194 Cntzccntz 19173 Cntrccntr 19174 mulGrpcmgp 19981 Ringcrg 20049 CRingccrg 20050 DivRingcdr 20307 Fieldcfield 20308 SubRingcsubrg 20351 SubDRingcsdrg 20394

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 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 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-subg 18997 df-cntz 19175 df-cntr 19176 df-cmn 19644 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-drng 20309 df-field 20310 df-subrg 20353 df-sdrg 20395

This theorem is referenced by: (None)

W3C validator