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Theorem primefld 20564
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 β†Ύs ∩ (SubDRingβ€˜π‘…))
Assertion
Ref Expression
primefld (𝑅 ∈ DivRing β†’ 𝑃 ∈ Field)

Proof of Theorem primefld
Dummy variables 𝑠 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 primefld.1 . . 3 𝑃 = (𝑅 β†Ύs ∩ (SubDRingβ€˜π‘…))
2 id 22 . . 3 (𝑅 ∈ DivRing β†’ 𝑅 ∈ DivRing)
3 issdrg 20547 . . . . . 6 (𝑠 ∈ (SubDRingβ€˜π‘…) ↔ (𝑅 ∈ DivRing ∧ 𝑠 ∈ (SubRingβ€˜π‘…) ∧ (𝑅 β†Ύs 𝑠) ∈ DivRing))
43simp2bi 1144 . . . . 5 (𝑠 ∈ (SubDRingβ€˜π‘…) β†’ 𝑠 ∈ (SubRingβ€˜π‘…))
54ssriv 3985 . . . 4 (SubDRingβ€˜π‘…) βŠ† (SubRingβ€˜π‘…)
65a1i 11 . . 3 (𝑅 ∈ DivRing β†’ (SubDRingβ€˜π‘…) βŠ† (SubRingβ€˜π‘…))
7 eqid 2730 . . . . 5 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
87sdrgid 20551 . . . 4 (𝑅 ∈ DivRing β†’ (Baseβ€˜π‘…) ∈ (SubDRingβ€˜π‘…))
98ne0d 4334 . . 3 (𝑅 ∈ DivRing β†’ (SubDRingβ€˜π‘…) β‰  βˆ…)
103simp3bi 1145 . . . 4 (𝑠 ∈ (SubDRingβ€˜π‘…) β†’ (𝑅 β†Ύs 𝑠) ∈ DivRing)
1110adantl 480 . . 3 ((𝑅 ∈ DivRing ∧ 𝑠 ∈ (SubDRingβ€˜π‘…)) β†’ (𝑅 β†Ύs 𝑠) ∈ DivRing)
121, 2, 6, 9, 11subdrgint 20562 . 2 (𝑅 ∈ DivRing β†’ 𝑃 ∈ DivRing)
13 drngring 20507 . . . 4 (𝑃 ∈ DivRing β†’ 𝑃 ∈ Ring)
1412, 13syl 17 . . 3 (𝑅 ∈ DivRing β†’ 𝑃 ∈ Ring)
15 ssidd 4004 . . . . . . . . . . . . . 14 (𝑅 ∈ DivRing β†’ (Baseβ€˜π‘…) βŠ† (Baseβ€˜π‘…))
16 eqid 2730 . . . . . . . . . . . . . . 15 (mulGrpβ€˜π‘…) = (mulGrpβ€˜π‘…)
17 eqid 2730 . . . . . . . . . . . . . . 15 (Cntzβ€˜(mulGrpβ€˜π‘…)) = (Cntzβ€˜(mulGrpβ€˜π‘…))
187, 16, 17cntzsdrg 20561 . . . . . . . . . . . . . 14 ((𝑅 ∈ DivRing ∧ (Baseβ€˜π‘…) βŠ† (Baseβ€˜π‘…)) β†’ ((Cntzβ€˜(mulGrpβ€˜π‘…))β€˜(Baseβ€˜π‘…)) ∈ (SubDRingβ€˜π‘…))
192, 15, 18syl2anc 582 . . . . . . . . . . . . 13 (𝑅 ∈ DivRing β†’ ((Cntzβ€˜(mulGrpβ€˜π‘…))β€˜(Baseβ€˜π‘…)) ∈ (SubDRingβ€˜π‘…))
20 intss1 4966 . . . . . . . . . . . . 13 (((Cntzβ€˜(mulGrpβ€˜π‘…))β€˜(Baseβ€˜π‘…)) ∈ (SubDRingβ€˜π‘…) β†’ ∩ (SubDRingβ€˜π‘…) βŠ† ((Cntzβ€˜(mulGrpβ€˜π‘…))β€˜(Baseβ€˜π‘…)))
2119, 20syl 17 . . . . . . . . . . . 12 (𝑅 ∈ DivRing β†’ ∩ (SubDRingβ€˜π‘…) βŠ† ((Cntzβ€˜(mulGrpβ€˜π‘…))β€˜(Baseβ€˜π‘…)))
2216, 7mgpbas 20034 . . . . . . . . . . . . 13 (Baseβ€˜π‘…) = (Baseβ€˜(mulGrpβ€˜π‘…))
2322, 17cntrval 19224 . . . . . . . . . . . 12 ((Cntzβ€˜(mulGrpβ€˜π‘…))β€˜(Baseβ€˜π‘…)) = (Cntrβ€˜(mulGrpβ€˜π‘…))
2421, 23sseqtrdi 4031 . . . . . . . . . . 11 (𝑅 ∈ DivRing β†’ ∩ (SubDRingβ€˜π‘…) βŠ† (Cntrβ€˜(mulGrpβ€˜π‘…)))
2522cntrss 19236 . . . . . . . . . . 11 (Cntrβ€˜(mulGrpβ€˜π‘…)) βŠ† (Baseβ€˜π‘…)
2624, 25sstrdi 3993 . . . . . . . . . 10 (𝑅 ∈ DivRing β†’ ∩ (SubDRingβ€˜π‘…) βŠ† (Baseβ€˜π‘…))
271, 7ressbas2 17186 . . . . . . . . . 10 (∩ (SubDRingβ€˜π‘…) βŠ† (Baseβ€˜π‘…) β†’ ∩ (SubDRingβ€˜π‘…) = (Baseβ€˜π‘ƒ))
2826, 27syl 17 . . . . . . . . 9 (𝑅 ∈ DivRing β†’ ∩ (SubDRingβ€˜π‘…) = (Baseβ€˜π‘ƒ))
2928, 24eqsstrrd 4020 . . . . . . . 8 (𝑅 ∈ DivRing β†’ (Baseβ€˜π‘ƒ) βŠ† (Cntrβ€˜(mulGrpβ€˜π‘…)))
3029adantr 479 . . . . . . 7 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ (Baseβ€˜π‘ƒ) βŠ† (Cntrβ€˜(mulGrpβ€˜π‘…)))
31 simprl 767 . . . . . . 7 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ π‘₯ ∈ (Baseβ€˜π‘ƒ))
3230, 31sseldd 3982 . . . . . 6 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ π‘₯ ∈ (Cntrβ€˜(mulGrpβ€˜π‘…)))
3328, 26eqsstrrd 4020 . . . . . . . 8 (𝑅 ∈ DivRing β†’ (Baseβ€˜π‘ƒ) βŠ† (Baseβ€˜π‘…))
3433adantr 479 . . . . . . 7 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ (Baseβ€˜π‘ƒ) βŠ† (Baseβ€˜π‘…))
35 simprr 769 . . . . . . 7 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ 𝑦 ∈ (Baseβ€˜π‘ƒ))
3634, 35sseldd 3982 . . . . . 6 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ 𝑦 ∈ (Baseβ€˜π‘…))
37 eqid 2730 . . . . . . . 8 (.rβ€˜π‘…) = (.rβ€˜π‘…)
3816, 37mgpplusg 20032 . . . . . . 7 (.rβ€˜π‘…) = (+gβ€˜(mulGrpβ€˜π‘…))
39 eqid 2730 . . . . . . 7 (Cntrβ€˜(mulGrpβ€˜π‘…)) = (Cntrβ€˜(mulGrpβ€˜π‘…))
4022, 38, 39cntri 19237 . . . . . 6 ((π‘₯ ∈ (Cntrβ€˜(mulGrpβ€˜π‘…)) ∧ 𝑦 ∈ (Baseβ€˜π‘…)) β†’ (π‘₯(.rβ€˜π‘…)𝑦) = (𝑦(.rβ€˜π‘…)π‘₯))
4132, 36, 40syl2anc 582 . . . . 5 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ (π‘₯(.rβ€˜π‘…)𝑦) = (𝑦(.rβ€˜π‘…)π‘₯))
428, 26ssexd 5323 . . . . . . 7 (𝑅 ∈ DivRing β†’ ∩ (SubDRingβ€˜π‘…) ∈ V)
431, 37ressmulr 17256 . . . . . . 7 (∩ (SubDRingβ€˜π‘…) ∈ V β†’ (.rβ€˜π‘…) = (.rβ€˜π‘ƒ))
4442, 43syl 17 . . . . . 6 (𝑅 ∈ DivRing β†’ (.rβ€˜π‘…) = (.rβ€˜π‘ƒ))
4544oveqdr 7439 . . . . 5 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ (π‘₯(.rβ€˜π‘…)𝑦) = (π‘₯(.rβ€˜π‘ƒ)𝑦))
4644oveqdr 7439 . . . . 5 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ (𝑦(.rβ€˜π‘…)π‘₯) = (𝑦(.rβ€˜π‘ƒ)π‘₯))
4741, 45, 463eqtr3d 2778 . . . 4 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ (π‘₯(.rβ€˜π‘ƒ)𝑦) = (𝑦(.rβ€˜π‘ƒ)π‘₯))
4847ralrimivva 3198 . . 3 (𝑅 ∈ DivRing β†’ βˆ€π‘₯ ∈ (Baseβ€˜π‘ƒ)βˆ€π‘¦ ∈ (Baseβ€˜π‘ƒ)(π‘₯(.rβ€˜π‘ƒ)𝑦) = (𝑦(.rβ€˜π‘ƒ)π‘₯))
49 eqid 2730 . . . 4 (Baseβ€˜π‘ƒ) = (Baseβ€˜π‘ƒ)
50 eqid 2730 . . . 4 (.rβ€˜π‘ƒ) = (.rβ€˜π‘ƒ)
5149, 50iscrng2 20146 . . 3 (𝑃 ∈ CRing ↔ (𝑃 ∈ Ring ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘ƒ)βˆ€π‘¦ ∈ (Baseβ€˜π‘ƒ)(π‘₯(.rβ€˜π‘ƒ)𝑦) = (𝑦(.rβ€˜π‘ƒ)π‘₯)))
5214, 48, 51sylanbrc 581 . 2 (𝑅 ∈ DivRing β†’ 𝑃 ∈ CRing)
53 isfld 20511 . 2 (𝑃 ∈ Field ↔ (𝑃 ∈ DivRing ∧ 𝑃 ∈ CRing))
5412, 52, 53sylanbrc 581 1 (𝑅 ∈ DivRing β†’ 𝑃 ∈ Field)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  Vcvv 3472   βŠ† wss 3947  βˆ© cint 4949  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148   β†Ύs cress 17177  .rcmulr 17202  Cntzccntz 19220  Cntrccntr 19221  mulGrpcmgp 20028  Ringcrg 20127  CRingccrg 20128  SubRingcsubrg 20457  DivRingcdr 20500  Fieldcfield 20501  SubDRingcsdrg 20545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18706  df-grp 18858  df-minusg 18859  df-subg 19039  df-cntz 19222  df-cntr 19223  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-cring 20130  df-oppr 20225  df-dvdsr 20248  df-unit 20249  df-invr 20279  df-dvr 20292  df-subrng 20434  df-subrg 20459  df-drng 20502  df-field 20503  df-sdrg 20546
This theorem is referenced by: (None)
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