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Theorem primefld 20421
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 20404 . . . . . 6 (𝑠 ∈ (SubDRingβ€˜π‘…) ↔ (𝑅 ∈ DivRing ∧ 𝑠 ∈ (SubRingβ€˜π‘…) ∧ (𝑅 β†Ύs 𝑠) ∈ DivRing))
43simp2bi 1147 . . . . 5 (𝑠 ∈ (SubDRingβ€˜π‘…) β†’ 𝑠 ∈ (SubRingβ€˜π‘…))
54ssriv 3987 . . . 4 (SubDRingβ€˜π‘…) βŠ† (SubRingβ€˜π‘…)
65a1i 11 . . 3 (𝑅 ∈ DivRing β†’ (SubDRingβ€˜π‘…) βŠ† (SubRingβ€˜π‘…))
7 eqid 2733 . . . . 5 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
87sdrgid 20408 . . . 4 (𝑅 ∈ DivRing β†’ (Baseβ€˜π‘…) ∈ (SubDRingβ€˜π‘…))
98ne0d 4336 . . 3 (𝑅 ∈ DivRing β†’ (SubDRingβ€˜π‘…) β‰  βˆ…)
103simp3bi 1148 . . . 4 (𝑠 ∈ (SubDRingβ€˜π‘…) β†’ (𝑅 β†Ύs 𝑠) ∈ DivRing)
1110adantl 483 . . 3 ((𝑅 ∈ DivRing ∧ 𝑠 ∈ (SubDRingβ€˜π‘…)) β†’ (𝑅 β†Ύs 𝑠) ∈ DivRing)
121, 2, 6, 9, 11subdrgint 20419 . 2 (𝑅 ∈ DivRing β†’ 𝑃 ∈ DivRing)
13 drngring 20364 . . . 4 (𝑃 ∈ DivRing β†’ 𝑃 ∈ Ring)
1412, 13syl 17 . . 3 (𝑅 ∈ DivRing β†’ 𝑃 ∈ Ring)
15 ssidd 4006 . . . . . . . . . . . . . 14 (𝑅 ∈ DivRing β†’ (Baseβ€˜π‘…) βŠ† (Baseβ€˜π‘…))
16 eqid 2733 . . . . . . . . . . . . . . 15 (mulGrpβ€˜π‘…) = (mulGrpβ€˜π‘…)
17 eqid 2733 . . . . . . . . . . . . . . 15 (Cntzβ€˜(mulGrpβ€˜π‘…)) = (Cntzβ€˜(mulGrpβ€˜π‘…))
187, 16, 17cntzsdrg 20418 . . . . . . . . . . . . . 14 ((𝑅 ∈ DivRing ∧ (Baseβ€˜π‘…) βŠ† (Baseβ€˜π‘…)) β†’ ((Cntzβ€˜(mulGrpβ€˜π‘…))β€˜(Baseβ€˜π‘…)) ∈ (SubDRingβ€˜π‘…))
192, 15, 18syl2anc 585 . . . . . . . . . . . . 13 (𝑅 ∈ DivRing β†’ ((Cntzβ€˜(mulGrpβ€˜π‘…))β€˜(Baseβ€˜π‘…)) ∈ (SubDRingβ€˜π‘…))
20 intss1 4968 . . . . . . . . . . . . 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 19993 . . . . . . . . . . . . 13 (Baseβ€˜π‘…) = (Baseβ€˜(mulGrpβ€˜π‘…))
2322, 17cntrval 19183 . . . . . . . . . . . 12 ((Cntzβ€˜(mulGrpβ€˜π‘…))β€˜(Baseβ€˜π‘…)) = (Cntrβ€˜(mulGrpβ€˜π‘…))
2421, 23sseqtrdi 4033 . . . . . . . . . . 11 (𝑅 ∈ DivRing β†’ ∩ (SubDRingβ€˜π‘…) βŠ† (Cntrβ€˜(mulGrpβ€˜π‘…)))
2522cntrss 19195 . . . . . . . . . . 11 (Cntrβ€˜(mulGrpβ€˜π‘…)) βŠ† (Baseβ€˜π‘…)
2624, 25sstrdi 3995 . . . . . . . . . 10 (𝑅 ∈ DivRing β†’ ∩ (SubDRingβ€˜π‘…) βŠ† (Baseβ€˜π‘…))
271, 7ressbas2 17182 . . . . . . . . . 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 4022 . . . . . . . 8 (𝑅 ∈ DivRing β†’ (Baseβ€˜π‘ƒ) βŠ† (Cntrβ€˜(mulGrpβ€˜π‘…)))
3029adantr 482 . . . . . . 7 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ (Baseβ€˜π‘ƒ) βŠ† (Cntrβ€˜(mulGrpβ€˜π‘…)))
31 simprl 770 . . . . . . 7 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ π‘₯ ∈ (Baseβ€˜π‘ƒ))
3230, 31sseldd 3984 . . . . . 6 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ π‘₯ ∈ (Cntrβ€˜(mulGrpβ€˜π‘…)))
3328, 26eqsstrrd 4022 . . . . . . . 8 (𝑅 ∈ DivRing β†’ (Baseβ€˜π‘ƒ) βŠ† (Baseβ€˜π‘…))
3433adantr 482 . . . . . . 7 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ (Baseβ€˜π‘ƒ) βŠ† (Baseβ€˜π‘…))
35 simprr 772 . . . . . . 7 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ 𝑦 ∈ (Baseβ€˜π‘ƒ))
3634, 35sseldd 3984 . . . . . 6 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ 𝑦 ∈ (Baseβ€˜π‘…))
37 eqid 2733 . . . . . . . 8 (.rβ€˜π‘…) = (.rβ€˜π‘…)
3816, 37mgpplusg 19991 . . . . . . 7 (.rβ€˜π‘…) = (+gβ€˜(mulGrpβ€˜π‘…))
39 eqid 2733 . . . . . . 7 (Cntrβ€˜(mulGrpβ€˜π‘…)) = (Cntrβ€˜(mulGrpβ€˜π‘…))
4022, 38, 39cntri 19196 . . . . . 6 ((π‘₯ ∈ (Cntrβ€˜(mulGrpβ€˜π‘…)) ∧ 𝑦 ∈ (Baseβ€˜π‘…)) β†’ (π‘₯(.rβ€˜π‘…)𝑦) = (𝑦(.rβ€˜π‘…)π‘₯))
4132, 36, 40syl2anc 585 . . . . 5 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ (π‘₯(.rβ€˜π‘…)𝑦) = (𝑦(.rβ€˜π‘…)π‘₯))
428, 26ssexd 5325 . . . . . . 7 (𝑅 ∈ DivRing β†’ ∩ (SubDRingβ€˜π‘…) ∈ V)
431, 37ressmulr 17252 . . . . . . 7 (∩ (SubDRingβ€˜π‘…) ∈ V β†’ (.rβ€˜π‘…) = (.rβ€˜π‘ƒ))
4442, 43syl 17 . . . . . 6 (𝑅 ∈ DivRing β†’ (.rβ€˜π‘…) = (.rβ€˜π‘ƒ))
4544oveqdr 7437 . . . . 5 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ (π‘₯(.rβ€˜π‘…)𝑦) = (π‘₯(.rβ€˜π‘ƒ)𝑦))
4644oveqdr 7437 . . . . 5 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ (𝑦(.rβ€˜π‘…)π‘₯) = (𝑦(.rβ€˜π‘ƒ)π‘₯))
4741, 45, 463eqtr3d 2781 . . . 4 ((𝑅 ∈ DivRing ∧ (π‘₯ ∈ (Baseβ€˜π‘ƒ) ∧ 𝑦 ∈ (Baseβ€˜π‘ƒ))) β†’ (π‘₯(.rβ€˜π‘ƒ)𝑦) = (𝑦(.rβ€˜π‘ƒ)π‘₯))
4847ralrimivva 3201 . . 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 2733 . . . 4 (Baseβ€˜π‘ƒ) = (Baseβ€˜π‘ƒ)
50 eqid 2733 . . . 4 (.rβ€˜π‘ƒ) = (.rβ€˜π‘ƒ)
5149, 50iscrng2 20075 . . 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 584 . 2 (𝑅 ∈ DivRing β†’ 𝑃 ∈ CRing)
53 isfld 20368 . 2 (𝑃 ∈ Field ↔ (𝑃 ∈ DivRing ∧ 𝑃 ∈ CRing))
5412, 52, 53sylanbrc 584 1 (𝑅 ∈ DivRing β†’ 𝑃 ∈ Field)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475   βŠ† wss 3949  βˆ© cint 4951  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144   β†Ύs cress 17173  .rcmulr 17198  Cntzccntz 19179  Cntrccntr 19180  mulGrpcmgp 19987  Ringcrg 20056  CRingccrg 20057  SubRingcsubrg 20315  DivRingcdr 20357  Fieldcfield 20358  SubDRingcsdrg 20402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-grp 18822  df-minusg 18823  df-subg 19003  df-cntz 19181  df-cntr 19182  df-cmn 19650  df-mgp 19988  df-ur 20005  df-ring 20058  df-cring 20059  df-oppr 20150  df-dvdsr 20171  df-unit 20172  df-invr 20202  df-dvr 20215  df-subrg 20317  df-drng 20359  df-field 20360  df-sdrg 20403
This theorem is referenced by: (None)
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