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Theorem fldgensdrg 32675
Description: A generated subfield is a sub-division-ring. (Contributed by Thierry Arnoux, 11-Jan-2025.)
Hypotheses
Ref Expression
fldgenval.1 𝐡 = (Baseβ€˜πΉ)
fldgenval.2 (πœ‘ β†’ 𝐹 ∈ DivRing)
fldgenval.3 (πœ‘ β†’ 𝑆 βŠ† 𝐡)
Assertion
Ref Expression
fldgensdrg (πœ‘ β†’ (𝐹 fldGen 𝑆) ∈ (SubDRingβ€˜πΉ))
Proof of Theorem fldgensdrg
Dummy variables π‘Ž π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fldgenval.1 . . 3 𝐡 = (Baseβ€˜πΉ)
2 fldgenval.2 . . 3 (πœ‘ β†’ 𝐹 ∈ DivRing)
3 fldgenval.3 . . 3 (πœ‘ β†’ 𝑆 βŠ† 𝐡)
41, 2, 3fldgenval 32673 . 2 (πœ‘ β†’ (𝐹 fldGen 𝑆) = ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž})
52drngringd 20509 . . . 4 (πœ‘ β†’ 𝐹 ∈ Ring)
6 eqid 2731 . . . . . 6 (𝐹 β†Ύs ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž}) = (𝐹 β†Ύs ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž})
7 sseq2 4008 . . . . . . . . . . . . 13 (π‘Ž = π‘₯ β†’ (𝑆 βŠ† π‘Ž ↔ 𝑆 βŠ† π‘₯))
87elrab 3683 . . . . . . . . . . . 12 (π‘₯ ∈ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} ↔ (π‘₯ ∈ (SubDRingβ€˜πΉ) ∧ 𝑆 βŠ† π‘₯))
98biimpi 215 . . . . . . . . . . 11 (π‘₯ ∈ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} β†’ (π‘₯ ∈ (SubDRingβ€˜πΉ) ∧ 𝑆 βŠ† π‘₯))
109adantl 481 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž}) β†’ (π‘₯ ∈ (SubDRingβ€˜πΉ) ∧ 𝑆 βŠ† π‘₯))
1110simpld 494 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž}) β†’ π‘₯ ∈ (SubDRingβ€˜πΉ))
12 issdrg 20548 . . . . . . . . . 10 (π‘₯ ∈ (SubDRingβ€˜πΉ) ↔ (𝐹 ∈ DivRing ∧ π‘₯ ∈ (SubRingβ€˜πΉ) ∧ (𝐹 β†Ύs π‘₯) ∈ DivRing))
1312simp2bi 1145 . . . . . . . . 9 (π‘₯ ∈ (SubDRingβ€˜πΉ) β†’ π‘₯ ∈ (SubRingβ€˜πΉ))
1411, 13syl 17 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž}) β†’ π‘₯ ∈ (SubRingβ€˜πΉ))
1514ex 412 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} β†’ π‘₯ ∈ (SubRingβ€˜πΉ)))
1615ssrdv 3988 . . . . . 6 (πœ‘ β†’ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} βŠ† (SubRingβ€˜πΉ))
17 sseq2 4008 . . . . . . . 8 (π‘Ž = 𝐡 β†’ (𝑆 βŠ† π‘Ž ↔ 𝑆 βŠ† 𝐡))
181sdrgid 20552 . . . . . . . . 9 (𝐹 ∈ DivRing β†’ 𝐡 ∈ (SubDRingβ€˜πΉ))
192, 18syl 17 . . . . . . . 8 (πœ‘ β†’ 𝐡 ∈ (SubDRingβ€˜πΉ))
2017, 19, 3elrabd 3685 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž})
2120ne0d 4335 . . . . . 6 (πœ‘ β†’ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} β‰  βˆ…)
2212simp3bi 1146 . . . . . . 7 (π‘₯ ∈ (SubDRingβ€˜πΉ) β†’ (𝐹 β†Ύs π‘₯) ∈ DivRing)
2311, 22syl 17 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž}) β†’ (𝐹 β†Ύs π‘₯) ∈ DivRing)
246, 2, 16, 21, 23subdrgint 20563 . . . . 5 (πœ‘ β†’ (𝐹 β†Ύs ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž}) ∈ DivRing)
2524drngringd 20509 . . . 4 (πœ‘ β†’ (𝐹 β†Ύs ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž}) ∈ Ring)
26 intss1 4967 . . . . 5 (𝐡 ∈ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} β†’ ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} βŠ† 𝐡)
2720, 26syl 17 . . . 4 (πœ‘ β†’ ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} βŠ† 𝐡)
28 issdrg 20548 . . . . . . . . . 10 (π‘Ž ∈ (SubDRingβ€˜πΉ) ↔ (𝐹 ∈ DivRing ∧ π‘Ž ∈ (SubRingβ€˜πΉ) ∧ (𝐹 β†Ύs π‘Ž) ∈ DivRing))
2928simp2bi 1145 . . . . . . . . 9 (π‘Ž ∈ (SubDRingβ€˜πΉ) β†’ π‘Ž ∈ (SubRingβ€˜πΉ))
30 eqid 2731 . . . . . . . . . 10 (1rβ€˜πΉ) = (1rβ€˜πΉ)
3130subrg1cl 20471 . . . . . . . . 9 (π‘Ž ∈ (SubRingβ€˜πΉ) β†’ (1rβ€˜πΉ) ∈ π‘Ž)
3229, 31syl 17 . . . . . . . 8 (π‘Ž ∈ (SubDRingβ€˜πΉ) β†’ (1rβ€˜πΉ) ∈ π‘Ž)
3332ad2antlr 724 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (SubDRingβ€˜πΉ)) ∧ 𝑆 βŠ† π‘Ž) β†’ (1rβ€˜πΉ) ∈ π‘Ž)
3433ex 412 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (SubDRingβ€˜πΉ)) β†’ (𝑆 βŠ† π‘Ž β†’ (1rβ€˜πΉ) ∈ π‘Ž))
3534ralrimiva 3145 . . . . 5 (πœ‘ β†’ βˆ€π‘Ž ∈ (SubDRingβ€˜πΉ)(𝑆 βŠ† π‘Ž β†’ (1rβ€˜πΉ) ∈ π‘Ž))
36 fvex 6904 . . . . . 6 (1rβ€˜πΉ) ∈ V
3736elintrab 4964 . . . . 5 ((1rβ€˜πΉ) ∈ ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} ↔ βˆ€π‘Ž ∈ (SubDRingβ€˜πΉ)(𝑆 βŠ† π‘Ž β†’ (1rβ€˜πΉ) ∈ π‘Ž))
3835, 37sylibr 233 . . . 4 (πœ‘ β†’ (1rβ€˜πΉ) ∈ ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž})
391, 30issubrg 20462 . . . . 5 (∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} ∈ (SubRingβ€˜πΉ) ↔ ((𝐹 ∈ Ring ∧ (𝐹 β†Ύs ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž}) ∈ Ring) ∧ (∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} βŠ† 𝐡 ∧ (1rβ€˜πΉ) ∈ ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž})))
4039biimpri 227 . . . 4 (((𝐹 ∈ Ring ∧ (𝐹 β†Ύs ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž}) ∈ Ring) ∧ (∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} βŠ† 𝐡 ∧ (1rβ€˜πΉ) ∈ ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž})) β†’ ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} ∈ (SubRingβ€˜πΉ))
415, 25, 27, 38, 40syl22anc 836 . . 3 (πœ‘ β†’ ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} ∈ (SubRingβ€˜πΉ))
42 issdrg 20548 . . 3 (∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} ∈ (SubDRingβ€˜πΉ) ↔ (𝐹 ∈ DivRing ∧ ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} ∈ (SubRingβ€˜πΉ) ∧ (𝐹 β†Ύs ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž}) ∈ DivRing))
432, 41, 24, 42syl3anbrc 1342 . 2 (πœ‘ β†’ ∩ {π‘Ž ∈ (SubDRingβ€˜πΉ) ∣ 𝑆 βŠ† π‘Ž} ∈ (SubDRingβ€˜πΉ))
444, 43eqeltrd 2832 1 (πœ‘ β†’ (𝐹 fldGen 𝑆) ∈ (SubDRingβ€˜πΉ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  {crab 3431   βŠ† wss 3948  βˆ© cint 4950  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149   β†Ύs cress 17178  1rcur 20076  Ringcrg 20128  SubRingcsubrg 20458  DivRingcdr 20501  SubDRingcsdrg 20546   fldGen cfldgen 32671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-tpos 8215  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-0g 17392  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-grp 18859  df-minusg 18860  df-subg 19040  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-ring 20130  df-oppr 20226  df-dvdsr 20249  df-unit 20250  df-invr 20280  df-dvr 20293  df-subrng 20435  df-subrg 20460  df-drng 20503  df-sdrg 20547  df-fldgen 32672
This theorem is referenced by:  fldgenfld  32681  1fldgenq  32683  algextdeglem2  33064  algextdeglem4  33066  algextdeglem5  33067
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