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Theorem subrgintm 13302
Description: The intersection of an inhabited collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
Assertion
Ref Expression
subrgintm ((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ ∩ 𝑆 ∈ (SubRingβ€˜π‘…))
Distinct variable groups:   𝑀,𝑅   𝑀,𝑆

Proof of Theorem subrgintm
Dummy variables π‘₯ π‘Ÿ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgsubg 13286 . . . . 5 (π‘Ÿ ∈ (SubRingβ€˜π‘…) β†’ π‘Ÿ ∈ (SubGrpβ€˜π‘…))
21ssriv 3159 . . . 4 (SubRingβ€˜π‘…) βŠ† (SubGrpβ€˜π‘…)
3 sstr 3163 . . . 4 ((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ (SubRingβ€˜π‘…) βŠ† (SubGrpβ€˜π‘…)) β†’ 𝑆 βŠ† (SubGrpβ€˜π‘…))
42, 3mpan2 425 . . 3 (𝑆 βŠ† (SubRingβ€˜π‘…) β†’ 𝑆 βŠ† (SubGrpβ€˜π‘…))
5 subgintm 12989 . . 3 ((𝑆 βŠ† (SubGrpβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ ∩ 𝑆 ∈ (SubGrpβ€˜π‘…))
64, 5sylan 283 . 2 ((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ ∩ 𝑆 ∈ (SubGrpβ€˜π‘…))
7 ssel2 3150 . . . . . 6 ((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ π‘Ÿ ∈ 𝑆) β†’ π‘Ÿ ∈ (SubRingβ€˜π‘…))
87adantlr 477 . . . . 5 (((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘Ÿ ∈ 𝑆) β†’ π‘Ÿ ∈ (SubRingβ€˜π‘…))
9 eqid 2177 . . . . . 6 (1rβ€˜π‘…) = (1rβ€˜π‘…)
109subrg1cl 13288 . . . . 5 (π‘Ÿ ∈ (SubRingβ€˜π‘…) β†’ (1rβ€˜π‘…) ∈ π‘Ÿ)
118, 10syl 14 . . . 4 (((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘Ÿ ∈ 𝑆) β†’ (1rβ€˜π‘…) ∈ π‘Ÿ)
1211ralrimiva 2550 . . 3 ((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ βˆ€π‘Ÿ ∈ 𝑆 (1rβ€˜π‘…) ∈ π‘Ÿ)
13 ssel 3149 . . . . . . 7 (𝑆 βŠ† (SubRingβ€˜π‘…) β†’ (𝑀 ∈ 𝑆 β†’ 𝑀 ∈ (SubRingβ€˜π‘…)))
14 subrgrcl 13285 . . . . . . 7 (𝑀 ∈ (SubRingβ€˜π‘…) β†’ 𝑅 ∈ Ring)
1513, 14syl6 33 . . . . . 6 (𝑆 βŠ† (SubRingβ€˜π‘…) β†’ (𝑀 ∈ 𝑆 β†’ 𝑅 ∈ Ring))
1615exlimdv 1819 . . . . 5 (𝑆 βŠ† (SubRingβ€˜π‘…) β†’ (βˆƒπ‘€ 𝑀 ∈ 𝑆 β†’ 𝑅 ∈ Ring))
1716imp 124 . . . 4 ((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ 𝑅 ∈ Ring)
18 ringsrg 13155 . . . 4 (𝑅 ∈ Ring β†’ 𝑅 ∈ SRing)
19 eqid 2177 . . . . . 6 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
2019, 9srgidcl 13090 . . . . 5 (𝑅 ∈ SRing β†’ (1rβ€˜π‘…) ∈ (Baseβ€˜π‘…))
21 elintg 3852 . . . . 5 ((1rβ€˜π‘…) ∈ (Baseβ€˜π‘…) β†’ ((1rβ€˜π‘…) ∈ ∩ 𝑆 ↔ βˆ€π‘Ÿ ∈ 𝑆 (1rβ€˜π‘…) ∈ π‘Ÿ))
2220, 21syl 14 . . . 4 (𝑅 ∈ SRing β†’ ((1rβ€˜π‘…) ∈ ∩ 𝑆 ↔ βˆ€π‘Ÿ ∈ 𝑆 (1rβ€˜π‘…) ∈ π‘Ÿ))
2317, 18, 223syl 17 . . 3 ((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ ((1rβ€˜π‘…) ∈ ∩ 𝑆 ↔ βˆ€π‘Ÿ ∈ 𝑆 (1rβ€˜π‘…) ∈ π‘Ÿ))
2412, 23mpbird 167 . 2 ((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ (1rβ€˜π‘…) ∈ ∩ 𝑆)
258adantlr 477 . . . . . 6 ((((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ π‘Ÿ ∈ 𝑆) β†’ π‘Ÿ ∈ (SubRingβ€˜π‘…))
26 simprl 529 . . . . . . 7 (((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ π‘₯ ∈ ∩ 𝑆)
27 elinti 3853 . . . . . . . 8 (π‘₯ ∈ ∩ 𝑆 β†’ (π‘Ÿ ∈ 𝑆 β†’ π‘₯ ∈ π‘Ÿ))
2827imp 124 . . . . . . 7 ((π‘₯ ∈ ∩ 𝑆 ∧ π‘Ÿ ∈ 𝑆) β†’ π‘₯ ∈ π‘Ÿ)
2926, 28sylan 283 . . . . . 6 ((((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ π‘Ÿ ∈ 𝑆) β†’ π‘₯ ∈ π‘Ÿ)
30 simprr 531 . . . . . . 7 (((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ 𝑦 ∈ ∩ 𝑆)
31 elinti 3853 . . . . . . . 8 (𝑦 ∈ ∩ 𝑆 β†’ (π‘Ÿ ∈ 𝑆 β†’ 𝑦 ∈ π‘Ÿ))
3231imp 124 . . . . . . 7 ((𝑦 ∈ ∩ 𝑆 ∧ π‘Ÿ ∈ 𝑆) β†’ 𝑦 ∈ π‘Ÿ)
3330, 32sylan 283 . . . . . 6 ((((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ π‘Ÿ ∈ 𝑆) β†’ 𝑦 ∈ π‘Ÿ)
34 eqid 2177 . . . . . . 7 (.rβ€˜π‘…) = (.rβ€˜π‘…)
3534subrgmcl 13292 . . . . . 6 ((π‘Ÿ ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ π‘Ÿ ∧ 𝑦 ∈ π‘Ÿ) β†’ (π‘₯(.rβ€˜π‘…)𝑦) ∈ π‘Ÿ)
3625, 29, 33, 35syl3anc 1238 . . . . 5 ((((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ π‘Ÿ ∈ 𝑆) β†’ (π‘₯(.rβ€˜π‘…)𝑦) ∈ π‘Ÿ)
3736ralrimiva 2550 . . . 4 (((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ βˆ€π‘Ÿ ∈ 𝑆 (π‘₯(.rβ€˜π‘…)𝑦) ∈ π‘Ÿ)
38 simplr 528 . . . . . 6 (((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ βˆƒπ‘€ 𝑀 ∈ 𝑆)
39 eleq1w 2238 . . . . . . . 8 (π‘Ÿ = 𝑀 β†’ (π‘Ÿ ∈ 𝑆 ↔ 𝑀 ∈ 𝑆))
4039cbvexv 1918 . . . . . . 7 (βˆƒπ‘Ÿ π‘Ÿ ∈ 𝑆 ↔ βˆƒπ‘€ 𝑀 ∈ 𝑆)
4136elexd 2750 . . . . . . . . 9 ((((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ π‘Ÿ ∈ 𝑆) β†’ (π‘₯(.rβ€˜π‘…)𝑦) ∈ V)
4241ex 115 . . . . . . . 8 (((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ (π‘Ÿ ∈ 𝑆 β†’ (π‘₯(.rβ€˜π‘…)𝑦) ∈ V))
4342exlimdv 1819 . . . . . . 7 (((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ (βˆƒπ‘Ÿ π‘Ÿ ∈ 𝑆 β†’ (π‘₯(.rβ€˜π‘…)𝑦) ∈ V))
4440, 43biimtrrid 153 . . . . . 6 (((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ (βˆƒπ‘€ 𝑀 ∈ 𝑆 β†’ (π‘₯(.rβ€˜π‘…)𝑦) ∈ V))
4538, 44mpd 13 . . . . 5 (((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ (π‘₯(.rβ€˜π‘…)𝑦) ∈ V)
46 elintg 3852 . . . . 5 ((π‘₯(.rβ€˜π‘…)𝑦) ∈ V β†’ ((π‘₯(.rβ€˜π‘…)𝑦) ∈ ∩ 𝑆 ↔ βˆ€π‘Ÿ ∈ 𝑆 (π‘₯(.rβ€˜π‘…)𝑦) ∈ π‘Ÿ))
4745, 46syl 14 . . . 4 (((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ ((π‘₯(.rβ€˜π‘…)𝑦) ∈ ∩ 𝑆 ↔ βˆ€π‘Ÿ ∈ 𝑆 (π‘₯(.rβ€˜π‘…)𝑦) ∈ π‘Ÿ))
4837, 47mpbird 167 . . 3 (((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ (π‘₯(.rβ€˜π‘…)𝑦) ∈ ∩ 𝑆)
4948ralrimivva 2559 . 2 ((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ βˆ€π‘₯ ∈ ∩ π‘†βˆ€π‘¦ ∈ ∩ 𝑆(π‘₯(.rβ€˜π‘…)𝑦) ∈ ∩ 𝑆)
5019, 9, 34issubrg2 13300 . . 3 (𝑅 ∈ Ring β†’ (∩ 𝑆 ∈ (SubRingβ€˜π‘…) ↔ (∩ 𝑆 ∈ (SubGrpβ€˜π‘…) ∧ (1rβ€˜π‘…) ∈ ∩ 𝑆 ∧ βˆ€π‘₯ ∈ ∩ π‘†βˆ€π‘¦ ∈ ∩ 𝑆(π‘₯(.rβ€˜π‘…)𝑦) ∈ ∩ 𝑆)))
5117, 50syl 14 . 2 ((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ (∩ 𝑆 ∈ (SubRingβ€˜π‘…) ↔ (∩ 𝑆 ∈ (SubGrpβ€˜π‘…) ∧ (1rβ€˜π‘…) ∈ ∩ 𝑆 ∧ βˆ€π‘₯ ∈ ∩ π‘†βˆ€π‘¦ ∈ ∩ 𝑆(π‘₯(.rβ€˜π‘…)𝑦) ∈ ∩ 𝑆)))
526, 24, 49, 51mpbir3and 1180 1 ((𝑆 βŠ† (SubRingβ€˜π‘…) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ ∩ 𝑆 ∈ (SubRingβ€˜π‘…))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978  βˆƒwex 1492   ∈ wcel 2148  βˆ€wral 2455  Vcvv 2737   βŠ† wss 3129  βˆ© cint 3844  β€˜cfv 5215  (class class class)co 5872  Basecbs 12454  .rcmulr 12529  SubGrpcsubg 12958  1rcur 13073  SRingcsrg 13077  Ringcrg 13110  SubRingcsubrg 13276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-subg 12961  df-cmn 13021  df-abl 13022  df-mgp 13062  df-ur 13074  df-srg 13078  df-ring 13112  df-subrg 13278
This theorem is referenced by:  subrgin  13303
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