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Theorem subrgint 17649
Description: The intersection of a nonempty collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
Assertion
Ref Expression
subrgint  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubRing `  R )
)

Proof of Theorem subrgint
Dummy variables  x  r  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgsubg 17633 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  r  e.  (SubGrp `  R ) )
21ssriv 3493 . . . 4  |-  (SubRing `  R
)  C_  (SubGrp `  R
)
3 sstr 3497 . . . 4  |-  ( ( S  C_  (SubRing `  R
)  /\  (SubRing `  R
)  C_  (SubGrp `  R
) )  ->  S  C_  (SubGrp `  R )
)
42, 3mpan2 669 . . 3  |-  ( S 
C_  (SubRing `  R )  ->  S  C_  (SubGrp `  R
) )
5 subgint 16427 . . 3  |-  ( ( S  C_  (SubGrp `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubGrp `  R )
)
64, 5sylan 469 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubGrp `  R )
)
7 ssel2 3484 . . . . . 6  |-  ( ( S  C_  (SubRing `  R
)  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
87adantlr 712 . . . . 5  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
9 eqid 2454 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
109subrg1cl 17635 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  r )
118, 10syl 16 . . . 4  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  r  e.  S )  ->  ( 1r `  R )  e.  r )
1211ralrimiva 2868 . . 3  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  A. r  e.  S  ( 1r `  R )  e.  r )
13 fvex 5858 . . . 4  |-  ( 1r
`  R )  e. 
_V
1413elint2 4278 . . 3  |-  ( ( 1r `  R )  e.  |^| S  <->  A. r  e.  S  ( 1r `  R )  e.  r )
1512, 14sylibr 212 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  ( 1r `  R )  e.  |^| S )
168adantlr 712 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
17 simprl 754 . . . . . . 7  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  x  e.  |^| S )
18 elinti 4280 . . . . . . . 8  |-  ( x  e.  |^| S  ->  (
r  e.  S  ->  x  e.  r )
)
1918imp 427 . . . . . . 7  |-  ( ( x  e.  |^| S  /\  r  e.  S
)  ->  x  e.  r )
2017, 19sylan 469 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  x  e.  r )
21 simprr 755 . . . . . . 7  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  y  e.  |^| S )
22 elinti 4280 . . . . . . . 8  |-  ( y  e.  |^| S  ->  (
r  e.  S  -> 
y  e.  r ) )
2322imp 427 . . . . . . 7  |-  ( ( y  e.  |^| S  /\  r  e.  S
)  ->  y  e.  r )
2421, 23sylan 469 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  y  e.  r )
25 eqid 2454 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
2625subrgmcl 17639 . . . . . 6  |-  ( ( r  e.  (SubRing `  R
)  /\  x  e.  r  /\  y  e.  r )  ->  ( x
( .r `  R
) y )  e.  r )
2716, 20, 24, 26syl3anc 1226 . . . . 5  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  (
x ( .r `  R ) y )  e.  r )
2827ralrimiva 2868 . . . 4  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  A. r  e.  S  ( x
( .r `  R
) y )  e.  r )
29 ovex 6298 . . . . 5  |-  ( x ( .r `  R
) y )  e. 
_V
3029elint2 4278 . . . 4  |-  ( ( x ( .r `  R ) y )  e.  |^| S  <->  A. r  e.  S  ( x
( .r `  R
) y )  e.  r )
3128, 30sylibr 212 . . 3  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  (
x ( .r `  R ) y )  e.  |^| S )
3231ralrimivva 2875 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  A. x  e.  |^| S A. y  e.  |^| S ( x ( .r `  R
) y )  e. 
|^| S )
33 ssn0 3817 . . 3  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  (SubRing `  R
)  =/=  (/) )
34 n0 3793 . . . 4  |-  ( (SubRing `  R )  =/=  (/)  <->  E. r 
r  e.  (SubRing `  R
) )
35 subrgrcl 17632 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  R  e.  Ring )
3635exlimiv 1727 . . . 4  |-  ( E. r  r  e.  (SubRing `  R )  ->  R  e.  Ring )
3734, 36sylbi 195 . . 3  |-  ( (SubRing `  R )  =/=  (/)  ->  R  e.  Ring )
38 eqid 2454 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
3938, 9, 25issubrg2 17647 . . 3  |-  ( R  e.  Ring  ->  ( |^| S  e.  (SubRing `  R
)  <->  ( |^| S  e.  (SubGrp `  R )  /\  ( 1r `  R
)  e.  |^| S  /\  A. x  e.  |^| S A. y  e.  |^| S ( x ( .r `  R ) y )  e.  |^| S ) ) )
4033, 37, 393syl 20 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  ( |^| S  e.  (SubRing `  R
)  <->  ( |^| S  e.  (SubGrp `  R )  /\  ( 1r `  R
)  e.  |^| S  /\  A. x  e.  |^| S A. y  e.  |^| S ( x ( .r `  R ) y )  e.  |^| S ) ) )
416, 15, 32, 40mpbir3and 1177 1  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubRing `  R )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971   E.wex 1617    e. wcel 1823    =/= wne 2649   A.wral 2804    C_ wss 3461   (/)c0 3783   |^|cint 4271   ` cfv 5570  (class class class)co 6270   Basecbs 14719   .rcmulr 14788  SubGrpcsubg 16397   1rcur 17351   Ringcrg 17396  SubRingcsubrg 17623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-grp 16259  df-minusg 16260  df-subg 16400  df-mgp 17340  df-ur 17352  df-ring 17398  df-subrg 17625
This theorem is referenced by:  subrgin  17650  subrgmre  17651  aspsubrg  18178  rgspncl  31362
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