Theorem rgspncl 37739

Description: The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.)

Hypotheses

Ref	Expression
rgspnval.r	|- ( ph -> R e. Ring )
rgspnval.b	|- ( ph -> B = ( Base ` R ) )
rgspnval.ss	|- ( ph -> A C_ B )
rgspnval.n	|- ( ph -> N = (RingSpan ` R ) )
rgspnval.sp	|- ( ph -> U = ( N ` A ) )

Assertion

Ref	Expression
rgspncl	|- ( ph -> U e. (SubRing ` R ) )

Proof of Theorem rgspncl

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rgspnval.r	. . 3 |- ( ph -> R e. Ring )
2		rgspnval.b	. . 3 |- ( ph -> B = ( Base ` R ) )
3		rgspnval.ss	. . 3 |- ( ph -> A C_ B )
4		rgspnval.n	. . 3 |- ( ph -> N = (RingSpan ` R ) )
5		rgspnval.sp	. . 3 |- ( ph -> U = ( N ` A ) )
6	1, 2, 3, 4, 5	rgspnval 37738	. 2 |- ( ph -> U = |^| { t e. (SubRing ` R ) | A C_ t } )
7		ssrab2 3687	. . 3 |- { t e. (SubRing ` R ) | A C_ t } C_ (SubRing ` R )
8		eqid 2622	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
9	8	subrgid 18782	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) e. (SubRing ` R ) )
10	1, 9	syl 17	. . . . . 6 |- ( ph -> ( Base ` R ) e. (SubRing ` R ) )
11	2, 10	eqeltrd 2701	. . . . 5 |- ( ph -> B e. (SubRing ` R ) )
12		sseq2 3627	. . . . . 6 |- ( t = B -> ( A C_ t <-> A C_ B ) )
13	12	rspcev 3309	. . . . 5 |- ( ( B e. (SubRing ` R ) /\ A C_ B ) -> E. t e. (SubRing ` R ) A C_ t )
14	11, 3, 13	syl2anc 693	. . . 4 |- ( ph -> E. t e. (SubRing ` R ) A C_ t )
15		rabn0 3958	. . . 4 |- ( { t e. (SubRing ` R ) | A C_ t } =/= (/) <-> E. t e. (SubRing ` R ) A C_ t )
16	14, 15	sylibr 224	. . 3 |- ( ph -> { t e. (SubRing ` R ) | A C_ t } =/= (/) )
17		subrgint 18802	. . 3 |- ( ( { t e. (SubRing ` R ) | A C_ t } C_ (SubRing ` R ) /\ { t e. (SubRing ` R ) | A C_ t } =/= (/) ) -> |^| { t e. (SubRing ` R ) | A C_ t } e. (SubRing ` R ) )
18	7, 16, 17	sylancr 695	. 2 |- ( ph -> |^| { t e. (SubRing ` R ) | A C_ t } e. (SubRing ` R ) )
19	6, 18	eqeltrd 2701	1 |- ( ph -> U e. (SubRing ` R ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   |^|cint 4475   ` cfv 5888   Basecbs 15857   Ringcrg 18547  SubRingcsubrg 18776  RingSpancrgspn 18777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-rgspn 18779

This theorem is referenced by:  rngunsnply  37743