Theorem rspsn 14366

Description: Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)

Hypotheses

Ref	Expression
rspsn.b	|- B = ( Base ` R )
rspsn.k	|- K = (RSpan ` R )
rspsn.d	|- .|| = ( ||r ` R )

Assertion

Ref	Expression
rspsn	|- ( ( R e. Ring /\ G e. B ) -> ( K ` { G } ) = { x | G .|| x } )

Distinct variable groups:   x, R   x, G   x, B   x, K   x, .||

Proof of Theorem rspsn

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqcom 2208	. . . . 5 |- ( x = ( a ( .r ` R ) G ) <-> ( a ( .r ` R ) G ) = x )
2	1	a1i 9	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> ( x = ( a ( .r ` R ) G ) <-> ( a ( .r ` R ) G ) = x ) )
3	2	rexbidv 2508	. . 3 |- ( ( R e. Ring /\ G e. B ) -> ( E. a e. B x = ( a ( .r ` R ) G ) <-> E. a e. B ( a ( .r ` R ) G ) = x ) )
4		rlmlmod 14296	. . . . 5 |- ( R e. Ring -> (ringLMod ` R ) e. LMod )
5		simpr 110	. . . . . 6 |- ( ( R e. Ring /\ G e. B ) -> G e. B )
6		rspsn.b	. . . . . . . 8 |- B = ( Base ` R )
7		rlmbasg 14287	. . . . . . . 8 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
8	6, 7	eqtrid 2251	. . . . . . 7 |- ( R e. Ring -> B = ( Base ` (ringLMod ` R ) ) )
9	8	adantr 276	. . . . . 6 |- ( ( R e. Ring /\ G e. B ) -> B = ( Base ` (ringLMod ` R ) ) )
10	5, 9	eleqtrd 2285	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> G e. ( Base ` (ringLMod ` R ) ) )
11		eqid 2206	. . . . . 6 |- (Scalar ` (ringLMod ` R ) ) = (Scalar ` (ringLMod ` R ) )
12		eqid 2206	. . . . . 6 |- ( Base ` (Scalar ` (ringLMod ` R ) ) ) = ( Base ` (Scalar ` (ringLMod ` R ) ) )
13		eqid 2206	. . . . . 6 |- ( Base ` (ringLMod ` R ) ) = ( Base ` (ringLMod ` R ) )
14		eqid 2206	. . . . . 6 |- ( .s ` (ringLMod ` R ) ) = ( .s ` (ringLMod ` R ) )
15		eqid 2206	. . . . . 6 |- ( LSpan ` (ringLMod ` R ) ) = ( LSpan ` (ringLMod ` R ) )
16	11, 12, 13, 14, 15	ellspsn 14249	. . . . 5 |- ( ( (ringLMod ` R ) e. LMod /\ G e. ( Base ` (ringLMod ` R ) ) ) -> ( x e. ( ( LSpan ` (ringLMod ` R ) ) ` { G } ) <-> E. a e. ( Base ` (Scalar ` (ringLMod ` R ) ) ) x = ( a ( .s ` (ringLMod ` R ) ) G ) ) )
17	4, 10, 16	syl2an2r 595	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> ( x e. ( ( LSpan ` (ringLMod ` R ) ) ` { G } ) <-> E. a e. ( Base ` (Scalar ` (ringLMod ` R ) ) ) x = ( a ( .s ` (ringLMod ` R ) ) G ) ) )
18		rspsn.k	. . . . . . . 8 |- K = (RSpan ` R )
19		rspvalg 14304	. . . . . . . 8 |- ( R e. Ring -> (RSpan ` R ) = ( LSpan ` (ringLMod ` R ) ) )
20	18, 19	eqtrid 2251	. . . . . . 7 |- ( R e. Ring -> K = ( LSpan ` (ringLMod ` R ) ) )
21	20	adantr 276	. . . . . 6 |- ( ( R e. Ring /\ G e. B ) -> K = ( LSpan ` (ringLMod ` R ) ) )
22	21	fveq1d 5590	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> ( K ` { G } ) = ( ( LSpan ` (ringLMod ` R ) ) ` { G } ) )
23	22	eleq2d 2276	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> ( x e. ( K ` { G } ) <-> x e. ( ( LSpan ` (ringLMod ` R ) ) ` { G } ) ) )
24		rlmscabas 14292	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
25	6, 24	eqtrid 2251	. . . . . 6 |- ( R e. Ring -> B = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
26	25	adantr 276	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> B = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
27		rlmvscag 14293	. . . . . . . 8 |- ( R e. Ring -> ( .r ` R ) = ( .s ` (ringLMod ` R ) ) )
28	27	adantr 276	. . . . . . 7 |- ( ( R e. Ring /\ G e. B ) -> ( .r ` R ) = ( .s ` (ringLMod ` R ) ) )
29	28	oveqd 5973	. . . . . 6 |- ( ( R e. Ring /\ G e. B ) -> ( a ( .r ` R ) G ) = ( a ( .s ` (ringLMod ` R ) ) G ) )
30	29	eqeq2d 2218	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> ( x = ( a ( .r ` R ) G ) <-> x = ( a ( .s ` (ringLMod ` R ) ) G ) ) )
31	26, 30	rexeqbidv 2720	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> ( E. a e. B x = ( a ( .r ` R ) G ) <-> E. a e. ( Base ` (Scalar ` (ringLMod ` R ) ) ) x = ( a ( .s ` (ringLMod ` R ) ) G ) ) )
32	17, 23, 31	3bitr4d 220	. . 3 |- ( ( R e. Ring /\ G e. B ) -> ( x e. ( K ` { G } ) <-> E. a e. B x = ( a ( .r ` R ) G ) ) )
33	6	a1i 9	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> B = ( Base ` R ) )
34		rspsn.d	. . . . 5 |- .|| = ( ||r ` R )
35	34	a1i 9	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> .|| = ( ||r ` R ) )
36		ringsrg 13879	. . . . 5 |- ( R e. Ring -> R e. SRing )
37	36	adantr 276	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> R e. SRing )
38		eqid 2206	. . . . 5 |- ( .r ` R ) = ( .r ` R )
39	38	a1i 9	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> ( .r ` R ) = ( .r ` R ) )
40	33, 35, 37, 39, 5	dvdsr2d 13927	. . 3 |- ( ( R e. Ring /\ G e. B ) -> ( G .|| x <-> E. a e. B ( a ( .r ` R ) G ) = x ) )
41	3, 32, 40	3bitr4d 220	. 2 |- ( ( R e. Ring /\ G e. B ) -> ( x e. ( K ` { G } ) <-> G .|| x ) )
42	41	eqabdv 2335	1 |- ( ( R e. Ring /\ G e. B ) -> ( K ` { G } ) = { x | G .|| x } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2177   {cab 2192   E.wrex 2486   {csn 3637   class class class wbr 4050   ` cfv 5279  (class class class)co 5956   Basecbs 12902   .rcmulr 12980  Scalarcsca 12982   .scvsca 12983  SRingcsrg 13795   Ringcrg 13828   ||_rcdsr 13918   LModclmod 14119   LSpanclspn 14218  ringLModcrglmod 14266  RSpancrsp 14300

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-cnex 8031  ax-resscn 8032  ax-1cn 8033  ax-1re 8034  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-addcom 8040  ax-addass 8042  ax-i2m1 8045  ax-0lt1 8046  ax-0id 8048  ax-rnegex 8049  ax-pre-ltirr 8052  ax-pre-lttrn 8054  ax-pre-ltadd 8056

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-id 4347  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-pnf 8124  df-mnf 8125  df-ltxr 8127  df-inn 9052  df-2 9110  df-3 9111  df-4 9112  df-5 9113  df-6 9114  df-7 9115  df-8 9116  df-ndx 12905  df-slot 12906  df-base 12908  df-sets 12909  df-iress 12910  df-plusg 12992  df-mulr 12993  df-sca 12995  df-vsca 12996  df-ip 12997  df-0g 13160  df-mgm 13258  df-sgrp 13304  df-mnd 13319  df-grp 13405  df-minusg 13406  df-sbg 13407  df-subg 13576  df-cmn 13692  df-abl 13693  df-mgp 13753  df-ur 13792  df-srg 13796  df-ring 13830  df-dvdsr 13921  df-subrg 14051  df-lmod 14121  df-lssm 14185  df-lsp 14219  df-sra 14267  df-rgmod 14268  df-rsp 14302

This theorem is referenced by:  zndvds  14481