Theorem rspsn 14030

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 2195	. . . . 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 2495	. . 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 13960	. . . . 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 13951	. . . . . . . 8 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
8	6, 7	eqtrid 2238	. . . . . . 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 2272	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> G e. ( Base ` (ringLMod ` R ) ) )
11		eqid 2193	. . . . . 6 |- (Scalar ` (ringLMod ` R ) ) = (Scalar ` (ringLMod ` R ) )
12		eqid 2193	. . . . . 6 |- ( Base ` (Scalar ` (ringLMod ` R ) ) ) = ( Base ` (Scalar ` (ringLMod ` R ) ) )
13		eqid 2193	. . . . . 6 |- ( Base ` (ringLMod ` R ) ) = ( Base ` (ringLMod ` R ) )
14		eqid 2193	. . . . . 6 |- ( .s ` (ringLMod ` R ) ) = ( .s ` (ringLMod ` R ) )
15		eqid 2193	. . . . . 6 |- ( LSpan ` (ringLMod ` R ) ) = ( LSpan ` (ringLMod ` R ) )
16	11, 12, 13, 14, 15	ellspsn 13913	. . . . 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 13968	. . . . . . . 8 |- ( R e. Ring -> (RSpan ` R ) = ( LSpan ` (ringLMod ` R ) ) )
20	18, 19	eqtrid 2238	. . . . . . 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 5556	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> ( K ` { G } ) = ( ( LSpan ` (ringLMod ` R ) ) ` { G } ) )
23	22	eleq2d 2263	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> ( x e. ( K ` { G } ) <-> x e. ( ( LSpan ` (ringLMod ` R ) ) ` { G } ) ) )
24		rlmscabas 13956	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
25	6, 24	eqtrid 2238	. . . . . 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 13957	. . . . . . . 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 5935	. . . . . 6 |- ( ( R e. Ring /\ G e. B ) -> ( a ( .r ` R ) G ) = ( a ( .s ` (ringLMod ` R ) ) G ) )
30	29	eqeq2d 2205	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> ( x = ( a ( .r ` R ) G ) <-> x = ( a ( .s ` (ringLMod ` R ) ) G ) ) )
31	26, 30	rexeqbidv 2707	. . . 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 13543	. . . . 5 |- ( R e. Ring -> R e. SRing )
37	36	adantr 276	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> R e. SRing )
38		eqid 2193	. . . . 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 13591	. . 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 2322	1 |- ( ( R e. Ring /\ G e. B ) -> ( K ` { G } ) = { x | G .|| x } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   {cab 2179   E.wrex 2473   {csn 3618   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   Basecbs 12618   .rcmulr 12696  Scalarcsca 12698   .scvsca 12699  SRingcsrg 13459   Ringcrg 13492   ||_rcdsr 13582   LModclmod 13783   LSpanclspn 13882  ringLModcrglmod 13930  RSpancrsp 13964

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-ip 12713  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-sbg 13077  df-subg 13240  df-cmn 13356  df-abl 13357  df-mgp 13417  df-ur 13456  df-srg 13460  df-ring 13494  df-dvdsr 13585  df-subrg 13715  df-lmod 13785  df-lssm 13849  df-lsp 13883  df-sra 13931  df-rgmod 13932  df-rsp 13966

This theorem is referenced by:  zndvds  14137