Theorem rspsn 14610

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 2233	. . . . 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 2534	. . 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 14540	. . . . 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 14531	. . . . . . . 8 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
8	6, 7	eqtrid 2276	. . . . . . 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 2310	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> G e. ( Base ` (ringLMod ` R ) ) )
11		eqid 2231	. . . . . 6 |- (Scalar ` (ringLMod ` R ) ) = (Scalar ` (ringLMod ` R ) )
12		eqid 2231	. . . . . 6 |- ( Base ` (Scalar ` (ringLMod ` R ) ) ) = ( Base ` (Scalar ` (ringLMod ` R ) ) )
13		eqid 2231	. . . . . 6 |- ( Base ` (ringLMod ` R ) ) = ( Base ` (ringLMod ` R ) )
14		eqid 2231	. . . . . 6 |- ( .s ` (ringLMod ` R ) ) = ( .s ` (ringLMod ` R ) )
15		eqid 2231	. . . . . 6 |- ( LSpan ` (ringLMod ` R ) ) = ( LSpan ` (ringLMod ` R ) )
16	11, 12, 13, 14, 15	ellspsn 14493	. . . . 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 599	. . . 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 14548	. . . . . . . 8 |- ( R e. Ring -> (RSpan ` R ) = ( LSpan ` (ringLMod ` R ) ) )
20	18, 19	eqtrid 2276	. . . . . . 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 5650	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> ( K ` { G } ) = ( ( LSpan ` (ringLMod ` R ) ) ` { G } ) )
23	22	eleq2d 2301	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> ( x e. ( K ` { G } ) <-> x e. ( ( LSpan ` (ringLMod ` R ) ) ` { G } ) ) )
24		rlmscabas 14536	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
25	6, 24	eqtrid 2276	. . . . . 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 14537	. . . . . . . 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 6045	. . . . . 6 |- ( ( R e. Ring /\ G e. B ) -> ( a ( .r ` R ) G ) = ( a ( .s ` (ringLMod ` R ) ) G ) )
30	29	eqeq2d 2243	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> ( x = ( a ( .r ` R ) G ) <-> x = ( a ( .s ` (ringLMod ` R ) ) G ) ) )
31	26, 30	rexeqbidv 2748	. . . 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 14122	. . . . 5 |- ( R e. Ring -> R e. SRing )
37	36	adantr 276	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> R e. SRing )
38		eqid 2231	. . . . 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 14171	. . 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 2361	1 |- ( ( R e. Ring /\ G e. B ) -> ( K ` { G } ) = { x | G .|| x } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   {cab 2217   E.wrex 2512   {csn 3673   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   Basecbs 13143   .rcmulr 13222  Scalarcsca 13224   .scvsca 13225  SRingcsrg 14038   Ringcrg 14071   ||_rcdsr 14161   LModclmod 14363   LSpanclspn 14462  ringLModcrglmod 14510  RSpancrsp 14544

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8259  df-mnf 8260  df-ltxr 8262  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-ndx 13146  df-slot 13147  df-base 13149  df-sets 13150  df-iress 13151  df-plusg 13234  df-mulr 13235  df-sca 13237  df-vsca 13238  df-ip 13239  df-0g 13402  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-grp 13647  df-minusg 13648  df-sbg 13649  df-subg 13818  df-cmn 13934  df-abl 13935  df-mgp 13996  df-ur 14035  df-srg 14039  df-ring 14073  df-dvdsr 14164  df-subrg 14295  df-lmod 14365  df-lssm 14429  df-lsp 14463  df-sra 14511  df-rgmod 14512  df-rsp 14546

This theorem is referenced by:  zndvds  14725