Theorem rspsn 14554

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 2533	. . 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 14484	. . . . 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 14475	. . . . . . . 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 14437	. . . . 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 14492	. . . . . . . 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 5641	. . . . 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 14480	. . . . . . 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 14481	. . . . . . . 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 6035	. . . . . 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 2747	. . . 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 14066	. . . . 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 14115	. . 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 2360	1 |- ( ( R e. Ring /\ G e. B ) -> ( K ` { G } ) = { x | G .|| x } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   {cab 2217   E.wrex 2511   {csn 3669   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   Basecbs 13087   .rcmulr 13166  Scalarcsca 13168   .scvsca 13169  SRingcsrg 13982   Ringcrg 14015   ||_rcdsr 14105   LModclmod 14307   LSpanclspn 14406  ringLModcrglmod 14454  RSpancrsp 14488

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-ndx 13090  df-slot 13091  df-base 13093  df-sets 13094  df-iress 13095  df-plusg 13178  df-mulr 13179  df-sca 13181  df-vsca 13182  df-ip 13183  df-0g 13346  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-grp 13591  df-minusg 13592  df-sbg 13593  df-subg 13762  df-cmn 13878  df-abl 13879  df-mgp 13940  df-ur 13979  df-srg 13983  df-ring 14017  df-dvdsr 14108  df-subrg 14239  df-lmod 14309  df-lssm 14373  df-lsp 14407  df-sra 14455  df-rgmod 14456  df-rsp 14490

This theorem is referenced by:  zndvds  14669