Theorem rspsn 14506

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 2231	. . . . 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 2531	. . 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 14436	. . . . 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 14427	. . . . . . . 8 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
8	6, 7	eqtrid 2274	. . . . . . 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 2308	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> G e. ( Base ` (ringLMod ` R ) ) )
11		eqid 2229	. . . . . 6 |- (Scalar ` (ringLMod ` R ) ) = (Scalar ` (ringLMod ` R ) )
12		eqid 2229	. . . . . 6 |- ( Base ` (Scalar ` (ringLMod ` R ) ) ) = ( Base ` (Scalar ` (ringLMod ` R ) ) )
13		eqid 2229	. . . . . 6 |- ( Base ` (ringLMod ` R ) ) = ( Base ` (ringLMod ` R ) )
14		eqid 2229	. . . . . 6 |- ( .s ` (ringLMod ` R ) ) = ( .s ` (ringLMod ` R ) )
15		eqid 2229	. . . . . 6 |- ( LSpan ` (ringLMod ` R ) ) = ( LSpan ` (ringLMod ` R ) )
16	11, 12, 13, 14, 15	ellspsn 14389	. . . . 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 597	. . . 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 14444	. . . . . . . 8 |- ( R e. Ring -> (RSpan ` R ) = ( LSpan ` (ringLMod ` R ) ) )
20	18, 19	eqtrid 2274	. . . . . . 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 5631	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> ( K ` { G } ) = ( ( LSpan ` (ringLMod ` R ) ) ` { G } ) )
23	22	eleq2d 2299	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> ( x e. ( K ` { G } ) <-> x e. ( ( LSpan ` (ringLMod ` R ) ) ` { G } ) ) )
24		rlmscabas 14432	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
25	6, 24	eqtrid 2274	. . . . . 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 14433	. . . . . . . 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 6024	. . . . . 6 |- ( ( R e. Ring /\ G e. B ) -> ( a ( .r ` R ) G ) = ( a ( .s ` (ringLMod ` R ) ) G ) )
30	29	eqeq2d 2241	. . . . 5 |- ( ( R e. Ring /\ G e. B ) -> ( x = ( a ( .r ` R ) G ) <-> x = ( a ( .s ` (ringLMod ` R ) ) G ) ) )
31	26, 30	rexeqbidv 2745	. . . 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 14018	. . . . 5 |- ( R e. Ring -> R e. SRing )
37	36	adantr 276	. . . 4 |- ( ( R e. Ring /\ G e. B ) -> R e. SRing )
38		eqid 2229	. . . . 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 14067	. . 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 2358	1 |- ( ( R e. Ring /\ G e. B ) -> ( K ` { G } ) = { x | G .|| x } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   E.wrex 2509   {csn 3666   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   Basecbs 13040   .rcmulr 13119  Scalarcsca 13121   .scvsca 13122  SRingcsrg 13934   Ringcrg 13967   ||_rcdsr 14057   LModclmod 14259   LSpanclspn 14358  ringLModcrglmod 14406  RSpancrsp 14440

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-pre-ltirr 8119  ax-pre-lttrn 8121  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pnf 8191  df-mnf 8192  df-ltxr 8194  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-iress 13048  df-plusg 13131  df-mulr 13132  df-sca 13134  df-vsca 13135  df-ip 13136  df-0g 13299  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-grp 13544  df-minusg 13545  df-sbg 13546  df-subg 13715  df-cmn 13831  df-abl 13832  df-mgp 13892  df-ur 13931  df-srg 13935  df-ring 13969  df-dvdsr 14060  df-subrg 14191  df-lmod 14261  df-lssm 14325  df-lsp 14359  df-sra 14407  df-rgmod 14408  df-rsp 14442

This theorem is referenced by:  zndvds  14621