Theorem lpival 19245

Description: Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)

Hypotheses

Ref	Expression
lpival.p	|- P = (LPIdeal ` R )
lpival.k	|- K = (RSpan ` R )
lpival.b	|- B = ( Base ` R )

Assertion

Ref	Expression
lpival	|- ( R e. Ring -> P = U_ g e. B { ( K ` { g } ) } )

Distinct variable groups:   R, g   P, g   B, g   g, K

Proof of Theorem lpival

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
2		fveq2 6191	. . . . . 6 |- ( r = R -> (RSpan ` r ) = (RSpan ` R ) )
3	2	fveq1d 6193	. . . . 5 |- ( r = R -> ( (RSpan ` r ) ` { g } ) = ( (RSpan ` R ) ` { g } ) )
4	3	sneqd 4189	. . . 4 |- ( r = R -> { ( (RSpan ` r ) ` { g } ) } = { ( (RSpan ` R ) ` { g } ) } )
5	1, 4	iuneq12d 4546	. . 3 |- ( r = R -> U_ g e. ( Base ` r ) { ( (RSpan ` r ) ` { g } ) } = U_ g e. ( Base ` R ) { ( (RSpan ` R ) ` { g } ) } )
6		df-lpidl 19243	. . 3 |- LPIdeal = ( r e. Ring |-> U_ g e. ( Base ` r ) { ( (RSpan ` r ) ` { g } ) } )
7		fvex 6201	. . . . . 6 |- (RSpan ` R ) e. _V
8	7	rnex 7100	. . . . 5 |- ran (RSpan ` R ) e. _V
9		p0ex 4853	. . . . 5 |- { (/) } e. _V
10	8, 9	unex 6956	. . . 4 |- ( ran (RSpan ` R ) u. { (/) } ) e. _V
11		iunss 4561	. . . . 5 |- ( U_ g e. ( Base ` R ) { ( (RSpan ` R ) ` { g } ) } C_ ( ran (RSpan ` R ) u. { (/) } ) <-> A. g e. ( Base ` R ) { ( (RSpan ` R ) ` { g } ) } C_ ( ran (RSpan ` R ) u. { (/) } ) )
12		fvrn0 6216	. . . . . . 7 |- ( (RSpan ` R ) ` { g } ) e. ( ran (RSpan ` R ) u. { (/) } )
13		snssi 4339	. . . . . . 7 |- ( ( (RSpan ` R ) ` { g } ) e. ( ran (RSpan ` R ) u. { (/) } ) -> { ( (RSpan ` R ) ` { g } ) } C_ ( ran (RSpan ` R ) u. { (/) } ) )
14	12, 13	ax-mp 5	. . . . . 6 |- { ( (RSpan ` R ) ` { g } ) } C_ ( ran (RSpan ` R ) u. { (/) } )
15	14	a1i 11	. . . . 5 |- ( g e. ( Base ` R ) -> { ( (RSpan ` R ) ` { g } ) } C_ ( ran (RSpan ` R ) u. { (/) } ) )
16	11, 15	mprgbir 2927	. . . 4 |- U_ g e. ( Base ` R ) { ( (RSpan ` R ) ` { g } ) } C_ ( ran (RSpan ` R ) u. { (/) } )
17	10, 16	ssexi 4803	. . 3 |- U_ g e. ( Base ` R ) { ( (RSpan ` R ) ` { g } ) } e. _V
18	5, 6, 17	fvmpt 6282	. 2 |- ( R e. Ring -> (LPIdeal ` R ) = U_ g e. ( Base ` R ) { ( (RSpan ` R ) ` { g } ) } )
19		lpival.p	. 2 |- P = (LPIdeal ` R )
20		lpival.b	. . . 4 |- B = ( Base ` R )
21		iuneq1 4534	. . . 4 |- ( B = ( Base ` R ) -> U_ g e. B { ( K ` { g } ) } = U_ g e. ( Base ` R ) { ( K ` { g } ) } )
22	20, 21	ax-mp 5	. . 3 |- U_ g e. B { ( K ` { g } ) } = U_ g e. ( Base ` R ) { ( K ` { g } ) }
23		lpival.k	. . . . . . 7 |- K = (RSpan ` R )
24	23	fveq1i 6192	. . . . . 6 |- ( K ` { g } ) = ( (RSpan ` R ) ` { g } )
25	24	sneqi 4188	. . . . 5 |- { ( K ` { g } ) } = { ( (RSpan ` R ) ` { g } ) }
26	25	a1i 11	. . . 4 |- ( g e. ( Base ` R ) -> { ( K ` { g } ) } = { ( (RSpan ` R ) ` { g } ) } )
27	26	iuneq2i 4539	. . 3 |- U_ g e. ( Base ` R ) { ( K ` { g } ) } = U_ g e. ( Base ` R ) { ( (RSpan ` R ) ` { g } ) }
28	22, 27	eqtri 2644	. 2 |- U_ g e. B { ( K ` { g } ) } = U_ g e. ( Base ` R ) { ( (RSpan ` R ) ` { g } ) }
29	18, 19, 28	3eqtr4g 2681	1 |- ( R e. Ring -> P = U_ g e. B { ( K ` { g } ) } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   ran crn 5115   ` cfv 5888   Basecbs 15857   Ringcrg 18547  RSpancrsp 19171  LPIdealclpidl 19241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-lpidl 19243

This theorem is referenced by:  islpidl  19246