Theorem igenval 33860

Description: The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.)

Hypotheses

Ref	Expression
igenval.1	|- G = ( 1st ` R )
igenval.2	|- X = ran G

Assertion

Ref	Expression
igenval	|- ( ( R e. RingOps /\ S C_ X ) -> ( R IdlGen S ) = |^| { j e. ( Idl ` R ) | S C_ j } )

Distinct variable groups:   R, j   S, j   j, X

Allowed substitution hint:   G( j)

Proof of Theorem igenval

Dummy variables r s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		igenval.1	. . . . . 6 |- G = ( 1st ` R )
2		igenval.2	. . . . . 6 |- X = ran G
3	1, 2	rngoidl 33823	. . . . 5 |- ( R e. RingOps -> X e. ( Idl ` R ) )
4		sseq2 3627	. . . . . 6 |- ( j = X -> ( S C_ j <-> S C_ X ) )
5	4	rspcev 3309	. . . . 5 |- ( ( X e. ( Idl ` R ) /\ S C_ X ) -> E. j e. ( Idl ` R ) S C_ j )
6	3, 5	sylan 488	. . . 4 |- ( ( R e. RingOps /\ S C_ X ) -> E. j e. ( Idl ` R ) S C_ j )
7		rabn0 3958	. . . 4 |- ( { j e. ( Idl ` R ) | S C_ j } =/= (/) <-> E. j e. ( Idl ` R ) S C_ j )
8	6, 7	sylibr 224	. . 3 |- ( ( R e. RingOps /\ S C_ X ) -> { j e. ( Idl ` R ) | S C_ j } =/= (/) )
9		intex 4820	. . 3 |- ( { j e. ( Idl ` R ) | S C_ j } =/= (/) <-> |^| { j e. ( Idl ` R ) | S C_ j } e. _V )
10	8, 9	sylib 208	. 2 |- ( ( R e. RingOps /\ S C_ X ) -> |^| { j e. ( Idl ` R ) | S C_ j } e. _V )
11		fvex 6201	. . . . . . 7 |- ( 1st ` R ) e. _V
12	1, 11	eqeltri 2697	. . . . . 6 |- G e. _V
13	12	rnex 7100	. . . . 5 |- ran G e. _V
14	2, 13	eqeltri 2697	. . . 4 |- X e. _V
15	14	elpw2 4828	. . 3 |- ( S e. ~P X <-> S C_ X )
16		simpl 473	. . . . . . 7 |- ( ( r = R /\ s = S ) -> r = R )
17	16	fveq2d 6195	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( Idl ` r ) = ( Idl ` R ) )
18		sseq1 3626	. . . . . . 7 |- ( s = S -> ( s C_ j <-> S C_ j ) )
19	18	adantl 482	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( s C_ j <-> S C_ j ) )
20	17, 19	rabeqbidv 3195	. . . . 5 |- ( ( r = R /\ s = S ) -> { j e. ( Idl ` r ) | s C_ j } = { j e. ( Idl ` R ) | S C_ j } )
21	20	inteqd 4480	. . . 4 |- ( ( r = R /\ s = S ) -> |^| { j e. ( Idl ` r ) | s C_ j } = |^| { j e. ( Idl ` R ) | S C_ j } )
22		fveq2 6191	. . . . . . . 8 |- ( r = R -> ( 1st ` r ) = ( 1st ` R ) )
23	22, 1	syl6eqr 2674	. . . . . . 7 |- ( r = R -> ( 1st ` r ) = G )
24	23	rneqd 5353	. . . . . 6 |- ( r = R -> ran ( 1st ` r ) = ran G )
25	24, 2	syl6eqr 2674	. . . . 5 |- ( r = R -> ran ( 1st ` r ) = X )
26	25	pweqd 4163	. . . 4 |- ( r = R -> ~P ran ( 1st ` r ) = ~P X )
27		df-igen 33859	. . . 4 |- IdlGen = ( r e. RingOps , s e. ~P ran ( 1st ` r ) |-> |^| { j e. ( Idl ` r ) | s C_ j } )
28	21, 26, 27	ovmpt2x 6789	. . 3 |- ( ( R e. RingOps /\ S e. ~P X /\ |^| { j e. ( Idl ` R ) | S C_ j } e. _V ) -> ( R IdlGen S ) = |^| { j e. ( Idl ` R ) | S C_ j } )
29	15, 28	syl3an2br 1366	. 2 |- ( ( R e. RingOps /\ S C_ X /\ |^| { j e. ( Idl ` R ) | S C_ j } e. _V ) -> ( R IdlGen S ) = |^| { j e. ( Idl ` R ) | S C_ j } )
30	10, 29	mpd3an3 1425	1 |- ( ( R e. RingOps /\ S C_ X ) -> ( R IdlGen S ) = |^| { j e. ( Idl ` R ) | S C_ j } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   RingOpscrngo 33693   Idlcidl 33806   IdlGen cigen 33858

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-int 4476  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-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ablo 27399  df-rngo 33694  df-idl 33809  df-igen 33859

This theorem is referenced by:  igenss  33861  igenidl  33862  igenmin  33863  igenidl2  33864  igenval2  33865