Theorem pridlval 33832

Description: The class of prime ideals of a ring R. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
pridlval.1	|- G = ( 1st ` R )
pridlval.2	|- H = ( 2nd ` R )
pridlval.3	|- X = ran G

Assertion

Ref	Expression
pridlval	|- ( R e. RingOps -> ( PrIdl ` R ) = { i e. ( Idl ` R ) | ( i =/= X /\ A. a e. ( Idl ` R ) A. b e. ( Idl ` R ) ( A. x e. a A. y e. b ( x H y ) e. i -> ( a C_ i \/ b C_ i ) ) ) } )

Distinct variable groups:   R, i, x, y, a, b   i, X   i, H

Allowed substitution hints:   G( x, y, i, a, b)   H( x, y, a, b)   X( x, y, a, b)

Proof of Theorem pridlval

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( r = R -> ( Idl ` r ) = ( Idl ` R ) )
2		fveq2 6191	. . . . . . . 8 |- ( r = R -> ( 1st ` r ) = ( 1st ` R ) )
3		pridlval.1	. . . . . . . 8 |- G = ( 1st ` R )
4	2, 3	syl6eqr 2674	. . . . . . 7 |- ( r = R -> ( 1st ` r ) = G )
5	4	rneqd 5353	. . . . . 6 |- ( r = R -> ran ( 1st ` r ) = ran G )
6		pridlval.3	. . . . . 6 |- X = ran G
7	5, 6	syl6eqr 2674	. . . . 5 |- ( r = R -> ran ( 1st ` r ) = X )
8	7	neeq2d 2854	. . . 4 |- ( r = R -> ( i =/= ran ( 1st ` r ) <-> i =/= X ) )
9		fveq2 6191	. . . . . . . . . . 11 |- ( r = R -> ( 2nd ` r ) = ( 2nd ` R ) )
10		pridlval.2	. . . . . . . . . . 11 |- H = ( 2nd ` R )
11	9, 10	syl6eqr 2674	. . . . . . . . . 10 |- ( r = R -> ( 2nd ` r ) = H )
12	11	oveqd 6667	. . . . . . . . 9 |- ( r = R -> ( x ( 2nd ` r ) y ) = ( x H y ) )
13	12	eleq1d 2686	. . . . . . . 8 |- ( r = R -> ( ( x ( 2nd ` r ) y ) e. i <-> ( x H y ) e. i ) )
14	13	2ralbidv 2989	. . . . . . 7 |- ( r = R -> ( A. x e. a A. y e. b ( x ( 2nd ` r ) y ) e. i <-> A. x e. a A. y e. b ( x H y ) e. i ) )
15	14	imbi1d 331	. . . . . 6 |- ( r = R -> ( ( A. x e. a A. y e. b ( x ( 2nd ` r ) y ) e. i -> ( a C_ i \/ b C_ i ) ) <-> ( A. x e. a A. y e. b ( x H y ) e. i -> ( a C_ i \/ b C_ i ) ) ) )
16	1, 15	raleqbidv 3152	. . . . 5 |- ( r = R -> ( A. b e. ( Idl ` r ) ( A. x e. a A. y e. b ( x ( 2nd ` r ) y ) e. i -> ( a C_ i \/ b C_ i ) ) <-> A. b e. ( Idl ` R ) ( A. x e. a A. y e. b ( x H y ) e. i -> ( a C_ i \/ b C_ i ) ) ) )
17	1, 16	raleqbidv 3152	. . . 4 |- ( r = R -> ( A. a e. ( Idl ` r ) A. b e. ( Idl ` r ) ( A. x e. a A. y e. b ( x ( 2nd ` r ) y ) e. i -> ( a C_ i \/ b C_ i ) ) <-> A. a e. ( Idl ` R ) A. b e. ( Idl ` R ) ( A. x e. a A. y e. b ( x H y ) e. i -> ( a C_ i \/ b C_ i ) ) ) )
18	8, 17	anbi12d 747	. . 3 |- ( r = R -> ( ( i =/= ran ( 1st ` r ) /\ A. a e. ( Idl ` r ) A. b e. ( Idl ` r ) ( A. x e. a A. y e. b ( x ( 2nd ` r ) y ) e. i -> ( a C_ i \/ b C_ i ) ) ) <-> ( i =/= X /\ A. a e. ( Idl ` R ) A. b e. ( Idl ` R ) ( A. x e. a A. y e. b ( x H y ) e. i -> ( a C_ i \/ b C_ i ) ) ) ) )
19	1, 18	rabeqbidv 3195	. 2 |- ( r = R -> { i e. ( Idl ` r ) | ( i =/= ran ( 1st ` r ) /\ A. a e. ( Idl ` r ) A. b e. ( Idl ` r ) ( A. x e. a A. y e. b ( x ( 2nd ` r ) y ) e. i -> ( a C_ i \/ b C_ i ) ) ) } = { i e. ( Idl ` R ) | ( i =/= X /\ A. a e. ( Idl ` R ) A. b e. ( Idl ` R ) ( A. x e. a A. y e. b ( x H y ) e. i -> ( a C_ i \/ b C_ i ) ) ) } )
20		df-pridl 33810	. 2 |- PrIdl = ( r e. RingOps |-> { i e. ( Idl ` r ) | ( i =/= ran ( 1st ` r ) /\ A. a e. ( Idl ` r ) A. b e. ( Idl ` r ) ( A. x e. a A. y e. b ( x ( 2nd ` r ) y ) e. i -> ( a C_ i \/ b C_ i ) ) ) } )
21		fvex 6201	. . 3 |- ( Idl ` R ) e. _V
22	21	rabex 4813	. 2 |- { i e. ( Idl ` R ) | ( i =/= X /\ A. a e. ( Idl ` R ) A. b e. ( Idl ` R ) ( A. x e. a A. y e. b ( x H y ) e. i -> ( a C_ i \/ b C_ i ) ) ) } e. _V
23	19, 20, 22	fvmpt 6282	1 |- ( R e. RingOps -> ( PrIdl ` R ) = { i e. ( Idl ` R ) | ( i =/= X /\ A. a e. ( Idl ` R ) A. b e. ( Idl ` R ) ( A. x e. a A. y e. b ( x H y ) e. i -> ( a C_ i \/ b C_ i ) ) ) } )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   C_ wss 3574   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   RingOpscrngo 33693   Idlcidl 33806   PrIdlcpridl 33807

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-ov 6653  df-pridl 33810

This theorem is referenced by:  ispridl  33833