Theorem isirred 18699

Description: An irreducible element of a ring is a non-unit that is not the product of two non-units. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
irred.1	|- B = ( Base ` R )
irred.2	|- U = (Unit ` R )
irred.3	|- I = (Irred ` R )
irred.4	|- N = ( B \ U )
irred.5	|- .x. = ( .r ` R )

Assertion

Ref	Expression
isirred	|- ( X e. I <-> ( X e. N /\ A. x e. N A. y e. N ( x .x. y ) =/= X ) )

Distinct variable groups:   x, y, N   x, R, y   x, X, y

Allowed substitution hints:   B( x, y)   .x. ( x, y)   U( x, y)   I( x, y)

Proof of Theorem isirred

Dummy variables r b z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6220	. . . 4 |- ( X e. (Irred ` R ) -> R e. dom Irred )
2		irred.3	. . . 4 |- I = (Irred ` R )
3	1, 2	eleq2s 2719	. . 3 |- ( X e. I -> R e. dom Irred )
4		elex 3212	. . 3 |- ( R e. dom Irred -> R e. _V )
5	3, 4	syl 17	. 2 |- ( X e. I -> R e. _V )
6		eldifi 3732	. . . . . 6 |- ( X e. ( B \ U ) -> X e. B )
7		irred.4	. . . . . 6 |- N = ( B \ U )
8	6, 7	eleq2s 2719	. . . . 5 |- ( X e. N -> X e. B )
9		irred.1	. . . . 5 |- B = ( Base ` R )
10	8, 9	syl6eleq 2711	. . . 4 |- ( X e. N -> X e. ( Base ` R ) )
11	10	elfvexd 6222	. . 3 |- ( X e. N -> R e. _V )
12	11	adantr 481	. 2 |- ( ( X e. N /\ A. x e. N A. y e. N ( x .x. y ) =/= X ) -> R e. _V )
13		fvex 6201	. . . . . . . 8 |- ( Base ` r ) e. _V
14		difexg 4808	. . . . . . . 8 |- ( ( Base ` r ) e. _V -> ( ( Base ` r ) \ (Unit ` r ) ) e. _V )
15	13, 14	mp1i 13	. . . . . . 7 |- ( r = R -> ( ( Base ` r ) \ (Unit ` r ) ) e. _V )
16		simpr 477	. . . . . . . . 9 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> b = ( ( Base ` r ) \ (Unit ` r ) ) )
17		simpl 473	. . . . . . . . . . . . 13 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> r = R )
18	17	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> ( Base ` r ) = ( Base ` R ) )
19	18, 9	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> ( Base ` r ) = B )
20	17	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> (Unit ` r ) = (Unit ` R ) )
21		irred.2	. . . . . . . . . . . 12 |- U = (Unit ` R )
22	20, 21	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> (Unit ` r ) = U )
23	19, 22	difeq12d 3729	. . . . . . . . . 10 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> ( ( Base ` r ) \ (Unit ` r ) ) = ( B \ U ) )
24	23, 7	syl6eqr 2674	. . . . . . . . 9 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> ( ( Base ` r ) \ (Unit ` r ) ) = N )
25	16, 24	eqtrd 2656	. . . . . . . 8 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> b = N )
26	17	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> ( .r ` r ) = ( .r ` R ) )
27		irred.5	. . . . . . . . . . . . 13 |- .x. = ( .r ` R )
28	26, 27	syl6eqr 2674	. . . . . . . . . . . 12 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> ( .r ` r ) = .x. )
29	28	oveqd 6667	. . . . . . . . . . 11 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> ( x ( .r ` r ) y ) = ( x .x. y ) )
30	29	neeq1d 2853	. . . . . . . . . 10 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> ( ( x ( .r ` r ) y ) =/= z <-> ( x .x. y ) =/= z ) )
31	25, 30	raleqbidv 3152	. . . . . . . . 9 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> ( A. y e. b ( x ( .r ` r ) y ) =/= z <-> A. y e. N ( x .x. y ) =/= z ) )
32	25, 31	raleqbidv 3152	. . . . . . . 8 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> ( A. x e. b A. y e. b ( x ( .r ` r ) y ) =/= z <-> A. x e. N A. y e. N ( x .x. y ) =/= z ) )
33	25, 32	rabeqbidv 3195	. . . . . . 7 |- ( ( r = R /\ b = ( ( Base ` r ) \ (Unit ` r ) ) ) -> { z e. b | A. x e. b A. y e. b ( x ( .r ` r ) y ) =/= z } = { z e. N | A. x e. N A. y e. N ( x .x. y ) =/= z } )
34	15, 33	csbied 3560	. . . . . 6 |- ( r = R -> [_ ( ( Base ` r ) \ (Unit ` r ) ) / b ]_ { z e. b | A. x e. b A. y e. b ( x ( .r ` r ) y ) =/= z } = { z e. N | A. x e. N A. y e. N ( x .x. y ) =/= z } )
35		df-irred 18643	. . . . . 6 |- Irred = ( r e. _V |-> [_ ( ( Base ` r ) \ (Unit ` r ) ) / b ]_ { z e. b | A. x e. b A. y e. b ( x ( .r ` r ) y ) =/= z } )
36		fvex 6201	. . . . . . . . . 10 |- ( Base ` R ) e. _V
37	9, 36	eqeltri 2697	. . . . . . . . 9 |- B e. _V
38		difexg 4808	. . . . . . . . 9 |- ( B e. _V -> ( B \ U ) e. _V )
39	37, 38	ax-mp 5	. . . . . . . 8 |- ( B \ U ) e. _V
40	7, 39	eqeltri 2697	. . . . . . 7 |- N e. _V
41	40	rabex 4813	. . . . . 6 |- { z e. N | A. x e. N A. y e. N ( x .x. y ) =/= z } e. _V
42	34, 35, 41	fvmpt 6282	. . . . 5 |- ( R e. _V -> (Irred ` R ) = { z e. N | A. x e. N A. y e. N ( x .x. y ) =/= z } )
43	2, 42	syl5eq 2668	. . . 4 |- ( R e. _V -> I = { z e. N | A. x e. N A. y e. N ( x .x. y ) =/= z } )
44	43	eleq2d 2687	. . 3 |- ( R e. _V -> ( X e. I <-> X e. { z e. N | A. x e. N A. y e. N ( x .x. y ) =/= z } ) )
45		neeq2 2857	. . . . 5 |- ( z = X -> ( ( x .x. y ) =/= z <-> ( x .x. y ) =/= X ) )
46	45	2ralbidv 2989	. . . 4 |- ( z = X -> ( A. x e. N A. y e. N ( x .x. y ) =/= z <-> A. x e. N A. y e. N ( x .x. y ) =/= X ) )
47	46	elrab 3363	. . 3 |- ( X e. { z e. N | A. x e. N A. y e. N ( x .x. y ) =/= z } <-> ( X e. N /\ A. x e. N A. y e. N ( x .x. y ) =/= X ) )
48	44, 47	syl6bb 276	. 2 |- ( R e. _V -> ( X e. I <-> ( X e. N /\ A. x e. N A. y e. N ( x .x. y ) =/= X ) ) )
49	5, 12, 48	pm5.21nii 368	1 |- ( X e. I <-> ( X e. N /\ A. x e. N A. y e. N ( x .x. y ) =/= X ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   [_csb 3533   \ cdif 3571   dom cdm 5114   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942  Unitcui 18639  Irredcir 18640

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

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-csb 3534  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-irred 18643

This theorem is referenced by:  isnirred  18700  isirred2  18701  opprirred  18702