Theorem irredmul 18709

Description: If product of two elements is irreducible, then one of the elements must be a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
irredn0.i	|- I = (Irred ` R )
irredmul.b	|- B = ( Base ` R )
irredmul.u	|- U = (Unit ` R )
irredmul.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
irredmul	|- ( ( X e. B /\ Y e. B /\ ( X .x. Y ) e. I ) -> ( X e. U \/ Y e. U ) )

Proof of Theorem irredmul

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		irredmul.b	. . . . 5 |- B = ( Base ` R )
2		irredmul.u	. . . . 5 |- U = (Unit ` R )
3		irredn0.i	. . . . 5 |- I = (Irred ` R )
4		irredmul.t	. . . . 5 |- .x. = ( .r ` R )
5	1, 2, 3, 4	isirred2 18701	. . . 4 |- ( ( X .x. Y ) e. I <-> ( ( X .x. Y ) e. B /\ -. ( X .x. Y ) e. U /\ A. x e. B A. y e. B ( ( x .x. y ) = ( X .x. Y ) -> ( x e. U \/ y e. U ) ) ) )
6	5	simp3bi 1078	. . 3 |- ( ( X .x. Y ) e. I -> A. x e. B A. y e. B ( ( x .x. y ) = ( X .x. Y ) -> ( x e. U \/ y e. U ) ) )
7		eqid 2622	. . . 4 |- ( X .x. Y ) = ( X .x. Y )
8		oveq1 6657	. . . . . . 7 |- ( x = X -> ( x .x. y ) = ( X .x. y ) )
9	8	eqeq1d 2624	. . . . . 6 |- ( x = X -> ( ( x .x. y ) = ( X .x. Y ) <-> ( X .x. y ) = ( X .x. Y ) ) )
10		eleq1 2689	. . . . . . 7 |- ( x = X -> ( x e. U <-> X e. U ) )
11	10	orbi1d 739	. . . . . 6 |- ( x = X -> ( ( x e. U \/ y e. U ) <-> ( X e. U \/ y e. U ) ) )
12	9, 11	imbi12d 334	. . . . 5 |- ( x = X -> ( ( ( x .x. y ) = ( X .x. Y ) -> ( x e. U \/ y e. U ) ) <-> ( ( X .x. y ) = ( X .x. Y ) -> ( X e. U \/ y e. U ) ) ) )
13		oveq2 6658	. . . . . . 7 |- ( y = Y -> ( X .x. y ) = ( X .x. Y ) )
14	13	eqeq1d 2624	. . . . . 6 |- ( y = Y -> ( ( X .x. y ) = ( X .x. Y ) <-> ( X .x. Y ) = ( X .x. Y ) ) )
15		eleq1 2689	. . . . . . 7 |- ( y = Y -> ( y e. U <-> Y e. U ) )
16	15	orbi2d 738	. . . . . 6 |- ( y = Y -> ( ( X e. U \/ y e. U ) <-> ( X e. U \/ Y e. U ) ) )
17	14, 16	imbi12d 334	. . . . 5 |- ( y = Y -> ( ( ( X .x. y ) = ( X .x. Y ) -> ( X e. U \/ y e. U ) ) <-> ( ( X .x. Y ) = ( X .x. Y ) -> ( X e. U \/ Y e. U ) ) ) )
18	12, 17	rspc2v 3322	. . . 4 |- ( ( X e. B /\ Y e. B ) -> ( A. x e. B A. y e. B ( ( x .x. y ) = ( X .x. Y ) -> ( x e. U \/ y e. U ) ) -> ( ( X .x. Y ) = ( X .x. Y ) -> ( X e. U \/ Y e. U ) ) ) )
19	7, 18	mpii 46	. . 3 |- ( ( X e. B /\ Y e. B ) -> ( A. x e. B A. y e. B ( ( x .x. y ) = ( X .x. Y ) -> ( x e. U \/ y e. U ) ) -> ( X e. U \/ Y e. U ) ) )
20	6, 19	syl5 34	. 2 |- ( ( X e. B /\ Y e. B ) -> ( ( X .x. Y ) e. I -> ( X e. U \/ Y e. U ) ) )
21	20	3impia 1261	1 |- ( ( X e. B /\ Y e. B /\ ( X .x. Y ) e. I ) -> ( X e. U \/ Y e. U ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ` 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:  prmirredlem  19841