Theorem islring 14069

Description: The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.)

Hypotheses

Ref	Expression
islring.b	|- B = ( Base ` R )
islring.a	|- .+ = ( +g ` R )
islring.1	|- .1. = ( 1r ` R )
islring.u	|- U = (Unit ` R )

Assertion

Ref	Expression
islring	|- ( R e. LRing <-> ( R e. NzRing /\ A. x e. B A. y e. B ( ( x .+ y ) = .1. -> ( x e. U \/ y e. U ) ) ) )

Distinct variable groups:   x, R, y   x, B, y

Allowed substitution hints:   .+ ( x, y)   U( x, y)   .1. ( x, y)

Proof of Theorem islring

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5599	. . . 4 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
2		islring.b	. . . 4 |- B = ( Base ` R )
3	1, 2	eqtr4di 2258	. . 3 |- ( r = R -> ( Base ` r ) = B )
4		fveq2 5599	. . . . . . . 8 |- ( r = R -> ( +g ` r ) = ( +g ` R ) )
5		islring.a	. . . . . . . 8 |- .+ = ( +g ` R )
6	4, 5	eqtr4di 2258	. . . . . . 7 |- ( r = R -> ( +g ` r ) = .+ )
7	6	oveqd 5984	. . . . . 6 |- ( r = R -> ( x ( +g ` r ) y ) = ( x .+ y ) )
8		fveq2 5599	. . . . . . 7 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
9		islring.1	. . . . . . 7 |- .1. = ( 1r ` R )
10	8, 9	eqtr4di 2258	. . . . . 6 |- ( r = R -> ( 1r ` r ) = .1. )
11	7, 10	eqeq12d 2222	. . . . 5 |- ( r = R -> ( ( x ( +g ` r ) y ) = ( 1r ` r ) <-> ( x .+ y ) = .1. ) )
12		fveq2 5599	. . . . . . . 8 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
13		islring.u	. . . . . . . 8 |- U = (Unit ` R )
14	12, 13	eqtr4di 2258	. . . . . . 7 |- ( r = R -> (Unit ` r ) = U )
15	14	eleq2d 2277	. . . . . 6 |- ( r = R -> ( x e. (Unit ` r ) <-> x e. U ) )
16	14	eleq2d 2277	. . . . . 6 |- ( r = R -> ( y e. (Unit ` r ) <-> y e. U ) )
17	15, 16	orbi12d 795	. . . . 5 |- ( r = R -> ( ( x e. (Unit ` r ) \/ y e. (Unit ` r ) ) <-> ( x e. U \/ y e. U ) ) )
18	11, 17	imbi12d 234	. . . 4 |- ( r = R -> ( ( ( x ( +g ` r ) y ) = ( 1r ` r ) -> ( x e. (Unit ` r ) \/ y e. (Unit ` r ) ) ) <-> ( ( x .+ y ) = .1. -> ( x e. U \/ y e. U ) ) ) )
19	3, 18	raleqbidv 2721	. . 3 |- ( r = R -> ( A. y e. ( Base ` r ) ( ( x ( +g ` r ) y ) = ( 1r ` r ) -> ( x e. (Unit ` r ) \/ y e. (Unit ` r ) ) ) <-> A. y e. B ( ( x .+ y ) = .1. -> ( x e. U \/ y e. U ) ) ) )
20	3, 19	raleqbidv 2721	. 2 |- ( r = R -> ( A. x e. ( Base ` r ) A. y e. ( Base ` r ) ( ( x ( +g ` r ) y ) = ( 1r ` r ) -> ( x e. (Unit ` r ) \/ y e. (Unit ` r ) ) ) <-> A. x e. B A. y e. B ( ( x .+ y ) = .1. -> ( x e. U \/ y e. U ) ) ) )
21		df-lring 14068	. 2 |- LRing = { r e. NzRing | A. x e. ( Base ` r ) A. y e. ( Base ` r ) ( ( x ( +g ` r ) y ) = ( 1r ` r ) -> ( x e. (Unit ` r ) \/ y e. (Unit ` r ) ) ) }
22	20, 21	elrab2 2939	1 |- ( R e. LRing <-> ( R e. NzRing /\ A. x e. B A. y e. B ( ( x .+ y ) = .1. -> ( x e. U \/ y e. U ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2178   A.wral 2486   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   1rcur 13836  Unitcui 13964  NzRingcnzr 14056  LRingclring 14067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970  df-lring 14068

This theorem is referenced by:  lringuplu  14073