Theorem isorng 29799

Description: An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)

Hypotheses

Ref	Expression
isorng.0	|- B = ( Base ` R )
isorng.1	|- .0. = ( 0g ` R )
isorng.2	|- .x. = ( .r ` R )
isorng.3	|- .<_ = ( le ` R )

Assertion

Ref	Expression
isorng	|- ( R e. oRing <-> ( R e. Ring /\ R e. oGrp /\ A. a e. B A. b e. B ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) ) )

Distinct variable groups:   a, b, B   R, a, b

Allowed substitution hints:   .x. ( a, b)   .<_ ( a, b)   .0. ( a, b)

Proof of Theorem isorng

Dummy variables l r t v z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3796	. . 3 |- ( R e. ( Ring i^i oGrp ) <-> ( R e. Ring /\ R e. oGrp ) )
2	1	anbi1i 731	. 2 |- ( ( R e. ( Ring i^i oGrp ) /\ A. a e. B A. b e. B ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) ) <-> ( ( R e. Ring /\ R e. oGrp ) /\ A. a e. B A. b e. B ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) ) )
3		fvexd 6203	. . . . 5 |- ( r = R -> ( .r ` r ) e. _V )
4		simpr 477	. . . . . . . . . . 11 |- ( ( r = R /\ t = ( .r ` r ) ) -> t = ( .r ` r ) )
5		simpl 473	. . . . . . . . . . . . 13 |- ( ( r = R /\ t = ( .r ` r ) ) -> r = R )
6	5	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( r = R /\ t = ( .r ` r ) ) -> ( .r ` r ) = ( .r ` R ) )
7		isorng.2	. . . . . . . . . . . 12 |- .x. = ( .r ` R )
8	6, 7	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( r = R /\ t = ( .r ` r ) ) -> ( .r ` r ) = .x. )
9	4, 8	eqtrd 2656	. . . . . . . . . 10 |- ( ( r = R /\ t = ( .r ` r ) ) -> t = .x. )
10	9	oveqd 6667	. . . . . . . . 9 |- ( ( r = R /\ t = ( .r ` r ) ) -> ( a t b ) = ( a .x. b ) )
11	10	breq2d 4665	. . . . . . . 8 |- ( ( r = R /\ t = ( .r ` r ) ) -> ( .0. l ( a t b ) <-> .0. l ( a .x. b ) ) )
12	11	imbi2d 330	. . . . . . 7 |- ( ( r = R /\ t = ( .r ` r ) ) -> ( ( ( .0. l a /\ .0. l b ) -> .0. l ( a t b ) ) <-> ( ( .0. l a /\ .0. l b ) -> .0. l ( a .x. b ) ) ) )
13	12	2ralbidv 2989	. . . . . 6 |- ( ( r = R /\ t = ( .r ` r ) ) -> ( A. a e. B A. b e. B ( ( .0. l a /\ .0. l b ) -> .0. l ( a t b ) ) <-> A. a e. B A. b e. B ( ( .0. l a /\ .0. l b ) -> .0. l ( a .x. b ) ) ) )
14	13	sbcbidv 3490	. . . . 5 |- ( ( r = R /\ t = ( .r ` r ) ) -> ( [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( .0. l a /\ .0. l b ) -> .0. l ( a t b ) ) <-> [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( .0. l a /\ .0. l b ) -> .0. l ( a .x. b ) ) ) )
15	3, 14	sbcied 3472	. . . 4 |- ( r = R -> ( [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( .0. l a /\ .0. l b ) -> .0. l ( a t b ) ) <-> [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( .0. l a /\ .0. l b ) -> .0. l ( a .x. b ) ) ) )
16		fvexd 6203	. . . . . 6 |- ( r = R -> ( Base ` r ) e. _V )
17		simpr 477	. . . . . . . . . . 11 |- ( ( r = R /\ v = ( Base ` r ) ) -> v = ( Base ` r ) )
18		fveq2 6191	. . . . . . . . . . . . 13 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
19		isorng.0	. . . . . . . . . . . . 13 |- B = ( Base ` R )
20	18, 19	syl6eqr 2674	. . . . . . . . . . . 12 |- ( r = R -> ( Base ` r ) = B )
21	20	adantr 481	. . . . . . . . . . 11 |- ( ( r = R /\ v = ( Base ` r ) ) -> ( Base ` r ) = B )
22	17, 21	eqtrd 2656	. . . . . . . . . 10 |- ( ( r = R /\ v = ( Base ` r ) ) -> v = B )
23		raleq 3138	. . . . . . . . . . 11 |- ( v = B -> ( A. b e. v ( ( z l a /\ z l b ) -> z l ( a t b ) ) <-> A. b e. B ( ( z l a /\ z l b ) -> z l ( a t b ) ) ) )
24	23	raleqbi1dv 3146	. . . . . . . . . 10 |- ( v = B -> ( A. a e. v A. b e. v ( ( z l a /\ z l b ) -> z l ( a t b ) ) <-> A. a e. B A. b e. B ( ( z l a /\ z l b ) -> z l ( a t b ) ) ) )
25	22, 24	syl 17	. . . . . . . . 9 |- ( ( r = R /\ v = ( Base ` r ) ) -> ( A. a e. v A. b e. v ( ( z l a /\ z l b ) -> z l ( a t b ) ) <-> A. a e. B A. b e. B ( ( z l a /\ z l b ) -> z l ( a t b ) ) ) )
26	25	sbcbidv 3490	. . . . . . . 8 |- ( ( r = R /\ v = ( Base ` r ) ) -> ( [. ( le ` r ) / l ]. A. a e. v A. b e. v ( ( z l a /\ z l b ) -> z l ( a t b ) ) <-> [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( z l a /\ z l b ) -> z l ( a t b ) ) ) )
27	26	sbcbidv 3490	. . . . . . 7 |- ( ( r = R /\ v = ( Base ` r ) ) -> ( [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. v A. b e. v ( ( z l a /\ z l b ) -> z l ( a t b ) ) <-> [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( z l a /\ z l b ) -> z l ( a t b ) ) ) )
28	27	sbcbidv 3490	. . . . . 6 |- ( ( r = R /\ v = ( Base ` r ) ) -> ( [. ( 0g ` r ) / z ]. [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. v A. b e. v ( ( z l a /\ z l b ) -> z l ( a t b ) ) <-> [. ( 0g ` r ) / z ]. [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( z l a /\ z l b ) -> z l ( a t b ) ) ) )
29	16, 28	sbcied 3472	. . . . 5 |- ( r = R -> ( [. ( Base ` r ) / v ]. [. ( 0g ` r ) / z ]. [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. v A. b e. v ( ( z l a /\ z l b ) -> z l ( a t b ) ) <-> [. ( 0g ` r ) / z ]. [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( z l a /\ z l b ) -> z l ( a t b ) ) ) )
30		fvexd 6203	. . . . . 6 |- ( r = R -> ( 0g ` r ) e. _V )
31		simpr 477	. . . . . . . . . . . . 13 |- ( ( r = R /\ z = ( 0g ` r ) ) -> z = ( 0g ` r ) )
32		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
33		isorng.1	. . . . . . . . . . . . . . 15 |- .0. = ( 0g ` R )
34	32, 33	syl6eqr 2674	. . . . . . . . . . . . . 14 |- ( r = R -> ( 0g ` r ) = .0. )
35	34	adantr 481	. . . . . . . . . . . . 13 |- ( ( r = R /\ z = ( 0g ` r ) ) -> ( 0g ` r ) = .0. )
36	31, 35	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( r = R /\ z = ( 0g ` r ) ) -> z = .0. )
37	36	breq1d 4663	. . . . . . . . . . 11 |- ( ( r = R /\ z = ( 0g ` r ) ) -> ( z l a <-> .0. l a ) )
38	36	breq1d 4663	. . . . . . . . . . 11 |- ( ( r = R /\ z = ( 0g ` r ) ) -> ( z l b <-> .0. l b ) )
39	37, 38	anbi12d 747	. . . . . . . . . 10 |- ( ( r = R /\ z = ( 0g ` r ) ) -> ( ( z l a /\ z l b ) <-> ( .0. l a /\ .0. l b ) ) )
40	36	breq1d 4663	. . . . . . . . . 10 |- ( ( r = R /\ z = ( 0g ` r ) ) -> ( z l ( a t b ) <-> .0. l ( a t b ) ) )
41	39, 40	imbi12d 334	. . . . . . . . 9 |- ( ( r = R /\ z = ( 0g ` r ) ) -> ( ( ( z l a /\ z l b ) -> z l ( a t b ) ) <-> ( ( .0. l a /\ .0. l b ) -> .0. l ( a t b ) ) ) )
42	41	2ralbidv 2989	. . . . . . . 8 |- ( ( r = R /\ z = ( 0g ` r ) ) -> ( A. a e. B A. b e. B ( ( z l a /\ z l b ) -> z l ( a t b ) ) <-> A. a e. B A. b e. B ( ( .0. l a /\ .0. l b ) -> .0. l ( a t b ) ) ) )
43	42	sbcbidv 3490	. . . . . . 7 |- ( ( r = R /\ z = ( 0g ` r ) ) -> ( [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( z l a /\ z l b ) -> z l ( a t b ) ) <-> [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( .0. l a /\ .0. l b ) -> .0. l ( a t b ) ) ) )
44	43	sbcbidv 3490	. . . . . 6 |- ( ( r = R /\ z = ( 0g ` r ) ) -> ( [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( z l a /\ z l b ) -> z l ( a t b ) ) <-> [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( .0. l a /\ .0. l b ) -> .0. l ( a t b ) ) ) )
45	30, 44	sbcied 3472	. . . . 5 |- ( r = R -> ( [. ( 0g ` r ) / z ]. [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( z l a /\ z l b ) -> z l ( a t b ) ) <-> [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( .0. l a /\ .0. l b ) -> .0. l ( a t b ) ) ) )
46	29, 45	bitr2d 269	. . . 4 |- ( r = R -> ( [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( .0. l a /\ .0. l b ) -> .0. l ( a t b ) ) <-> [. ( Base ` r ) / v ]. [. ( 0g ` r ) / z ]. [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. v A. b e. v ( ( z l a /\ z l b ) -> z l ( a t b ) ) ) )
47		fvexd 6203	. . . . 5 |- ( r = R -> ( le ` r ) e. _V )
48		simpr 477	. . . . . . . . . 10 |- ( ( r = R /\ l = ( le ` r ) ) -> l = ( le ` r ) )
49		simpl 473	. . . . . . . . . . . 12 |- ( ( r = R /\ l = ( le ` r ) ) -> r = R )
50	49	fveq2d 6195	. . . . . . . . . . 11 |- ( ( r = R /\ l = ( le ` r ) ) -> ( le ` r ) = ( le ` R ) )
51		isorng.3	. . . . . . . . . . 11 |- .<_ = ( le ` R )
52	50, 51	syl6eqr 2674	. . . . . . . . . 10 |- ( ( r = R /\ l = ( le ` r ) ) -> ( le ` r ) = .<_ )
53	48, 52	eqtrd 2656	. . . . . . . . 9 |- ( ( r = R /\ l = ( le ` r ) ) -> l = .<_ )
54	53	breqd 4664	. . . . . . . 8 |- ( ( r = R /\ l = ( le ` r ) ) -> ( .0. l a <-> .0. .<_ a ) )
55	53	breqd 4664	. . . . . . . 8 |- ( ( r = R /\ l = ( le ` r ) ) -> ( .0. l b <-> .0. .<_ b ) )
56	54, 55	anbi12d 747	. . . . . . 7 |- ( ( r = R /\ l = ( le ` r ) ) -> ( ( .0. l a /\ .0. l b ) <-> ( .0. .<_ a /\ .0. .<_ b ) ) )
57	53	breqd 4664	. . . . . . 7 |- ( ( r = R /\ l = ( le ` r ) ) -> ( .0. l ( a .x. b ) <-> .0. .<_ ( a .x. b ) ) )
58	56, 57	imbi12d 334	. . . . . 6 |- ( ( r = R /\ l = ( le ` r ) ) -> ( ( ( .0. l a /\ .0. l b ) -> .0. l ( a .x. b ) ) <-> ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) ) )
59	58	2ralbidv 2989	. . . . 5 |- ( ( r = R /\ l = ( le ` r ) ) -> ( A. a e. B A. b e. B ( ( .0. l a /\ .0. l b ) -> .0. l ( a .x. b ) ) <-> A. a e. B A. b e. B ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) ) )
60	47, 59	sbcied 3472	. . . 4 |- ( r = R -> ( [. ( le ` r ) / l ]. A. a e. B A. b e. B ( ( .0. l a /\ .0. l b ) -> .0. l ( a .x. b ) ) <-> A. a e. B A. b e. B ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) ) )
61	15, 46, 60	3bitr3d 298	. . 3 |- ( r = R -> ( [. ( Base ` r ) / v ]. [. ( 0g ` r ) / z ]. [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. v A. b e. v ( ( z l a /\ z l b ) -> z l ( a t b ) ) <-> A. a e. B A. b e. B ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) ) )
62		df-orng 29797	. . 3 |- oRing = { r e. ( Ring i^i oGrp ) | [. ( Base ` r ) / v ]. [. ( 0g ` r ) / z ]. [. ( .r ` r ) / t ]. [. ( le ` r ) / l ]. A. a e. v A. b e. v ( ( z l a /\ z l b ) -> z l ( a t b ) ) }
63	61, 62	elrab2 3366	. 2 |- ( R e. oRing <-> ( R e. ( Ring i^i oGrp ) /\ A. a e. B A. b e. B ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) ) )
64		df-3an 1039	. 2 |- ( ( R e. Ring /\ R e. oGrp /\ A. a e. B A. b e. B ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) ) <-> ( ( R e. Ring /\ R e. oGrp ) /\ A. a e. B A. b e. B ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) ) )
65	2, 63, 64	3bitr4i 292	1 |- ( R e. oRing <-> ( R e. Ring /\ R e. oGrp /\ A. a e. B A. b e. B ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   i^i cin 3573   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   lecple 15948   0gc0g 16100   Ringcrg 18547  oGrpcogrp 29698  oRingcorng 29795

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-nul 4789

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-iota 5851  df-fv 5896  df-ov 6653  df-orng 29797

This theorem is referenced by:  orngring  29800  orngogrp  29801  orngmul  29803  suborng  29815  reofld  29840