Theorem orngmul 29803

Description: In an ordered ring, the ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 20-Jan-2018.)

Hypotheses

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

Assertion

Ref	Expression
orngmul	|- ( ( R e. oRing /\ ( X e. B /\ .0. .<_ X ) /\ ( Y e. B /\ .0. .<_ Y ) ) -> .0. .<_ ( X .x. Y ) )

Proof of Theorem orngmul

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2r 1088	. 2 |- ( ( R e. oRing /\ ( X e. B /\ .0. .<_ X ) /\ ( Y e. B /\ .0. .<_ Y ) ) -> .0. .<_ X )
2		simp3r 1090	. 2 |- ( ( R e. oRing /\ ( X e. B /\ .0. .<_ X ) /\ ( Y e. B /\ .0. .<_ Y ) ) -> .0. .<_ Y )
3		simp2l 1087	. . 3 |- ( ( R e. oRing /\ ( X e. B /\ .0. .<_ X ) /\ ( Y e. B /\ .0. .<_ Y ) ) -> X e. B )
4		simp3l 1089	. . 3 |- ( ( R e. oRing /\ ( X e. B /\ .0. .<_ X ) /\ ( Y e. B /\ .0. .<_ Y ) ) -> Y e. B )
5		orngmul.0	. . . . . 6 |- B = ( Base ` R )
6		orngmul.2	. . . . . 6 |- .0. = ( 0g ` R )
7		orngmul.3	. . . . . 6 |- .x. = ( .r ` R )
8		orngmul.1	. . . . . 6 |- .<_ = ( le ` R )
9	5, 6, 7, 8	isorng 29799	. . . . 5 |- ( 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 ) ) ) )
10	9	simp3bi 1078	. . . 4 |- ( R e. oRing -> A. a e. B A. b e. B ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) )
11	10	3ad2ant1 1082	. . 3 |- ( ( R e. oRing /\ ( X e. B /\ .0. .<_ X ) /\ ( Y e. B /\ .0. .<_ Y ) ) -> A. a e. B A. b e. B ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) )
12		breq2 4657	. . . . . 6 |- ( a = X -> ( .0. .<_ a <-> .0. .<_ X ) )
13	12	anbi1d 741	. . . . 5 |- ( a = X -> ( ( .0. .<_ a /\ .0. .<_ b ) <-> ( .0. .<_ X /\ .0. .<_ b ) ) )
14		oveq1 6657	. . . . . 6 |- ( a = X -> ( a .x. b ) = ( X .x. b ) )
15	14	breq2d 4665	. . . . 5 |- ( a = X -> ( .0. .<_ ( a .x. b ) <-> .0. .<_ ( X .x. b ) ) )
16	13, 15	imbi12d 334	. . . 4 |- ( a = X -> ( ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) <-> ( ( .0. .<_ X /\ .0. .<_ b ) -> .0. .<_ ( X .x. b ) ) ) )
17		breq2 4657	. . . . . 6 |- ( b = Y -> ( .0. .<_ b <-> .0. .<_ Y ) )
18	17	anbi2d 740	. . . . 5 |- ( b = Y -> ( ( .0. .<_ X /\ .0. .<_ b ) <-> ( .0. .<_ X /\ .0. .<_ Y ) ) )
19		oveq2 6658	. . . . . 6 |- ( b = Y -> ( X .x. b ) = ( X .x. Y ) )
20	19	breq2d 4665	. . . . 5 |- ( b = Y -> ( .0. .<_ ( X .x. b ) <-> .0. .<_ ( X .x. Y ) ) )
21	18, 20	imbi12d 334	. . . 4 |- ( b = Y -> ( ( ( .0. .<_ X /\ .0. .<_ b ) -> .0. .<_ ( X .x. b ) ) <-> ( ( .0. .<_ X /\ .0. .<_ Y ) -> .0. .<_ ( X .x. Y ) ) ) )
22	16, 21	rspc2va 3323	. . 3 |- ( ( ( X e. B /\ Y e. B ) /\ A. a e. B A. b e. B ( ( .0. .<_ a /\ .0. .<_ b ) -> .0. .<_ ( a .x. b ) ) ) -> ( ( .0. .<_ X /\ .0. .<_ Y ) -> .0. .<_ ( X .x. Y ) ) )
23	3, 4, 11, 22	syl21anc 1325	. 2 |- ( ( R e. oRing /\ ( X e. B /\ .0. .<_ X ) /\ ( Y e. B /\ .0. .<_ Y ) ) -> ( ( .0. .<_ X /\ .0. .<_ Y ) -> .0. .<_ ( X .x. Y ) ) )
24	1, 2, 23	mp2and 715	1 |- ( ( R e. oRing /\ ( X e. B /\ .0. .<_ X ) /\ ( Y e. B /\ .0. .<_ Y ) ) -> .0. .<_ ( X .x. Y ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   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:  orngsqr  29804  ornglmulle  29805  orngrmulle  29806  orngmullt  29809  suborng  29815