Theorem orngrmullt 29808

Description: In an ordered ring, multiplication with a positive does not change strict comparison. (Contributed by Thierry Arnoux, 9-Apr-2018.)

Hypotheses

Ref	Expression
ornglmullt.b	|- B = ( Base ` R )
ornglmullt.t	|- .x. = ( .r ` R )
ornglmullt.0	|- .0. = ( 0g ` R )
ornglmullt.1	|- ( ph -> R e. oRing )
ornglmullt.2	|- ( ph -> X e. B )
ornglmullt.3	|- ( ph -> Y e. B )
ornglmullt.4	|- ( ph -> Z e. B )
ornglmullt.l	|- .< = ( lt ` R )
ornglmullt.d	|- ( ph -> R e. DivRing )
ornglmullt.5	|- ( ph -> X .< Y )
ornglmullt.6	|- ( ph -> .0. .< Z )

Assertion

Ref	Expression
orngrmullt	|- ( ph -> ( X .x. Z ) .< ( Y .x. Z ) )

Proof of Theorem orngrmullt

Step	Hyp	Ref	Expression
1		ornglmullt.b	. . 3 |- B = ( Base ` R )
2		ornglmullt.t	. . 3 |- .x. = ( .r ` R )
3		ornglmullt.0	. . 3 |- .0. = ( 0g ` R )
4		ornglmullt.1	. . 3 |- ( ph -> R e. oRing )
5		ornglmullt.2	. . 3 |- ( ph -> X e. B )
6		ornglmullt.3	. . 3 |- ( ph -> Y e. B )
7		ornglmullt.4	. . 3 |- ( ph -> Z e. B )
8		eqid 2622	. . 3 |- ( le ` R ) = ( le ` R )
9		ornglmullt.5	. . . 4 |- ( ph -> X .< Y )
10		ornglmullt.l	. . . . . 6 |- .< = ( lt ` R )
11	8, 10	pltle 16961	. . . . 5 |- ( ( R e. oRing /\ X e. B /\ Y e. B ) -> ( X .< Y -> X ( le ` R ) Y ) )
12	11	imp 445	. . . 4 |- ( ( ( R e. oRing /\ X e. B /\ Y e. B ) /\ X .< Y ) -> X ( le ` R ) Y )
13	4, 5, 6, 9, 12	syl31anc 1329	. . 3 |- ( ph -> X ( le ` R ) Y )
14		orngring 29800	. . . . . 6 |- ( R e. oRing -> R e. Ring )
15	4, 14	syl 17	. . . . 5 |- ( ph -> R e. Ring )
16		ringgrp 18552	. . . . 5 |- ( R e. Ring -> R e. Grp )
17	1, 3	grpidcl 17450	. . . . 5 |- ( R e. Grp -> .0. e. B )
18	15, 16, 17	3syl 18	. . . 4 |- ( ph -> .0. e. B )
19		ornglmullt.6	. . . 4 |- ( ph -> .0. .< Z )
20	8, 10	pltle 16961	. . . . 5 |- ( ( R e. oRing /\ .0. e. B /\ Z e. B ) -> ( .0. .< Z -> .0. ( le ` R ) Z ) )
21	20	imp 445	. . . 4 |- ( ( ( R e. oRing /\ .0. e. B /\ Z e. B ) /\ .0. .< Z ) -> .0. ( le ` R ) Z )
22	4, 18, 7, 19, 21	syl31anc 1329	. . 3 |- ( ph -> .0. ( le ` R ) Z )
23	1, 2, 3, 4, 5, 6, 7, 8, 13, 22	orngrmulle 29806	. 2 |- ( ph -> ( X .x. Z ) ( le ` R ) ( Y .x. Z ) )
24		simpr 477	. . . . . 6 |- ( ( ph /\ ( X .x. Z ) = ( Y .x. Z ) ) -> ( X .x. Z ) = ( Y .x. Z ) )
25	24	oveq1d 6665	. . . . 5 |- ( ( ph /\ ( X .x. Z ) = ( Y .x. Z ) ) -> ( ( X .x. Z ) (/r ` R ) Z ) = ( ( Y .x. Z ) (/r ` R ) Z ) )
26		ornglmullt.d	. . . . . . . 8 |- ( ph -> R e. DivRing )
27	10	pltne 16962	. . . . . . . . . . 11 |- ( ( R e. oRing /\ .0. e. B /\ Z e. B ) -> ( .0. .< Z -> .0. =/= Z ) )
28	27	imp 445	. . . . . . . . . 10 |- ( ( ( R e. oRing /\ .0. e. B /\ Z e. B ) /\ .0. .< Z ) -> .0. =/= Z )
29	4, 18, 7, 19, 28	syl31anc 1329	. . . . . . . . 9 |- ( ph -> .0. =/= Z )
30	29	necomd 2849	. . . . . . . 8 |- ( ph -> Z =/= .0. )
31		eqid 2622	. . . . . . . . . 10 |- (Unit ` R ) = (Unit ` R )
32	1, 31, 3	drngunit 18752	. . . . . . . . 9 |- ( R e. DivRing -> ( Z e. (Unit ` R ) <-> ( Z e. B /\ Z =/= .0. ) ) )
33	32	biimpar 502	. . . . . . . 8 |- ( ( R e. DivRing /\ ( Z e. B /\ Z =/= .0. ) ) -> Z e. (Unit ` R ) )
34	26, 7, 30, 33	syl12anc 1324	. . . . . . 7 |- ( ph -> Z e. (Unit ` R ) )
35		eqid 2622	. . . . . . . 8 |- (/r ` R ) = (/r ` R )
36	1, 31, 35, 2	dvrcan3 18692	. . . . . . 7 |- ( ( R e. Ring /\ X e. B /\ Z e. (Unit ` R ) ) -> ( ( X .x. Z ) (/r ` R ) Z ) = X )
37	15, 5, 34, 36	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( X .x. Z ) (/r ` R ) Z ) = X )
38	37	adantr 481	. . . . 5 |- ( ( ph /\ ( X .x. Z ) = ( Y .x. Z ) ) -> ( ( X .x. Z ) (/r ` R ) Z ) = X )
39	1, 31, 35, 2	dvrcan3 18692	. . . . . . 7 |- ( ( R e. Ring /\ Y e. B /\ Z e. (Unit ` R ) ) -> ( ( Y .x. Z ) (/r ` R ) Z ) = Y )
40	15, 6, 34, 39	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( Y .x. Z ) (/r ` R ) Z ) = Y )
41	40	adantr 481	. . . . 5 |- ( ( ph /\ ( X .x. Z ) = ( Y .x. Z ) ) -> ( ( Y .x. Z ) (/r ` R ) Z ) = Y )
42	25, 38, 41	3eqtr3d 2664	. . . 4 |- ( ( ph /\ ( X .x. Z ) = ( Y .x. Z ) ) -> X = Y )
43	10	pltne 16962	. . . . . . . 8 |- ( ( R e. oRing /\ X e. B /\ Y e. B ) -> ( X .< Y -> X =/= Y ) )
44	43	imp 445	. . . . . . 7 |- ( ( ( R e. oRing /\ X e. B /\ Y e. B ) /\ X .< Y ) -> X =/= Y )
45	4, 5, 6, 9, 44	syl31anc 1329	. . . . . 6 |- ( ph -> X =/= Y )
46	45	adantr 481	. . . . 5 |- ( ( ph /\ ( X .x. Z ) = ( Y .x. Z ) ) -> X =/= Y )
47	46	neneqd 2799	. . . 4 |- ( ( ph /\ ( X .x. Z ) = ( Y .x. Z ) ) -> -. X = Y )
48	42, 47	pm2.65da 600	. . 3 |- ( ph -> -. ( X .x. Z ) = ( Y .x. Z ) )
49	48	neqned 2801	. 2 |- ( ph -> ( X .x. Z ) =/= ( Y .x. Z ) )
50	1, 2	ringcl 18561	. . . 4 |- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
51	15, 5, 7, 50	syl3anc 1326	. . 3 |- ( ph -> ( X .x. Z ) e. B )
52	1, 2	ringcl 18561	. . . 4 |- ( ( R e. Ring /\ Y e. B /\ Z e. B ) -> ( Y .x. Z ) e. B )
53	15, 6, 7, 52	syl3anc 1326	. . 3 |- ( ph -> ( Y .x. Z ) e. B )
54	8, 10	pltval 16960	. . 3 |- ( ( R e. oRing /\ ( X .x. Z ) e. B /\ ( Y .x. Z ) e. B ) -> ( ( X .x. Z ) .< ( Y .x. Z ) <-> ( ( X .x. Z ) ( le ` R ) ( Y .x. Z ) /\ ( X .x. Z ) =/= ( Y .x. Z ) ) ) )
55	4, 51, 53, 54	syl3anc 1326	. 2 |- ( ph -> ( ( X .x. Z ) .< ( Y .x. Z ) <-> ( ( X .x. Z ) ( le ` R ) ( Y .x. Z ) /\ ( X .x. Z ) =/= ( Y .x. Z ) ) ) )
56	23, 49, 55	mpbir2and 957	1 |- ( ph -> ( X .x. Z ) .< ( Y .x. Z ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   lecple 15948   0gc0g 16100   ltcplt 16941   Grpcgrp 17422   Ringcrg 18547  Unitcui 18639  /_rcdvr 18682   DivRingcdr 18747  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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-plt 16958  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-omnd 29699  df-ogrp 29700  df-orng 29797

This theorem is referenced by:  isarchiofld  29817