Theorem orngmullt 29809

Description: In an ordered ring, the strict ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 9-Sep-2018.)

Hypotheses

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

Assertion

Ref	Expression
orngmullt	|- ( ph -> .0. .< ( X .x. Y ) )

Proof of Theorem orngmullt

Step	Hyp	Ref	Expression
1		orngmullt.1	. . 3 |- ( ph -> R e. oRing )
2		orngmullt.2	. . 3 |- ( ph -> X e. B )
3		orngmullt.x	. . . . 5 |- ( ph -> .0. .< X )
4		orngring 29800	. . . . . . 7 |- ( R e. oRing -> R e. Ring )
5		ringgrp 18552	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
6		orngmullt.b	. . . . . . . 8 |- B = ( Base ` R )
7		orngmullt.0	. . . . . . . 8 |- .0. = ( 0g ` R )
8	6, 7	grpidcl 17450	. . . . . . 7 |- ( R e. Grp -> .0. e. B )
9	1, 4, 5, 8	4syl 19	. . . . . 6 |- ( ph -> .0. e. B )
10		eqid 2622	. . . . . . 7 |- ( le ` R ) = ( le ` R )
11		orngmullt.l	. . . . . . 7 |- .< = ( lt ` R )
12	10, 11	pltval 16960	. . . . . 6 |- ( ( R e. oRing /\ .0. e. B /\ X e. B ) -> ( .0. .< X <-> ( .0. ( le ` R ) X /\ .0. =/= X ) ) )
13	1, 9, 2, 12	syl3anc 1326	. . . . 5 |- ( ph -> ( .0. .< X <-> ( .0. ( le ` R ) X /\ .0. =/= X ) ) )
14	3, 13	mpbid 222	. . . 4 |- ( ph -> ( .0. ( le ` R ) X /\ .0. =/= X ) )
15	14	simpld 475	. . 3 |- ( ph -> .0. ( le ` R ) X )
16		orngmullt.3	. . 3 |- ( ph -> Y e. B )
17		orngmullt.y	. . . . 5 |- ( ph -> .0. .< Y )
18	10, 11	pltval 16960	. . . . . 6 |- ( ( R e. oRing /\ .0. e. B /\ Y e. B ) -> ( .0. .< Y <-> ( .0. ( le ` R ) Y /\ .0. =/= Y ) ) )
19	1, 9, 16, 18	syl3anc 1326	. . . . 5 |- ( ph -> ( .0. .< Y <-> ( .0. ( le ` R ) Y /\ .0. =/= Y ) ) )
20	17, 19	mpbid 222	. . . 4 |- ( ph -> ( .0. ( le ` R ) Y /\ .0. =/= Y ) )
21	20	simpld 475	. . 3 |- ( ph -> .0. ( le ` R ) Y )
22		orngmullt.t	. . . 4 |- .x. = ( .r ` R )
23	6, 10, 7, 22	orngmul 29803	. . 3 |- ( ( R e. oRing /\ ( X e. B /\ .0. ( le ` R ) X ) /\ ( Y e. B /\ .0. ( le ` R ) Y ) ) -> .0. ( le ` R ) ( X .x. Y ) )
24	1, 2, 15, 16, 21, 23	syl122anc 1335	. 2 |- ( ph -> .0. ( le ` R ) ( X .x. Y ) )
25	14	simprd 479	. . . . 5 |- ( ph -> .0. =/= X )
26	25	necomd 2849	. . . 4 |- ( ph -> X =/= .0. )
27	20	simprd 479	. . . . 5 |- ( ph -> .0. =/= Y )
28	27	necomd 2849	. . . 4 |- ( ph -> Y =/= .0. )
29		orngmullt.4	. . . . 5 |- ( ph -> R e. DivRing )
30	6, 7, 22, 29, 2, 16	drngmulne0 18769	. . . 4 |- ( ph -> ( ( X .x. Y ) =/= .0. <-> ( X =/= .0. /\ Y =/= .0. ) ) )
31	26, 28, 30	mpbir2and 957	. . 3 |- ( ph -> ( X .x. Y ) =/= .0. )
32	31	necomd 2849	. 2 |- ( ph -> .0. =/= ( X .x. Y ) )
33	1, 4	syl 17	. . . 4 |- ( ph -> R e. Ring )
34	6, 22	ringcl 18561	. . . 4 |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
35	33, 2, 16, 34	syl3anc 1326	. . 3 |- ( ph -> ( X .x. Y ) e. B )
36	10, 11	pltval 16960	. . 3 |- ( ( R e. oRing /\ .0. e. B /\ ( X .x. Y ) e. B ) -> ( .0. .< ( X .x. Y ) <-> ( .0. ( le ` R ) ( X .x. Y ) /\ .0. =/= ( X .x. Y ) ) ) )
37	1, 9, 35, 36	syl3anc 1326	. 2 |- ( ph -> ( .0. .< ( X .x. Y ) <-> ( .0. ( le ` R ) ( X .x. Y ) /\ .0. =/= ( X .x. Y ) ) ) )
38	24, 32, 37	mpbir2and 957	1 |- ( ph -> .0. .< ( X .x. Y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = 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   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-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-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749  df-orng 29797

This theorem is referenced by: (None)