Theorem orngsqr 29804

Description: In an ordered ring, all squares are positive. (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
orngsqr	|- ( ( R e. oRing /\ X e. B ) -> .0. .<_ ( X .x. X ) )

Proof of Theorem orngsqr

Step	Hyp	Ref	Expression
1		simpll 790	. . 3 |- ( ( ( R e. oRing /\ X e. B ) /\ .0. .<_ X ) -> R e. oRing )
2		simplr 792	. . 3 |- ( ( ( R e. oRing /\ X e. B ) /\ .0. .<_ X ) -> X e. B )
3		simpr 477	. . 3 |- ( ( ( R e. oRing /\ X e. B ) /\ .0. .<_ X ) -> .0. .<_ X )
4		orngmul.0	. . . 4 |- B = ( Base ` R )
5		orngmul.1	. . . 4 |- .<_ = ( le ` R )
6		orngmul.2	. . . 4 |- .0. = ( 0g ` R )
7		orngmul.3	. . . 4 |- .x. = ( .r ` R )
8	4, 5, 6, 7	orngmul 29803	. . 3 |- ( ( R e. oRing /\ ( X e. B /\ .0. .<_ X ) /\ ( X e. B /\ .0. .<_ X ) ) -> .0. .<_ ( X .x. X ) )
9	1, 2, 3, 2, 3, 8	syl122anc 1335	. 2 |- ( ( ( R e. oRing /\ X e. B ) /\ .0. .<_ X ) -> .0. .<_ ( X .x. X ) )
10		simpll 790	. . . 4 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> R e. oRing )
11		orngring 29800	. . . . . . 7 |- ( R e. oRing -> R e. Ring )
12	11	ad2antrr 762	. . . . . 6 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> R e. Ring )
13		ringgrp 18552	. . . . . 6 |- ( R e. Ring -> R e. Grp )
14	12, 13	syl 17	. . . . 5 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> R e. Grp )
15		simplr 792	. . . . 5 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> X e. B )
16		eqid 2622	. . . . . 6 |- ( invg ` R ) = ( invg ` R )
17	4, 16	grpinvcl 17467	. . . . 5 |- ( ( R e. Grp /\ X e. B ) -> ( ( invg ` R ) ` X ) e. B )
18	14, 15, 17	syl2anc 693	. . . 4 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> ( ( invg ` R ) ` X ) e. B )
19		orngogrp 29801	. . . . . . . 8 |- ( R e. oRing -> R e. oGrp )
20		isogrp 29702	. . . . . . . . 9 |- ( R e. oGrp <-> ( R e. Grp /\ R e. oMnd ) )
21	20	simprbi 480	. . . . . . . 8 |- ( R e. oGrp -> R e. oMnd )
22	19, 21	syl 17	. . . . . . 7 |- ( R e. oRing -> R e. oMnd )
23	10, 22	syl 17	. . . . . 6 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> R e. oMnd )
24	4, 6	grpidcl 17450	. . . . . . 7 |- ( R e. Grp -> .0. e. B )
25	14, 24	syl 17	. . . . . 6 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> .0. e. B )
26		simpl 473	. . . . . . . . . . 11 |- ( ( R e. oRing /\ X e. B ) -> R e. oRing )
27	11, 13, 24	3syl 18	. . . . . . . . . . . 12 |- ( R e. oRing -> .0. e. B )
28	26, 27	syl 17	. . . . . . . . . . 11 |- ( ( R e. oRing /\ X e. B ) -> .0. e. B )
29		simpr 477	. . . . . . . . . . 11 |- ( ( R e. oRing /\ X e. B ) -> X e. B )
30	26, 28, 29	3jca 1242	. . . . . . . . . 10 |- ( ( R e. oRing /\ X e. B ) -> ( R e. oRing /\ .0. e. B /\ X e. B ) )
31		eqid 2622	. . . . . . . . . . . 12 |- ( lt ` R ) = ( lt ` R )
32	5, 31	pltle 16961	. . . . . . . . . . 11 |- ( ( R e. oRing /\ .0. e. B /\ X e. B ) -> ( .0. ( lt ` R ) X -> .0. .<_ X ) )
33	32	con3dimp 457	. . . . . . . . . 10 |- ( ( ( R e. oRing /\ .0. e. B /\ X e. B ) /\ -. .0. .<_ X ) -> -. .0. ( lt ` R ) X )
34	30, 33	sylan 488	. . . . . . . . 9 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> -. .0. ( lt ` R ) X )
35		omndtos 29705	. . . . . . . . . . . . 13 |- ( R e. oMnd -> R e. Toset )
36	22, 35	syl 17	. . . . . . . . . . . 12 |- ( R e. oRing -> R e. Toset )
37	4, 5, 31	tosso 17036	. . . . . . . . . . . . . 14 |- ( R e. Toset -> ( R e. Toset <-> ( ( lt ` R ) Or B /\ ( _I |` B ) C_ .<_ ) ) )
38	37	ibi 256	. . . . . . . . . . . . 13 |- ( R e. Toset -> ( ( lt ` R ) Or B /\ ( _I |` B ) C_ .<_ ) )
39	38	simpld 475	. . . . . . . . . . . 12 |- ( R e. Toset -> ( lt ` R ) Or B )
40	10, 36, 39	3syl 18	. . . . . . . . . . 11 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> ( lt ` R ) Or B )
41		solin 5058	. . . . . . . . . . 11 |- ( ( ( lt ` R ) Or B /\ ( .0. e. B /\ X e. B ) ) -> ( .0. ( lt ` R ) X \/ .0. = X \/ X ( lt ` R ) .0. ) )
42	40, 25, 15, 41	syl12anc 1324	. . . . . . . . . 10 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> ( .0. ( lt ` R ) X \/ .0. = X \/ X ( lt ` R ) .0. ) )
43		3orass 1040	. . . . . . . . . 10 |- ( ( .0. ( lt ` R ) X \/ .0. = X \/ X ( lt ` R ) .0. ) <-> ( .0. ( lt ` R ) X \/ ( .0. = X \/ X ( lt ` R ) .0. ) ) )
44	42, 43	sylib 208	. . . . . . . . 9 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> ( .0. ( lt ` R ) X \/ ( .0. = X \/ X ( lt ` R ) .0. ) ) )
45		orel1 397	. . . . . . . . 9 |- ( -. .0. ( lt ` R ) X -> ( ( .0. ( lt ` R ) X \/ ( .0. = X \/ X ( lt ` R ) .0. ) ) -> ( .0. = X \/ X ( lt ` R ) .0. ) ) )
46	34, 44, 45	sylc 65	. . . . . . . 8 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> ( .0. = X \/ X ( lt ` R ) .0. ) )
47		orcom 402	. . . . . . . . 9 |- ( ( .0. = X \/ X ( lt ` R ) .0. ) <-> ( X ( lt ` R ) .0. \/ .0. = X ) )
48		eqcom 2629	. . . . . . . . . 10 |- ( .0. = X <-> X = .0. )
49	48	orbi2i 541	. . . . . . . . 9 |- ( ( X ( lt ` R ) .0. \/ .0. = X ) <-> ( X ( lt ` R ) .0. \/ X = .0. ) )
50	47, 49	bitri 264	. . . . . . . 8 |- ( ( .0. = X \/ X ( lt ` R ) .0. ) <-> ( X ( lt ` R ) .0. \/ X = .0. ) )
51	46, 50	sylib 208	. . . . . . 7 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> ( X ( lt ` R ) .0. \/ X = .0. ) )
52		tospos 29658	. . . . . . . . 9 |- ( R e. Toset -> R e. Poset )
53	10, 36, 52	3syl 18	. . . . . . . 8 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> R e. Poset )
54	4, 5, 31	pleval2 16965	. . . . . . . 8 |- ( ( R e. Poset /\ X e. B /\ .0. e. B ) -> ( X .<_ .0. <-> ( X ( lt ` R ) .0. \/ X = .0. ) ) )
55	53, 15, 25, 54	syl3anc 1326	. . . . . . 7 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> ( X .<_ .0. <-> ( X ( lt ` R ) .0. \/ X = .0. ) ) )
56	51, 55	mpbird 247	. . . . . 6 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> X .<_ .0. )
57		eqid 2622	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
58	4, 5, 57	omndadd 29706	. . . . . 6 |- ( ( R e. oMnd /\ ( X e. B /\ .0. e. B /\ ( ( invg ` R ) ` X ) e. B ) /\ X .<_ .0. ) -> ( X ( +g ` R ) ( ( invg ` R ) ` X ) ) .<_ ( .0. ( +g ` R ) ( ( invg ` R ) ` X ) ) )
59	23, 15, 25, 18, 56, 58	syl131anc 1339	. . . . 5 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> ( X ( +g ` R ) ( ( invg ` R ) ` X ) ) .<_ ( .0. ( +g ` R ) ( ( invg ` R ) ` X ) ) )
60	4, 57, 6, 16	grprinv 17469	. . . . . 6 |- ( ( R e. Grp /\ X e. B ) -> ( X ( +g ` R ) ( ( invg ` R ) ` X ) ) = .0. )
61	14, 15, 60	syl2anc 693	. . . . 5 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> ( X ( +g ` R ) ( ( invg ` R ) ` X ) ) = .0. )
62	4, 57, 6	grplid 17452	. . . . . 6 |- ( ( R e. Grp /\ ( ( invg ` R ) ` X ) e. B ) -> ( .0. ( +g ` R ) ( ( invg ` R ) ` X ) ) = ( ( invg ` R ) ` X ) )
63	14, 18, 62	syl2anc 693	. . . . 5 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> ( .0. ( +g ` R ) ( ( invg ` R ) ` X ) ) = ( ( invg ` R ) ` X ) )
64	59, 61, 63	3brtr3d 4684	. . . 4 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> .0. .<_ ( ( invg ` R ) ` X ) )
65	4, 5, 6, 7	orngmul 29803	. . . 4 |- ( ( R e. oRing /\ ( ( ( invg ` R ) ` X ) e. B /\ .0. .<_ ( ( invg ` R ) ` X ) ) /\ ( ( ( invg ` R ) ` X ) e. B /\ .0. .<_ ( ( invg ` R ) ` X ) ) ) -> .0. .<_ ( ( ( invg ` R ) ` X ) .x. ( ( invg ` R ) ` X ) ) )
66	10, 18, 64, 18, 64, 65	syl122anc 1335	. . 3 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> .0. .<_ ( ( ( invg ` R ) ` X ) .x. ( ( invg ` R ) ` X ) ) )
67	4, 7, 16, 12, 15, 15	ringm2neg 18598	. . 3 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> ( ( ( invg ` R ) ` X ) .x. ( ( invg ` R ) ` X ) ) = ( X .x. X ) )
68	66, 67	breqtrd 4679	. 2 |- ( ( ( R e. oRing /\ X e. B ) /\ -. .0. .<_ X ) -> .0. .<_ ( X .x. X ) )
69	9, 68	pm2.61dan 832	1 |- ( ( R e. oRing /\ X e. B ) -> .0. .<_ ( X .x. X ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   _I cid 5023   Or wor 5034   |` cres 5116   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   lecple 15948   0gc0g 16100   Posetcpo 16940   ltcplt 16941  Tosetctos 17033   Grpcgrp 17422   invgcminusg 17423   Ringcrg 18547  oMndcomnd 29697  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-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-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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-preset 16928  df-poset 16946  df-plt 16958  df-toset 17034  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-omnd 29699  df-ogrp 29700  df-orng 29797

This theorem is referenced by:  orng0le1  29812