Theorem archiabllem2a 29748

Description: Lemma for archiabl 29752, which requires the group to be both left- and right-ordered. (Contributed by Thierry Arnoux, 13-Apr-2018.)

Hypotheses

Ref	Expression
archiabllem.b	|- B = ( Base ` W )
archiabllem.0	|- .0. = ( 0g ` W )
archiabllem.e	|- .<_ = ( le ` W )
archiabllem.t	|- .< = ( lt ` W )
archiabllem.m	|- .x. = (.g ` W )
archiabllem.g	|- ( ph -> W e. oGrp )
archiabllem.a	|- ( ph -> W e. Archi )
archiabllem2.1	|- .+ = ( +g ` W )
archiabllem2.2	|- ( ph -> (oppg ` W ) e. oGrp )
archiabllem2.3	|- ( ( ph /\ a e. B /\ .0. .< a ) -> E. b e. B ( .0. .< b /\ b .< a ) )
archiabllem2a.4	|- ( ph -> X e. B )
archiabllem2a.5	|- ( ph -> .0. .< X )

Assertion

Ref	Expression
archiabllem2a	|- ( ph -> E. c e. B ( .0. .< c /\ ( c .+ c ) .<_ X ) )

Distinct variable groups:   a, b, c, B   W, a, b, c   X, a, b, c   ph, a, b   .+ , a, b, c   .<_ , a, b, c   .< , a, b, c   .0. , a, b, c

Allowed substitution hints:   ph( c)   .x. ( a, b, c)

Proof of Theorem archiabllem2a

Step	Hyp	Ref	Expression
1		simpllr 799	. . . 4 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ ( b .+ b ) .<_ X ) -> b e. B )
2		simplrl 800	. . . 4 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ ( b .+ b ) .<_ X ) -> .0. .< b )
3		simpr 477	. . . 4 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ ( b .+ b ) .<_ X ) -> ( b .+ b ) .<_ X )
4		breq2 4657	. . . . . 6 |- ( c = b -> ( .0. .< c <-> .0. .< b ) )
5		id 22	. . . . . . . 8 |- ( c = b -> c = b )
6	5, 5	oveq12d 6668	. . . . . . 7 |- ( c = b -> ( c .+ c ) = ( b .+ b ) )
7	6	breq1d 4663	. . . . . 6 |- ( c = b -> ( ( c .+ c ) .<_ X <-> ( b .+ b ) .<_ X ) )
8	4, 7	anbi12d 747	. . . . 5 |- ( c = b -> ( ( .0. .< c /\ ( c .+ c ) .<_ X ) <-> ( .0. .< b /\ ( b .+ b ) .<_ X ) ) )
9	8	rspcev 3309	. . . 4 |- ( ( b e. B /\ ( .0. .< b /\ ( b .+ b ) .<_ X ) ) -> E. c e. B ( .0. .< c /\ ( c .+ c ) .<_ X ) )
10	1, 2, 3, 9	syl12anc 1324	. . 3 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ ( b .+ b ) .<_ X ) -> E. c e. B ( .0. .< c /\ ( c .+ c ) .<_ X ) )
11		simplll 798	. . . . . 6 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ph )
12		archiabllem.g	. . . . . 6 |- ( ph -> W e. oGrp )
13		ogrpgrp 29703	. . . . . 6 |- ( W e. oGrp -> W e. Grp )
14	11, 12, 13	3syl 18	. . . . 5 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> W e. Grp )
15		archiabllem2a.4	. . . . . 6 |- ( ph -> X e. B )
16	11, 15	syl 17	. . . . 5 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> X e. B )
17		simpllr 799	. . . . 5 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> b e. B )
18		archiabllem.b	. . . . . 6 |- B = ( Base ` W )
19		eqid 2622	. . . . . 6 |- ( -g ` W ) = ( -g ` W )
20	18, 19	grpsubcl 17495	. . . . 5 |- ( ( W e. Grp /\ X e. B /\ b e. B ) -> ( X ( -g ` W ) b ) e. B )
21	14, 16, 17, 20	syl3anc 1326	. . . 4 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( X ( -g ` W ) b ) e. B )
22		archiabllem.0	. . . . . . 7 |- .0. = ( 0g ` W )
23	18, 22, 19	grpsubid 17499	. . . . . 6 |- ( ( W e. Grp /\ b e. B ) -> ( b ( -g ` W ) b ) = .0. )
24	14, 17, 23	syl2anc 693	. . . . 5 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( b ( -g ` W ) b ) = .0. )
25	11, 12	syl 17	. . . . . 6 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> W e. oGrp )
26		simplrr 801	. . . . . 6 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> b .< X )
27		archiabllem.t	. . . . . . 7 |- .< = ( lt ` W )
28	18, 27, 19	ogrpsublt 29722	. . . . . 6 |- ( ( W e. oGrp /\ ( b e. B /\ X e. B /\ b e. B ) /\ b .< X ) -> ( b ( -g ` W ) b ) .< ( X ( -g ` W ) b ) )
29	25, 17, 16, 17, 26, 28	syl131anc 1339	. . . . 5 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( b ( -g ` W ) b ) .< ( X ( -g ` W ) b ) )
30	24, 29	eqbrtrrd 4677	. . . 4 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> .0. .< ( X ( -g ` W ) b ) )
31		archiabllem2.1	. . . . . . 7 |- .+ = ( +g ` W )
32		archiabllem2.2	. . . . . . . 8 |- ( ph -> (oppg ` W ) e. oGrp )
33	11, 32	syl 17	. . . . . . 7 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> (oppg ` W ) e. oGrp )
34	18, 31	grpcl 17430	. . . . . . . . . 10 |- ( ( W e. Grp /\ b e. B /\ b e. B ) -> ( b .+ b ) e. B )
35	14, 17, 17, 34	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( b .+ b ) e. B )
36		simpr 477	. . . . . . . . 9 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> X .< ( b .+ b ) )
37	18, 27, 19	ogrpsublt 29722	. . . . . . . . 9 |- ( ( W e. oGrp /\ ( X e. B /\ ( b .+ b ) e. B /\ b e. B ) /\ X .< ( b .+ b ) ) -> ( X ( -g ` W ) b ) .< ( ( b .+ b ) ( -g ` W ) b ) )
38	25, 16, 35, 17, 36, 37	syl131anc 1339	. . . . . . . 8 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( X ( -g ` W ) b ) .< ( ( b .+ b ) ( -g ` W ) b ) )
39	18, 31, 19	grpaddsubass 17505	. . . . . . . . . 10 |- ( ( W e. Grp /\ ( b e. B /\ b e. B /\ b e. B ) ) -> ( ( b .+ b ) ( -g ` W ) b ) = ( b .+ ( b ( -g ` W ) b ) ) )
40	14, 17, 17, 17, 39	syl13anc 1328	. . . . . . . . 9 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( b .+ b ) ( -g ` W ) b ) = ( b .+ ( b ( -g ` W ) b ) ) )
41	24	oveq2d 6666	. . . . . . . . 9 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( b .+ ( b ( -g ` W ) b ) ) = ( b .+ .0. ) )
42	18, 31, 22	grprid 17453	. . . . . . . . . 10 |- ( ( W e. Grp /\ b e. B ) -> ( b .+ .0. ) = b )
43	14, 17, 42	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( b .+ .0. ) = b )
44	40, 41, 43	3eqtrd 2660	. . . . . . . 8 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( b .+ b ) ( -g ` W ) b ) = b )
45	38, 44	breqtrd 4679	. . . . . . 7 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( X ( -g ` W ) b ) .< b )
46	18, 27, 31, 14, 33, 21, 17, 21, 45	ogrpaddltrd 29720	. . . . . 6 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .< ( ( X ( -g ` W ) b ) .+ b ) )
47	18, 31, 19	grpnpcan 17507	. . . . . . 7 |- ( ( W e. Grp /\ X e. B /\ b e. B ) -> ( ( X ( -g ` W ) b ) .+ b ) = X )
48	14, 16, 17, 47	syl3anc 1326	. . . . . 6 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( X ( -g ` W ) b ) .+ b ) = X )
49	46, 48	breqtrd 4679	. . . . 5 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .< X )
50		ovexd 6680	. . . . . 6 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) e. _V )
51		archiabllem.e	. . . . . . 7 |- .<_ = ( le ` W )
52	51, 27	pltle 16961	. . . . . 6 |- ( ( W e. Grp /\ ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) e. _V /\ X e. B ) -> ( ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .< X -> ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .<_ X ) )
53	14, 50, 16, 52	syl3anc 1326	. . . . 5 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .< X -> ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .<_ X ) )
54	49, 53	mpd 15	. . . 4 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .<_ X )
55		breq2 4657	. . . . . 6 |- ( c = ( X ( -g ` W ) b ) -> ( .0. .< c <-> .0. .< ( X ( -g ` W ) b ) ) )
56		id 22	. . . . . . . 8 |- ( c = ( X ( -g ` W ) b ) -> c = ( X ( -g ` W ) b ) )
57	56, 56	oveq12d 6668	. . . . . . 7 |- ( c = ( X ( -g ` W ) b ) -> ( c .+ c ) = ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) )
58	57	breq1d 4663	. . . . . 6 |- ( c = ( X ( -g ` W ) b ) -> ( ( c .+ c ) .<_ X <-> ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .<_ X ) )
59	55, 58	anbi12d 747	. . . . 5 |- ( c = ( X ( -g ` W ) b ) -> ( ( .0. .< c /\ ( c .+ c ) .<_ X ) <-> ( .0. .< ( X ( -g ` W ) b ) /\ ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .<_ X ) ) )
60	59	rspcev 3309	. . . 4 |- ( ( ( X ( -g ` W ) b ) e. B /\ ( .0. .< ( X ( -g ` W ) b ) /\ ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .<_ X ) ) -> E. c e. B ( .0. .< c /\ ( c .+ c ) .<_ X ) )
61	21, 30, 54, 60	syl12anc 1324	. . 3 |- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> E. c e. B ( .0. .< c /\ ( c .+ c ) .<_ X ) )
62	12	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> W e. oGrp )
63		isogrp 29702	. . . . . 6 |- ( W e. oGrp <-> ( W e. Grp /\ W e. oMnd ) )
64	63	simprbi 480	. . . . 5 |- ( W e. oGrp -> W e. oMnd )
65		omndtos 29705	. . . . 5 |- ( W e. oMnd -> W e. Toset )
66	62, 64, 65	3syl 18	. . . 4 |- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> W e. Toset )
67	62, 13	syl 17	. . . . 5 |- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> W e. Grp )
68		simplr 792	. . . . 5 |- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> b e. B )
69	67, 68, 68, 34	syl3anc 1326	. . . 4 |- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> ( b .+ b ) e. B )
70	15	ad2antrr 762	. . . 4 |- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> X e. B )
71	18, 51, 27	tlt2 29664	. . . 4 |- ( ( W e. Toset /\ ( b .+ b ) e. B /\ X e. B ) -> ( ( b .+ b ) .<_ X \/ X .< ( b .+ b ) ) )
72	66, 69, 70, 71	syl3anc 1326	. . 3 |- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> ( ( b .+ b ) .<_ X \/ X .< ( b .+ b ) ) )
73	10, 61, 72	mpjaodan 827	. 2 |- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> E. c e. B ( .0. .< c /\ ( c .+ c ) .<_ X ) )
74		archiabllem2.3	. . . . 5 |- ( ( ph /\ a e. B /\ .0. .< a ) -> E. b e. B ( .0. .< b /\ b .< a ) )
75	74	3expia 1267	. . . 4 |- ( ( ph /\ a e. B ) -> ( .0. .< a -> E. b e. B ( .0. .< b /\ b .< a ) ) )
76	75	ralrimiva 2966	. . 3 |- ( ph -> A. a e. B ( .0. .< a -> E. b e. B ( .0. .< b /\ b .< a ) ) )
77		archiabllem2a.5	. . 3 |- ( ph -> .0. .< X )
78		breq2 4657	. . . . 5 |- ( a = X -> ( .0. .< a <-> .0. .< X ) )
79		breq2 4657	. . . . . . 7 |- ( a = X -> ( b .< a <-> b .< X ) )
80	79	anbi2d 740	. . . . . 6 |- ( a = X -> ( ( .0. .< b /\ b .< a ) <-> ( .0. .< b /\ b .< X ) ) )
81	80	rexbidv 3052	. . . . 5 |- ( a = X -> ( E. b e. B ( .0. .< b /\ b .< a ) <-> E. b e. B ( .0. .< b /\ b .< X ) ) )
82	78, 81	imbi12d 334	. . . 4 |- ( a = X -> ( ( .0. .< a -> E. b e. B ( .0. .< b /\ b .< a ) ) <-> ( .0. .< X -> E. b e. B ( .0. .< b /\ b .< X ) ) ) )
83	82	rspcv 3305	. . 3 |- ( X e. B -> ( A. a e. B ( .0. .< a -> E. b e. B ( .0. .< b /\ b .< a ) ) -> ( .0. .< X -> E. b e. B ( .0. .< b /\ b .< X ) ) ) )
84	15, 76, 77, 83	syl3c 66	. 2 |- ( ph -> E. b e. B ( .0. .< b /\ b .< X ) )
85	73, 84	r19.29a 3078	1 |- ( ph -> E. c e. B ( .0. .< c /\ ( c .+ c ) .<_ X ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   lecple 15948   0gc0g 16100   ltcplt 16941  Tosetctos 17033   Grpcgrp 17422   -gcsg 17424  ._gcmg 17540  opp_gcoppg 17775  oMndcomnd 29697  oGrpcogrp 29698  Archicarchi 29731

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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-dec 11494  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-ple 15961  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-sbg 17427  df-oppg 17776  df-omnd 29699  df-ogrp 29700

This theorem is referenced by:  archiabllem2c  29749