Theorem archiabllem1b 29746

Description: Lemma for archiabl 29752. (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 )
archiabllem1.u	|- ( ph -> U e. B )
archiabllem1.p	|- ( ph -> .0. .< U )
archiabllem1.s	|- ( ( ph /\ x e. B /\ .0. .< x ) -> U .<_ x )

Assertion

Ref	Expression
archiabllem1b	|- ( ( ph /\ y e. B ) -> E. n e. ZZ y = ( n .x. U ) )

Distinct variable groups:   x, n, y, B   U, n, x   n, W, x, y   ph, n, x, y   .x. , n, x   .0. , n, x   .< , n, x   x, .<_

Allowed substitution hints:   .< ( y)   .x. ( y)   U( y)   .<_ ( y, n)   .0. ( y)

Proof of Theorem archiabllem1b

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0zd 11389	. . 3 |- ( ( ( ph /\ y e. B ) /\ y = .0. ) -> 0 e. ZZ )
2		simpr 477	. . . 4 |- ( ( ( ph /\ y e. B ) /\ y = .0. ) -> y = .0. )
3		archiabllem1.u	. . . . . 6 |- ( ph -> U e. B )
4		archiabllem.b	. . . . . . 7 |- B = ( Base ` W )
5		archiabllem.0	. . . . . . 7 |- .0. = ( 0g ` W )
6		archiabllem.m	. . . . . . 7 |- .x. = (.g ` W )
7	4, 5, 6	mulg0 17546	. . . . . 6 |- ( U e. B -> ( 0 .x. U ) = .0. )
8	3, 7	syl 17	. . . . 5 |- ( ph -> ( 0 .x. U ) = .0. )
9	8	ad2antrr 762	. . . 4 |- ( ( ( ph /\ y e. B ) /\ y = .0. ) -> ( 0 .x. U ) = .0. )
10	2, 9	eqtr4d 2659	. . 3 |- ( ( ( ph /\ y e. B ) /\ y = .0. ) -> y = ( 0 .x. U ) )
11		oveq1 6657	. . . . 5 |- ( n = 0 -> ( n .x. U ) = ( 0 .x. U ) )
12	11	eqeq2d 2632	. . . 4 |- ( n = 0 -> ( y = ( n .x. U ) <-> y = ( 0 .x. U ) ) )
13	12	rspcev 3309	. . 3 |- ( ( 0 e. ZZ /\ y = ( 0 .x. U ) ) -> E. n e. ZZ y = ( n .x. U ) )
14	1, 10, 13	syl2anc 693	. 2 |- ( ( ( ph /\ y e. B ) /\ y = .0. ) -> E. n e. ZZ y = ( n .x. U ) )
15		simplr 792	. . . . . . 7 |- ( ( ( ( ph /\ y e. B /\ y .< .0. ) /\ m e. NN ) /\ ( ( invg ` W ) ` y ) = ( m .x. U ) ) -> m e. NN )
16	15	nnzd 11481	. . . . . 6 |- ( ( ( ( ph /\ y e. B /\ y .< .0. ) /\ m e. NN ) /\ ( ( invg ` W ) ` y ) = ( m .x. U ) ) -> m e. ZZ )
17	16	znegcld 11484	. . . . 5 |- ( ( ( ( ph /\ y e. B /\ y .< .0. ) /\ m e. NN ) /\ ( ( invg ` W ) ` y ) = ( m .x. U ) ) -> -u m e. ZZ )
18	3	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ y e. B /\ y .< .0. ) -> U e. B )
19	18	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ph /\ y e. B /\ y .< .0. ) /\ m e. NN ) /\ ( ( invg ` W ) ` y ) = ( m .x. U ) ) -> U e. B )
20		eqid 2622	. . . . . . . 8 |- ( invg ` W ) = ( invg ` W )
21	4, 6, 20	mulgnegnn 17551	. . . . . . 7 |- ( ( m e. NN /\ U e. B ) -> ( -u m .x. U ) = ( ( invg ` W ) ` ( m .x. U ) ) )
22	15, 19, 21	syl2anc 693	. . . . . 6 |- ( ( ( ( ph /\ y e. B /\ y .< .0. ) /\ m e. NN ) /\ ( ( invg ` W ) ` y ) = ( m .x. U ) ) -> ( -u m .x. U ) = ( ( invg ` W ) ` ( m .x. U ) ) )
23		simpr 477	. . . . . . 7 |- ( ( ( ( ph /\ y e. B /\ y .< .0. ) /\ m e. NN ) /\ ( ( invg ` W ) ` y ) = ( m .x. U ) ) -> ( ( invg ` W ) ` y ) = ( m .x. U ) )
24	23	fveq2d 6195	. . . . . 6 |- ( ( ( ( ph /\ y e. B /\ y .< .0. ) /\ m e. NN ) /\ ( ( invg ` W ) ` y ) = ( m .x. U ) ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` y ) ) = ( ( invg ` W ) ` ( m .x. U ) ) )
25		archiabllem.g	. . . . . . . . . 10 |- ( ph -> W e. oGrp )
26	25	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ y e. B /\ y .< .0. ) -> W e. oGrp )
27		ogrpgrp 29703	. . . . . . . . 9 |- ( W e. oGrp -> W e. Grp )
28	26, 27	syl 17	. . . . . . . 8 |- ( ( ph /\ y e. B /\ y .< .0. ) -> W e. Grp )
29		simp2 1062	. . . . . . . 8 |- ( ( ph /\ y e. B /\ y .< .0. ) -> y e. B )
30	4, 20	grpinvinv 17482	. . . . . . . 8 |- ( ( W e. Grp /\ y e. B ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` y ) ) = y )
31	28, 29, 30	syl2anc 693	. . . . . . 7 |- ( ( ph /\ y e. B /\ y .< .0. ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` y ) ) = y )
32	31	ad2antrr 762	. . . . . 6 |- ( ( ( ( ph /\ y e. B /\ y .< .0. ) /\ m e. NN ) /\ ( ( invg ` W ) ` y ) = ( m .x. U ) ) -> ( ( invg ` W ) ` ( ( invg ` W ) ` y ) ) = y )
33	22, 24, 32	3eqtr2rd 2663	. . . . 5 |- ( ( ( ( ph /\ y e. B /\ y .< .0. ) /\ m e. NN ) /\ ( ( invg ` W ) ` y ) = ( m .x. U ) ) -> y = ( -u m .x. U ) )
34		oveq1 6657	. . . . . . 7 |- ( n = -u m -> ( n .x. U ) = ( -u m .x. U ) )
35	34	eqeq2d 2632	. . . . . 6 |- ( n = -u m -> ( y = ( n .x. U ) <-> y = ( -u m .x. U ) ) )
36	35	rspcev 3309	. . . . 5 |- ( ( -u m e. ZZ /\ y = ( -u m .x. U ) ) -> E. n e. ZZ y = ( n .x. U ) )
37	17, 33, 36	syl2anc 693	. . . 4 |- ( ( ( ( ph /\ y e. B /\ y .< .0. ) /\ m e. NN ) /\ ( ( invg ` W ) ` y ) = ( m .x. U ) ) -> E. n e. ZZ y = ( n .x. U ) )
38		archiabllem.e	. . . . 5 |- .<_ = ( le ` W )
39		archiabllem.t	. . . . 5 |- .< = ( lt ` W )
40		archiabllem.a	. . . . . 6 |- ( ph -> W e. Archi )
41	40	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ y e. B /\ y .< .0. ) -> W e. Archi )
42		archiabllem1.p	. . . . . 6 |- ( ph -> .0. .< U )
43	42	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ y e. B /\ y .< .0. ) -> .0. .< U )
44		simp1 1061	. . . . . 6 |- ( ( ph /\ y e. B /\ y .< .0. ) -> ph )
45		archiabllem1.s	. . . . . 6 |- ( ( ph /\ x e. B /\ .0. .< x ) -> U .<_ x )
46	44, 45	syl3an1 1359	. . . . 5 |- ( ( ( ph /\ y e. B /\ y .< .0. ) /\ x e. B /\ .0. .< x ) -> U .<_ x )
47	4, 20	grpinvcl 17467	. . . . . 6 |- ( ( W e. Grp /\ y e. B ) -> ( ( invg ` W ) ` y ) e. B )
48	28, 29, 47	syl2anc 693	. . . . 5 |- ( ( ph /\ y e. B /\ y .< .0. ) -> ( ( invg ` W ) ` y ) e. B )
49	4, 5	grpidcl 17450	. . . . . . . 8 |- ( W e. Grp -> .0. e. B )
50	28, 49	syl 17	. . . . . . 7 |- ( ( ph /\ y e. B /\ y .< .0. ) -> .0. e. B )
51		simp3 1063	. . . . . . 7 |- ( ( ph /\ y e. B /\ y .< .0. ) -> y .< .0. )
52		eqid 2622	. . . . . . . 8 |- ( +g ` W ) = ( +g ` W )
53	4, 39, 52	ogrpaddlt 29718	. . . . . . 7 |- ( ( W e. oGrp /\ ( y e. B /\ .0. e. B /\ ( ( invg ` W ) ` y ) e. B ) /\ y .< .0. ) -> ( y ( +g ` W ) ( ( invg ` W ) ` y ) ) .< ( .0. ( +g ` W ) ( ( invg ` W ) ` y ) ) )
54	26, 29, 50, 48, 51, 53	syl131anc 1339	. . . . . 6 |- ( ( ph /\ y e. B /\ y .< .0. ) -> ( y ( +g ` W ) ( ( invg ` W ) ` y ) ) .< ( .0. ( +g ` W ) ( ( invg ` W ) ` y ) ) )
55	4, 52, 5, 20	grprinv 17469	. . . . . . 7 |- ( ( W e. Grp /\ y e. B ) -> ( y ( +g ` W ) ( ( invg ` W ) ` y ) ) = .0. )
56	28, 29, 55	syl2anc 693	. . . . . 6 |- ( ( ph /\ y e. B /\ y .< .0. ) -> ( y ( +g ` W ) ( ( invg ` W ) ` y ) ) = .0. )
57	4, 52, 5	grplid 17452	. . . . . . 7 |- ( ( W e. Grp /\ ( ( invg ` W ) ` y ) e. B ) -> ( .0. ( +g ` W ) ( ( invg ` W ) ` y ) ) = ( ( invg ` W ) ` y ) )
58	28, 48, 57	syl2anc 693	. . . . . 6 |- ( ( ph /\ y e. B /\ y .< .0. ) -> ( .0. ( +g ` W ) ( ( invg ` W ) ` y ) ) = ( ( invg ` W ) ` y ) )
59	54, 56, 58	3brtr3d 4684	. . . . 5 |- ( ( ph /\ y e. B /\ y .< .0. ) -> .0. .< ( ( invg ` W ) ` y ) )
60	4, 5, 38, 39, 6, 26, 41, 18, 43, 46, 48, 59	archiabllem1a 29745	. . . 4 |- ( ( ph /\ y e. B /\ y .< .0. ) -> E. m e. NN ( ( invg ` W ) ` y ) = ( m .x. U ) )
61	37, 60	r19.29a 3078	. . 3 |- ( ( ph /\ y e. B /\ y .< .0. ) -> E. n e. ZZ y = ( n .x. U ) )
62	61	3expa 1265	. 2 |- ( ( ( ph /\ y e. B ) /\ y .< .0. ) -> E. n e. ZZ y = ( n .x. U ) )
63		nnssz 11397	. . 3 |- NN C_ ZZ
64	25	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ y e. B /\ .0. .< y ) -> W e. oGrp )
65	40	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ y e. B /\ .0. .< y ) -> W e. Archi )
66	3	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ y e. B /\ .0. .< y ) -> U e. B )
67	42	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ y e. B /\ .0. .< y ) -> .0. .< U )
68		simp1 1061	. . . . . 6 |- ( ( ph /\ y e. B /\ .0. .< y ) -> ph )
69	68, 45	syl3an1 1359	. . . . 5 |- ( ( ( ph /\ y e. B /\ .0. .< y ) /\ x e. B /\ .0. .< x ) -> U .<_ x )
70		simp2 1062	. . . . 5 |- ( ( ph /\ y e. B /\ .0. .< y ) -> y e. B )
71		simp3 1063	. . . . 5 |- ( ( ph /\ y e. B /\ .0. .< y ) -> .0. .< y )
72	4, 5, 38, 39, 6, 64, 65, 66, 67, 69, 70, 71	archiabllem1a 29745	. . . 4 |- ( ( ph /\ y e. B /\ .0. .< y ) -> E. n e. NN y = ( n .x. U ) )
73	72	3expa 1265	. . 3 |- ( ( ( ph /\ y e. B ) /\ .0. .< y ) -> E. n e. NN y = ( n .x. U ) )
74		ssrexv 3667	. . 3 |- ( NN C_ ZZ -> ( E. n e. NN y = ( n .x. U ) -> E. n e. ZZ y = ( n .x. U ) ) )
75	63, 73, 74	mpsyl 68	. 2 |- ( ( ( ph /\ y e. B ) /\ .0. .< y ) -> E. n e. ZZ y = ( n .x. U ) )
76		isogrp 29702	. . . . . 6 |- ( W e. oGrp <-> ( W e. Grp /\ W e. oMnd ) )
77	76	simprbi 480	. . . . 5 |- ( W e. oGrp -> W e. oMnd )
78		omndtos 29705	. . . . 5 |- ( W e. oMnd -> W e. Toset )
79	25, 77, 78	3syl 18	. . . 4 |- ( ph -> W e. Toset )
80	79	adantr 481	. . 3 |- ( ( ph /\ y e. B ) -> W e. Toset )
81		simpr 477	. . 3 |- ( ( ph /\ y e. B ) -> y e. B )
82	25, 27, 49	3syl 18	. . . 4 |- ( ph -> .0. e. B )
83	82	adantr 481	. . 3 |- ( ( ph /\ y e. B ) -> .0. e. B )
84	4, 39	tlt3 29665	. . 3 |- ( ( W e. Toset /\ y e. B /\ .0. e. B ) -> ( y = .0. \/ y .< .0. \/ .0. .< y ) )
85	80, 81, 83, 84	syl3anc 1326	. 2 |- ( ( ph /\ y e. B ) -> ( y = .0. \/ y .< .0. \/ .0. .< y ) )
86	14, 62, 75, 85	mpjao3dan 1395	1 |- ( ( ph /\ y e. B ) -> E. n e. ZZ y = ( n .x. U ) )

Syntax hints:   -> wi 4   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   -ucneg 10267   NNcn 11020   ZZcz 11377   Basecbs 15857   +g cplusg 15941   lecple 15948   0gc0g 16100   ltcplt 16941  Tosetctos 17033   Grpcgrp 17422   invgcminusg 17423  ._gcmg 17540  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-inf2 8538  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-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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  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-mulg 17541  df-omnd 29699  df-ogrp 29700  df-inftm 29732  df-archi 29733

This theorem is referenced by:  archiabllem1  29747