Theorem archirng 29742

Description: Property of Archimedean ordered groups, framing positive Y between multiples of X. (Contributed by Thierry Arnoux, 12-Apr-2018.)

Hypotheses

Ref	Expression
archirng.b	|- B = ( Base ` W )
archirng.0	|- .0. = ( 0g ` W )
archirng.i	|- .< = ( lt ` W )
archirng.l	|- .<_ = ( le ` W )
archirng.x	|- .x. = (.g ` W )
archirng.1	|- ( ph -> W e. oGrp )
archirng.2	|- ( ph -> W e. Archi )
archirng.3	|- ( ph -> X e. B )
archirng.4	|- ( ph -> Y e. B )
archirng.5	|- ( ph -> .0. .< X )
archirng.6	|- ( ph -> .0. .< Y )

Assertion

Ref	Expression
archirng	|- ( ph -> E. n e. NN0 ( ( n .x. X ) .< Y /\ Y .<_ ( ( n + 1 ) .x. X ) ) )

Distinct variable groups:   n, X   n, Y   ph, n   .0. , n   .<_ , n   .< , n   .x. , n

Allowed substitution hints:   B( n)   W( n)

Proof of Theorem archirng

Dummy variables x m y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . 4 |- ( m = 0 -> ( m .x. X ) = ( 0 .x. X ) )
2	1	breq2d 4665	. . 3 |- ( m = 0 -> ( Y .<_ ( m .x. X ) <-> Y .<_ ( 0 .x. X ) ) )
3		oveq1 6657	. . . 4 |- ( m = n -> ( m .x. X ) = ( n .x. X ) )
4	3	breq2d 4665	. . 3 |- ( m = n -> ( Y .<_ ( m .x. X ) <-> Y .<_ ( n .x. X ) ) )
5		oveq1 6657	. . . 4 |- ( m = ( n + 1 ) -> ( m .x. X ) = ( ( n + 1 ) .x. X ) )
6	5	breq2d 4665	. . 3 |- ( m = ( n + 1 ) -> ( Y .<_ ( m .x. X ) <-> Y .<_ ( ( n + 1 ) .x. X ) ) )
7		archirng.6	. . . . 5 |- ( ph -> .0. .< Y )
8		archirng.1	. . . . . . 7 |- ( ph -> W e. oGrp )
9		isogrp 29702	. . . . . . . 8 |- ( W e. oGrp <-> ( W e. Grp /\ W e. oMnd ) )
10	9	simprbi 480	. . . . . . 7 |- ( W e. oGrp -> W e. oMnd )
11		omndtos 29705	. . . . . . 7 |- ( W e. oMnd -> W e. Toset )
12	8, 10, 11	3syl 18	. . . . . 6 |- ( ph -> W e. Toset )
13		ogrpgrp 29703	. . . . . . . 8 |- ( W e. oGrp -> W e. Grp )
14	8, 13	syl 17	. . . . . . 7 |- ( ph -> W e. Grp )
15		archirng.b	. . . . . . . 8 |- B = ( Base ` W )
16		archirng.0	. . . . . . . 8 |- .0. = ( 0g ` W )
17	15, 16	grpidcl 17450	. . . . . . 7 |- ( W e. Grp -> .0. e. B )
18	14, 17	syl 17	. . . . . 6 |- ( ph -> .0. e. B )
19		archirng.4	. . . . . 6 |- ( ph -> Y e. B )
20		archirng.l	. . . . . . 7 |- .<_ = ( le ` W )
21		archirng.i	. . . . . . 7 |- .< = ( lt ` W )
22	15, 20, 21	tltnle 29662	. . . . . 6 |- ( ( W e. Toset /\ .0. e. B /\ Y e. B ) -> ( .0. .< Y <-> -. Y .<_ .0. ) )
23	12, 18, 19, 22	syl3anc 1326	. . . . 5 |- ( ph -> ( .0. .< Y <-> -. Y .<_ .0. ) )
24	7, 23	mpbid 222	. . . 4 |- ( ph -> -. Y .<_ .0. )
25		archirng.3	. . . . . 6 |- ( ph -> X e. B )
26		archirng.x	. . . . . . 7 |- .x. = (.g ` W )
27	15, 16, 26	mulg0 17546	. . . . . 6 |- ( X e. B -> ( 0 .x. X ) = .0. )
28	25, 27	syl 17	. . . . 5 |- ( ph -> ( 0 .x. X ) = .0. )
29	28	breq2d 4665	. . . 4 |- ( ph -> ( Y .<_ ( 0 .x. X ) <-> Y .<_ .0. ) )
30	24, 29	mtbird 315	. . 3 |- ( ph -> -. Y .<_ ( 0 .x. X ) )
31	25, 19	jca 554	. . . 4 |- ( ph -> ( X e. B /\ Y e. B ) )
32		omndmnd 29704	. . . . . 6 |- ( W e. oMnd -> W e. Mnd )
33	8, 10, 32	3syl 18	. . . . 5 |- ( ph -> W e. Mnd )
34		archirng.2	. . . . 5 |- ( ph -> W e. Archi )
35	15, 16, 26, 20, 21	isarchi2 29739	. . . . . 6 |- ( ( W e. Toset /\ W e. Mnd ) -> ( W e. Archi <-> A. x e. B A. y e. B ( .0. .< x -> E. m e. NN y .<_ ( m .x. x ) ) ) )
36	35	biimpa 501	. . . . 5 |- ( ( ( W e. Toset /\ W e. Mnd ) /\ W e. Archi ) -> A. x e. B A. y e. B ( .0. .< x -> E. m e. NN y .<_ ( m .x. x ) ) )
37	12, 33, 34, 36	syl21anc 1325	. . . 4 |- ( ph -> A. x e. B A. y e. B ( .0. .< x -> E. m e. NN y .<_ ( m .x. x ) ) )
38		archirng.5	. . . 4 |- ( ph -> .0. .< X )
39		breq2 4657	. . . . . 6 |- ( x = X -> ( .0. .< x <-> .0. .< X ) )
40		oveq2 6658	. . . . . . . 8 |- ( x = X -> ( m .x. x ) = ( m .x. X ) )
41	40	breq2d 4665	. . . . . . 7 |- ( x = X -> ( y .<_ ( m .x. x ) <-> y .<_ ( m .x. X ) ) )
42	41	rexbidv 3052	. . . . . 6 |- ( x = X -> ( E. m e. NN y .<_ ( m .x. x ) <-> E. m e. NN y .<_ ( m .x. X ) ) )
43	39, 42	imbi12d 334	. . . . 5 |- ( x = X -> ( ( .0. .< x -> E. m e. NN y .<_ ( m .x. x ) ) <-> ( .0. .< X -> E. m e. NN y .<_ ( m .x. X ) ) ) )
44		breq1 4656	. . . . . . 7 |- ( y = Y -> ( y .<_ ( m .x. X ) <-> Y .<_ ( m .x. X ) ) )
45	44	rexbidv 3052	. . . . . 6 |- ( y = Y -> ( E. m e. NN y .<_ ( m .x. X ) <-> E. m e. NN Y .<_ ( m .x. X ) ) )
46	45	imbi2d 330	. . . . 5 |- ( y = Y -> ( ( .0. .< X -> E. m e. NN y .<_ ( m .x. X ) ) <-> ( .0. .< X -> E. m e. NN Y .<_ ( m .x. X ) ) ) )
47	43, 46	rspc2v 3322	. . . 4 |- ( ( X e. B /\ Y e. B ) -> ( A. x e. B A. y e. B ( .0. .< x -> E. m e. NN y .<_ ( m .x. x ) ) -> ( .0. .< X -> E. m e. NN Y .<_ ( m .x. X ) ) ) )
48	31, 37, 38, 47	syl3c 66	. . 3 |- ( ph -> E. m e. NN Y .<_ ( m .x. X ) )
49	2, 4, 6, 30, 48	nn0min 29567	. 2 |- ( ph -> E. n e. NN0 ( -. Y .<_ ( n .x. X ) /\ Y .<_ ( ( n + 1 ) .x. X ) ) )
50	12	adantr 481	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> W e. Toset )
51	14	adantr 481	. . . . . 6 |- ( ( ph /\ n e. NN0 ) -> W e. Grp )
52		simpr 477	. . . . . . 7 |- ( ( ph /\ n e. NN0 ) -> n e. NN0 )
53	52	nn0zd 11480	. . . . . 6 |- ( ( ph /\ n e. NN0 ) -> n e. ZZ )
54	25	adantr 481	. . . . . 6 |- ( ( ph /\ n e. NN0 ) -> X e. B )
55	15, 26	mulgcl 17559	. . . . . 6 |- ( ( W e. Grp /\ n e. ZZ /\ X e. B ) -> ( n .x. X ) e. B )
56	51, 53, 54, 55	syl3anc 1326	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> ( n .x. X ) e. B )
57	19	adantr 481	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> Y e. B )
58	15, 20, 21	tltnle 29662	. . . . 5 |- ( ( W e. Toset /\ ( n .x. X ) e. B /\ Y e. B ) -> ( ( n .x. X ) .< Y <-> -. Y .<_ ( n .x. X ) ) )
59	50, 56, 57, 58	syl3anc 1326	. . . 4 |- ( ( ph /\ n e. NN0 ) -> ( ( n .x. X ) .< Y <-> -. Y .<_ ( n .x. X ) ) )
60	59	anbi1d 741	. . 3 |- ( ( ph /\ n e. NN0 ) -> ( ( ( n .x. X ) .< Y /\ Y .<_ ( ( n + 1 ) .x. X ) ) <-> ( -. Y .<_ ( n .x. X ) /\ Y .<_ ( ( n + 1 ) .x. X ) ) ) )
61	60	rexbidva 3049	. 2 |- ( ph -> ( E. n e. NN0 ( ( n .x. X ) .< Y /\ Y .<_ ( ( n + 1 ) .x. X ) ) <-> E. n e. NN0 ( -. Y .<_ ( n .x. X ) /\ Y .<_ ( ( n + 1 ) .x. X ) ) ) )
62	49, 61	mpbird 247	1 |- ( ph -> E. n e. NN0 ( ( n .x. X ) .< Y /\ Y .<_ ( ( n + 1 ) .x. X ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   NNcn 11020   NN0cn0 11292   ZZcz 11377   Basecbs 15857   lecple 15948   0gc0g 16100   ltcplt 16941  Tosetctos 17033   Mndcmnd 17294   Grpcgrp 17422  ._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-mulg 17541  df-omnd 29699  df-ogrp 29700  df-inftm 29732  df-archi 29733

This theorem is referenced by:  archirngz  29743  archiabllem1a  29745