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Theorem archiabllem1b 28347
Description: Lemma for archiabl 28353 (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.
StepHypRef Expression
1 0zd 10949 . . 3  |-  ( ( ( ph  /\  y  e.  B )  /\  y  =  .0.  )  ->  0  e.  ZZ )
2 simpr 462 . . . 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 )
74, 5, 6mulg0 16714 . . . . . 6  |-  ( U  e.  B  ->  (
0  .x.  U )  =  .0.  )
83, 7syl 17 . . . . 5  |-  ( ph  ->  ( 0  .x.  U
)  =  .0.  )
98ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  y  e.  B )  /\  y  =  .0.  )  ->  (
0  .x.  U )  =  .0.  )
102, 9eqtr4d 2473 . . 3  |-  ( ( ( ph  /\  y  e.  B )  /\  y  =  .0.  )  ->  y  =  ( 0  .x. 
U ) )
11 oveq1 6312 . . . . 5  |-  ( n  =  0  ->  (
n  .x.  U )  =  ( 0  .x. 
U ) )
1211eqeq2d 2443 . . . 4  |-  ( n  =  0  ->  (
y  =  ( n 
.x.  U )  <->  y  =  ( 0  .x.  U
) ) )
1312rspcev 3188 . . 3  |-  ( ( 0  e.  ZZ  /\  y  =  ( 0 
.x.  U ) )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
141, 10, 13syl2anc 665 . 2  |-  ( ( ( ph  /\  y  e.  B )  /\  y  =  .0.  )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
15 simplr 760 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  m  e.  NN )
1615nnzd 11039 . . . . . 6  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  m  e.  ZZ )
1716znegcld 11042 . . . . 5  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  -u m  e.  ZZ )
1833ad2ant1 1026 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  U  e.  B )
1918ad2antrr 730 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  U  e.  B
)
20 eqid 2429 . . . . . . . 8  |-  ( invg `  W )  =  ( invg `  W )
214, 6, 20mulgnegnn 16719 . . . . . . 7  |-  ( ( m  e.  NN  /\  U  e.  B )  ->  ( -u m  .x.  U )  =  ( ( invg `  W ) `  (
m  .x.  U )
) )
2215, 19, 21syl2anc 665 . . . . . 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 462 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  ( ( invg `  W ) `
 y )  =  ( m  .x.  U
) )
2423fveq2d 5885 . . . . . 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 )
26253ad2ant1 1026 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  W  e. oGrp )
27 ogrpgrp 28304 . . . . . . . . 9  |-  ( W  e. oGrp  ->  W  e.  Grp )
2826, 27syl 17 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  W  e.  Grp )
29 simp2 1006 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  y  e.  B )
304, 20grpinvinv 16672 . . . . . . . 8  |-  ( ( W  e.  Grp  /\  y  e.  B )  ->  ( ( invg `  W ) `  (
( invg `  W ) `  y
) )  =  y )
3128, 29, 30syl2anc 665 . . . . . . 7  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  ( ( invg `  W ) `  (
( invg `  W ) `  y
) )  =  y )
3231ad2antrr 730 . . . . . 6  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  ( ( invg `  W ) `
 ( ( invg `  W ) `
 y ) )  =  y )
3322, 24, 323eqtr2rd 2477 . . . . 5  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  y  =  (
-u m  .x.  U
) )
34 oveq1 6312 . . . . . . 7  |-  ( n  =  -u m  ->  (
n  .x.  U )  =  ( -u m  .x.  U ) )
3534eqeq2d 2443 . . . . . 6  |-  ( n  =  -u m  ->  (
y  =  ( n 
.x.  U )  <->  y  =  ( -u m  .x.  U
) ) )
3635rspcev 3188 . . . . 5  |-  ( (
-u m  e.  ZZ  /\  y  =  ( -u m  .x.  U ) )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
3717, 33, 36syl2anc 665 . . . 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 )
41403ad2ant1 1026 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  W  e. Archi )
42 archiabllem1.p . . . . . 6  |-  ( ph  ->  .0.  .<  U )
43423ad2ant1 1026 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  .0.  .<  U )
44 simp1 1005 . . . . . 6  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  ph )
45 archiabllem1.s . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  .0.  .<  x
)  ->  U  .<_  x )
4644, 45syl3an1 1297 . . . . 5  |-  ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  x  e.  B  /\  .0.  .<  x )  ->  U  .<_  x )
474, 20grpinvcl 16662 . . . . . 6  |-  ( ( W  e.  Grp  /\  y  e.  B )  ->  ( ( invg `  W ) `  y
)  e.  B )
4828, 29, 47syl2anc 665 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  ( ( invg `  W ) `  y
)  e.  B )
494, 5grpidcl 16645 . . . . . . . 8  |-  ( W  e.  Grp  ->  .0.  e.  B )
5028, 49syl 17 . . . . . . 7  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  .0.  e.  B )
51 simp3 1007 . . . . . . 7  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  y  .<  .0.  )
52 eqid 2429 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
534, 39, 52ogrpaddlt 28319 . . . . . . 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
) ) )
5426, 29, 50, 48, 51, 53syl131anc 1277 . . . . . 6  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  ( y ( +g  `  W ) ( ( invg `  W
) `  y )
)  .<  (  .0.  ( +g  `  W ) ( ( invg `  W ) `  y
) ) )
554, 52, 5, 20grprinv 16664 . . . . . . 7  |-  ( ( W  e.  Grp  /\  y  e.  B )  ->  ( y ( +g  `  W ) ( ( invg `  W
) `  y )
)  =  .0.  )
5628, 29, 55syl2anc 665 . . . . . 6  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  ( y ( +g  `  W ) ( ( invg `  W
) `  y )
)  =  .0.  )
574, 52, 5grplid 16647 . . . . . . 7  |-  ( ( W  e.  Grp  /\  ( ( invg `  W ) `  y
)  e.  B )  ->  (  .0.  ( +g  `  W ) ( ( invg `  W ) `  y
) )  =  ( ( invg `  W ) `  y
) )
5828, 48, 57syl2anc 665 . . . . . 6  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  (  .0.  ( +g  `  W ) ( ( invg `  W
) `  y )
)  =  ( ( invg `  W
) `  y )
)
5954, 56, 583brtr3d 4455 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  .0.  .<  ( ( invg `  W ) `
 y ) )
604, 5, 38, 39, 6, 26, 41, 18, 43, 46, 48, 59archiabllem1a 28346 . . . 4  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  E. m  e.  NN  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )
6137, 60r19.29a 2977 . . 3  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
62613expa 1205 . 2  |-  ( ( ( ph  /\  y  e.  B )  /\  y  .<  .0.  )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
63 nnssz 10957 . . 3  |-  NN  C_  ZZ
64253ad2ant1 1026 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  W  e. oGrp )
65403ad2ant1 1026 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  W  e. Archi )
6633ad2ant1 1026 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  U  e.  B )
67423ad2ant1 1026 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  .0.  .<  U )
68 simp1 1005 . . . . . 6  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  ph )
6968, 45syl3an1 1297 . . . . 5  |-  ( ( ( ph  /\  y  e.  B  /\  .0.  .<  y )  /\  x  e.  B  /\  .0.  .<  x )  ->  U  .<_  x )
70 simp2 1006 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  y  e.  B )
71 simp3 1007 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  .0.  .<  y
)
724, 5, 38, 39, 6, 64, 65, 66, 67, 69, 70, 71archiabllem1a 28346 . . . 4  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  E. n  e.  NN  y  =  ( n  .x.  U ) )
73723expa 1205 . . 3  |-  ( ( ( ph  /\  y  e.  B )  /\  .0.  .< 
y )  ->  E. n  e.  NN  y  =  ( n  .x.  U ) )
74 ssrexv 3532 . . 3  |-  ( NN  C_  ZZ  ->  ( E. n  e.  NN  y  =  ( n  .x.  U )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) ) )
7563, 73, 74mpsyl 65 . 2  |-  ( ( ( ph  /\  y  e.  B )  /\  .0.  .< 
y )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
76 isogrp 28303 . . . . . 6  |-  ( W  e. oGrp 
<->  ( W  e.  Grp  /\  W  e. oMnd ) )
7776simprbi 465 . . . . 5  |-  ( W  e. oGrp  ->  W  e. oMnd )
78 omndtos 28306 . . . . 5  |-  ( W  e. oMnd  ->  W  e. Toset )
7925, 77, 783syl 18 . . . 4  |-  ( ph  ->  W  e. Toset )
8079adantr 466 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  W  e. Toset )
81 simpr 462 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  y  e.  B )
8225, 27, 493syl 18 . . . 4  |-  ( ph  ->  .0.  e.  B )
8382adantr 466 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  .0.  e.  B )
844, 39tlt3 28264 . . 3  |-  ( ( W  e. Toset  /\  y  e.  B  /\  .0.  e.  B )  ->  (
y  =  .0.  \/  y  .<  .0.  \/  .0.  .< 
y ) )
8580, 81, 83, 84syl3anc 1264 . 2  |-  ( (
ph  /\  y  e.  B )  ->  (
y  =  .0.  \/  y  .<  .0.  \/  .0.  .< 
y ) )
8614, 62, 75, 85mpjao3dan 1331 1  |-  ( (
ph  /\  y  e.  B )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1870   E.wrex 2783    C_ wss 3442   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   0cc0 9538   -ucneg 9860   NNcn 10609   ZZcz 10937   Basecbs 15084   +g cplusg 15152   lecple 15159   0gc0g 15297   ltcplt 16137  Tosetctos 16230   Grpcgrp 16620   invgcminusg 16621  .gcmg 16623  oMndcomnd 28298  oGrpcogrp 28299  Archicarchi 28332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-seq 12211  df-0g 15299  df-preset 16124  df-poset 16142  df-plt 16155  df-toset 16231  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-sbg 16626  df-mulg 16627  df-omnd 28300  df-ogrp 28301  df-inftm 28333  df-archi 28334
This theorem is referenced by:  archiabllem1  28348
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