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Theorem archiabllem1b 26214
Description: Lemma for archiabl 26220 (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 10663 . . 3  |-  ( ( ( ph  /\  y  e.  B )  /\  y  =  .0.  )  ->  0  e.  ZZ )
2 simpr 461 . . . 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 15637 . . . . . 6  |-  ( U  e.  B  ->  (
0  .x.  U )  =  .0.  )
83, 7syl 16 . . . . 5  |-  ( ph  ->  ( 0  .x.  U
)  =  .0.  )
98ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  y  e.  B )  /\  y  =  .0.  )  ->  (
0  .x.  U )  =  .0.  )
102, 9eqtr4d 2478 . . 3  |-  ( ( ( ph  /\  y  e.  B )  /\  y  =  .0.  )  ->  y  =  ( 0  .x. 
U ) )
11 oveq1 6103 . . . . 5  |-  ( n  =  0  ->  (
n  .x.  U )  =  ( 0  .x. 
U ) )
1211eqeq2d 2454 . . . 4  |-  ( n  =  0  ->  (
y  =  ( n 
.x.  U )  <->  y  =  ( 0  .x.  U
) ) )
1312rspcev 3078 . . 3  |-  ( ( 0  e.  ZZ  /\  y  =  ( 0 
.x.  U ) )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
141, 10, 13syl2anc 661 . 2  |-  ( ( ( ph  /\  y  e.  B )  /\  y  =  .0.  )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
15 nnssz 10671 . . . . . . 7  |-  NN  C_  ZZ
16 simplr 754 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  m  e.  NN )
1715, 16sseldi 3359 . . . . . 6  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  m  e.  ZZ )
1817znegcld 10754 . . . . 5  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  -u m  e.  ZZ )
1933ad2ant1 1009 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  U  e.  B )
2019ad2antrr 725 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  U  e.  B
)
21 eqid 2443 . . . . . . . 8  |-  ( invg `  W )  =  ( invg `  W )
224, 6, 21mulgnegnn 15642 . . . . . . 7  |-  ( ( m  e.  NN  /\  U  e.  B )  ->  ( -u m  .x.  U )  =  ( ( invg `  W ) `  (
m  .x.  U )
) )
2316, 20, 22syl2anc 661 . . . . . 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 )
) )
24 simpr 461 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  ( ( invg `  W ) `
 y )  =  ( m  .x.  U
) )
2524fveq2d 5700 . . . . . 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 ) ) )
26 archiabllem.g . . . . . . . . . 10  |-  ( ph  ->  W  e. oGrp )
27263ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  W  e. oGrp )
28 isogrp 26170 . . . . . . . . . 10  |-  ( W  e. oGrp 
<->  ( W  e.  Grp  /\  W  e. oMnd ) )
2928simplbi 460 . . . . . . . . 9  |-  ( W  e. oGrp  ->  W  e.  Grp )
3027, 29syl 16 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  W  e.  Grp )
31 simp2 989 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  y  e.  B )
324, 21grpinvinv 15598 . . . . . . . 8  |-  ( ( W  e.  Grp  /\  y  e.  B )  ->  ( ( invg `  W ) `  (
( invg `  W ) `  y
) )  =  y )
3330, 31, 32syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  ( ( invg `  W ) `  (
( invg `  W ) `  y
) )  =  y )
3433ad2antrr 725 . . . . . 6  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  ( ( invg `  W ) `
 ( ( invg `  W ) `
 y ) )  =  y )
3523, 25, 343eqtr2rd 2482 . . . . 5  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  y  =  (
-u m  .x.  U
) )
36 oveq1 6103 . . . . . . 7  |-  ( n  =  -u m  ->  (
n  .x.  U )  =  ( -u m  .x.  U ) )
3736eqeq2d 2454 . . . . . 6  |-  ( n  =  -u m  ->  (
y  =  ( n 
.x.  U )  <->  y  =  ( -u m  .x.  U
) ) )
3837rspcev 3078 . . . . 5  |-  ( (
-u m  e.  ZZ  /\  y  =  ( -u m  .x.  U ) )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
3918, 35, 38syl2anc 661 . . . 4  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
40 archiabllem.e . . . . 5  |-  .<_  =  ( le `  W )
41 archiabllem.t . . . . 5  |-  .<  =  ( lt `  W )
42 archiabllem.a . . . . . 6  |-  ( ph  ->  W  e. Archi )
43423ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  W  e. Archi )
44 archiabllem1.p . . . . . 6  |-  ( ph  ->  .0.  .<  U )
45443ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  .0.  .<  U )
46 simp1 988 . . . . . 6  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  ph )
47 archiabllem1.s . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  .0.  .<  x
)  ->  U  .<_  x )
4846, 47syl3an1 1251 . . . . 5  |-  ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  x  e.  B  /\  .0.  .<  x )  ->  U  .<_  x )
494, 21grpinvcl 15588 . . . . . 6  |-  ( ( W  e.  Grp  /\  y  e.  B )  ->  ( ( invg `  W ) `  y
)  e.  B )
5030, 31, 49syl2anc 661 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  ( ( invg `  W ) `  y
)  e.  B )
514, 5grpidcl 15571 . . . . . . . 8  |-  ( W  e.  Grp  ->  .0.  e.  B )
5230, 51syl 16 . . . . . . 7  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  .0.  e.  B )
53 simp3 990 . . . . . . 7  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  y  .<  .0.  )
54 eqid 2443 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
554, 41, 54ogrpaddlt 26186 . . . . . . 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
) ) )
5627, 31, 52, 50, 53, 55syl131anc 1231 . . . . . 6  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  ( y ( +g  `  W ) ( ( invg `  W
) `  y )
)  .<  (  .0.  ( +g  `  W ) ( ( invg `  W ) `  y
) ) )
574, 54, 5, 21grprinv 15590 . . . . . . 7  |-  ( ( W  e.  Grp  /\  y  e.  B )  ->  ( y ( +g  `  W ) ( ( invg `  W
) `  y )
)  =  .0.  )
5830, 31, 57syl2anc 661 . . . . . 6  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  ( y ( +g  `  W ) ( ( invg `  W
) `  y )
)  =  .0.  )
594, 54, 5grplid 15573 . . . . . . 7  |-  ( ( W  e.  Grp  /\  ( ( invg `  W ) `  y
)  e.  B )  ->  (  .0.  ( +g  `  W ) ( ( invg `  W ) `  y
) )  =  ( ( invg `  W ) `  y
) )
6030, 50, 59syl2anc 661 . . . . . 6  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  (  .0.  ( +g  `  W ) ( ( invg `  W
) `  y )
)  =  ( ( invg `  W
) `  y )
)
6156, 58, 603brtr3d 4326 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  .0.  .<  ( ( invg `  W ) `
 y ) )
624, 5, 40, 41, 6, 27, 43, 19, 45, 48, 50, 61archiabllem1a 26213 . . . 4  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  E. m  e.  NN  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )
6339, 62r19.29a 2867 . . 3  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
64633expa 1187 . 2  |-  ( ( ( ph  /\  y  e.  B )  /\  y  .<  .0.  )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
65263ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  W  e. oGrp )
66423ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  W  e. Archi )
6733ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  U  e.  B )
68443ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  .0.  .<  U )
69 simp1 988 . . . . . 6  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  ph )
7069, 47syl3an1 1251 . . . . 5  |-  ( ( ( ph  /\  y  e.  B  /\  .0.  .<  y )  /\  x  e.  B  /\  .0.  .<  x )  ->  U  .<_  x )
71 simp2 989 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  y  e.  B )
72 simp3 990 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  .0.  .<  y
)
734, 5, 40, 41, 6, 65, 66, 67, 68, 70, 71, 72archiabllem1a 26213 . . . 4  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  E. n  e.  NN  y  =  ( n  .x.  U ) )
74733expa 1187 . . 3  |-  ( ( ( ph  /\  y  e.  B )  /\  .0.  .< 
y )  ->  E. n  e.  NN  y  =  ( n  .x.  U ) )
75 ssrexv 3422 . . 3  |-  ( NN  C_  ZZ  ->  ( E. n  e.  NN  y  =  ( n  .x.  U )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) ) )
7615, 74, 75mpsyl 63 . 2  |-  ( ( ( ph  /\  y  e.  B )  /\  .0.  .< 
y )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
7728simprbi 464 . . . . 5  |-  ( W  e. oGrp  ->  W  e. oMnd )
78 omndtos 26173 . . . . 5  |-  ( W  e. oMnd  ->  W  e. Toset )
7926, 77, 783syl 20 . . . 4  |-  ( ph  ->  W  e. Toset )
8079adantr 465 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  W  e. Toset )
81 simpr 461 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  y  e.  B )
8226, 29, 513syl 20 . . . 4  |-  ( ph  ->  .0.  e.  B )
8382adantr 465 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  .0.  e.  B )
844, 41tlt3 26131 . . 3  |-  ( ( W  e. Toset  /\  y  e.  B  /\  .0.  e.  B )  ->  (
y  =  .0.  \/  y  .<  .0.  \/  .0.  .< 
y ) )
8580, 81, 83, 84syl3anc 1218 . 2  |-  ( (
ph  /\  y  e.  B )  ->  (
y  =  .0.  \/  y  .<  .0.  \/  .0.  .< 
y ) )
8614, 64, 76, 85mpjao3dan 1285 1  |-  ( (
ph  /\  y  e.  B )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2721    C_ wss 3333   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   0cc0 9287   -ucneg 9601   NNcn 10327   ZZcz 10651   Basecbs 14179   +g cplusg 14243   lecple 14250   0gc0g 14383   ltcplt 15116  Tosetctos 15208   Grpcgrp 15415   invgcminusg 15416  .gcmg 15419  oMndcomnd 26165  oGrpcogrp 26166  Archicarchi 26199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-seq 11812  df-0g 14385  df-poset 15121  df-plt 15133  df-toset 15209  df-mnd 15420  df-grp 15550  df-minusg 15551  df-sbg 15552  df-mulg 15553  df-omnd 26167  df-ogrp 26168  df-inftm 26200  df-archi 26201
This theorem is referenced by:  archiabllem1  26215
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