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Theorem archiabllem1b 27426
Description: Lemma for archiabl 27432 (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 10876 . . 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 15957 . . . . . 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 2511 . . 3  |-  ( ( ( ph  /\  y  e.  B )  /\  y  =  .0.  )  ->  y  =  ( 0  .x. 
U ) )
11 oveq1 6291 . . . . 5  |-  ( n  =  0  ->  (
n  .x.  U )  =  ( 0  .x. 
U ) )
1211eqeq2d 2481 . . . 4  |-  ( n  =  0  ->  (
y  =  ( n 
.x.  U )  <->  y  =  ( 0  .x.  U
) ) )
1312rspcev 3214 . . 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 10884 . . . . . . 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 3502 . . . . . 6  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  m  e.  ZZ )
1817znegcld 10968 . . . . 5  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  -u m  e.  ZZ )
1933ad2ant1 1017 . . . . . . . 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 2467 . . . . . . . 8  |-  ( invg `  W )  =  ( invg `  W )
224, 6, 21mulgnegnn 15962 . . . . . . 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 5870 . . . . . 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 1017 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  W  e. oGrp )
28 isogrp 27382 . . . . . . . . . 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 997 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  y  e.  B )
324, 21grpinvinv 15915 . . . . . . . 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 2515 . . . . 5  |-  ( ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  m  e.  NN )  /\  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )  ->  y  =  (
-u m  .x.  U
) )
36 oveq1 6291 . . . . . . 7  |-  ( n  =  -u m  ->  (
n  .x.  U )  =  ( -u m  .x.  U ) )
3736eqeq2d 2481 . . . . . 6  |-  ( n  =  -u m  ->  (
y  =  ( n 
.x.  U )  <->  y  =  ( -u m  .x.  U
) ) )
3837rspcev 3214 . . . . 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 1017 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  W  e. Archi )
44 archiabllem1.p . . . . . 6  |-  ( ph  ->  .0.  .<  U )
45443ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  .0.  .<  U )
46 simp1 996 . . . . . 6  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  ph )
47 archiabllem1.s . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  .0.  .<  x
)  ->  U  .<_  x )
4846, 47syl3an1 1261 . . . . 5  |-  ( ( ( ph  /\  y  e.  B  /\  y  .<  .0.  )  /\  x  e.  B  /\  .0.  .<  x )  ->  U  .<_  x )
494, 21grpinvcl 15905 . . . . . 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 15888 . . . . . . . 8  |-  ( W  e.  Grp  ->  .0.  e.  B )
5230, 51syl 16 . . . . . . 7  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  .0.  e.  B )
53 simp3 998 . . . . . . 7  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  y  .<  .0.  )
54 eqid 2467 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
554, 41, 54ogrpaddlt 27398 . . . . . . 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 1241 . . . . . 6  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  ( y ( +g  `  W ) ( ( invg `  W
) `  y )
)  .<  (  .0.  ( +g  `  W ) ( ( invg `  W ) `  y
) ) )
574, 54, 5, 21grprinv 15907 . . . . . . 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 15890 . . . . . . 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 4476 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  .0.  .<  ( ( invg `  W ) `
 y ) )
624, 5, 40, 41, 6, 27, 43, 19, 45, 48, 50, 61archiabllem1a 27425 . . . 4  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  E. m  e.  NN  ( ( invg `  W ) `  y
)  =  ( m 
.x.  U ) )
6339, 62r19.29a 3003 . . 3  |-  ( (
ph  /\  y  e.  B  /\  y  .<  .0.  )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
64633expa 1196 . 2  |-  ( ( ( ph  /\  y  e.  B )  /\  y  .<  .0.  )  ->  E. n  e.  ZZ  y  =  ( n  .x.  U ) )
65263ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  W  e. oGrp )
66423ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  W  e. Archi )
6733ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  U  e.  B )
68443ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  .0.  .<  U )
69 simp1 996 . . . . . 6  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  ph )
7069, 47syl3an1 1261 . . . . 5  |-  ( ( ( ph  /\  y  e.  B  /\  .0.  .<  y )  /\  x  e.  B  /\  .0.  .<  x )  ->  U  .<_  x )
71 simp2 997 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  y  e.  B )
72 simp3 998 . . . . 5  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  .0.  .<  y
)
734, 5, 40, 41, 6, 65, 66, 67, 68, 70, 71, 72archiabllem1a 27425 . . . 4  |-  ( (
ph  /\  y  e.  B  /\  .0.  .<  y
)  ->  E. n  e.  NN  y  =  ( n  .x.  U ) )
74733expa 1196 . . 3  |-  ( ( ( ph  /\  y  e.  B )  /\  .0.  .< 
y )  ->  E. n  e.  NN  y  =  ( n  .x.  U ) )
75 ssrexv 3565 . . 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 27385 . . . . 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 27343 . . 3  |-  ( ( W  e. Toset  /\  y  e.  B  /\  .0.  e.  B )  ->  (
y  =  .0.  \/  y  .<  .0.  \/  .0.  .< 
y ) )
8580, 81, 83, 84syl3anc 1228 . 2  |-  ( (
ph  /\  y  e.  B )  ->  (
y  =  .0.  \/  y  .<  .0.  \/  .0.  .< 
y ) )
8614, 64, 76, 85mpjao3dan 1295 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 972    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815    C_ wss 3476   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   0cc0 9492   -ucneg 9806   NNcn 10536   ZZcz 10864   Basecbs 14490   +g cplusg 14555   lecple 14562   0gc0g 14695   ltcplt 15428  Tosetctos 15520   Grpcgrp 15727   invgcminusg 15728  .gcmg 15731  oMndcomnd 27377  oGrpcogrp 27378  Archicarchi 27411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-seq 12076  df-0g 14697  df-poset 15433  df-plt 15445  df-toset 15521  df-mnd 15732  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-omnd 27379  df-ogrp 27380  df-inftm 27412  df-archi 27413
This theorem is referenced by:  archiabllem1  27427
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