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Theorem archiabllem2a 28190
Description: Lemma for archiabl 28194, which requires the group to be both left- and right-ordered. (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 )
archiabllem2.1  |-  .+  =  ( +g  `  W )
archiabllem2.2  |-  ( ph  ->  (oppg
`  W )  e. oGrp
)
archiabllem2.3  |-  ( (
ph  /\  a  e.  B  /\  .0.  .<  a
)  ->  E. b  e.  B  (  .0.  .< 
b  /\  b  .<  a ) )
archiabllem2a.4  |-  ( ph  ->  X  e.  B )
archiabllem2a.5  |-  ( ph  ->  .0.  .<  X )
Assertion
Ref Expression
archiabllem2a  |-  ( ph  ->  E. c  e.  B  (  .0.  .<  c  /\  ( c  .+  c
)  .<_  X ) )
Distinct variable groups:    a, b,
c, B    W, a,
b, c    X, a,
b, c    ph, a, b    .+ , a, b, c    .<_ , a, b, c    .< , a, b, c    .0. , a, b,
c
Allowed substitution hints:    ph( c)    .x. ( a,
b, c)

Proof of Theorem archiabllem2a
StepHypRef Expression
1 simpllr 761 . . . 4  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  ( b 
.+  b )  .<_  X )  ->  b  e.  B )
2 simplrl 762 . . . 4  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  ( b 
.+  b )  .<_  X )  ->  .0.  .< 
b )
3 simpr 459 . . . 4  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  ( b 
.+  b )  .<_  X )  ->  (
b  .+  b )  .<_  X )
4 breq2 4399 . . . . . 6  |-  ( c  =  b  ->  (  .0.  .<  c  <->  .0.  .<  b
) )
5 id 22 . . . . . . . 8  |-  ( c  =  b  ->  c  =  b )
65, 5oveq12d 6296 . . . . . . 7  |-  ( c  =  b  ->  (
c  .+  c )  =  ( b  .+  b ) )
76breq1d 4405 . . . . . 6  |-  ( c  =  b  ->  (
( c  .+  c
)  .<_  X  <->  ( b  .+  b )  .<_  X ) )
84, 7anbi12d 709 . . . . 5  |-  ( c  =  b  ->  (
(  .0.  .<  c  /\  ( c  .+  c
)  .<_  X )  <->  (  .0.  .< 
b  /\  ( b  .+  b )  .<_  X ) ) )
98rspcev 3160 . . . 4  |-  ( ( b  e.  B  /\  (  .0.  .<  b  /\  ( b  .+  b
)  .<_  X ) )  ->  E. c  e.  B  (  .0.  .<  c  /\  ( c  .+  c
)  .<_  X ) )
101, 2, 3, 9syl12anc 1228 . . 3  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  ( b 
.+  b )  .<_  X )  ->  E. c  e.  B  (  .0.  .< 
c  /\  ( c  .+  c )  .<_  X ) )
11 simplll 760 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ph )
12 archiabllem.g . . . . . 6  |-  ( ph  ->  W  e. oGrp )
13 ogrpgrp 28145 . . . . . 6  |-  ( W  e. oGrp  ->  W  e.  Grp )
1411, 12, 133syl 18 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  W  e.  Grp )
15 archiabllem2a.4 . . . . . 6  |-  ( ph  ->  X  e.  B )
1611, 15syl 17 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  X  e.  B )
17 simpllr 761 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  b  e.  B )
18 archiabllem.b . . . . . 6  |-  B  =  ( Base `  W
)
19 eqid 2402 . . . . . 6  |-  ( -g `  W )  =  (
-g `  W )
2018, 19grpsubcl 16442 . . . . 5  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  b  e.  B )  ->  ( X ( -g `  W ) b )  e.  B )
2114, 16, 17, 20syl3anc 1230 . . . 4  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( X
( -g `  W ) b )  e.  B
)
22 archiabllem.0 . . . . . . 7  |-  .0.  =  ( 0g `  W )
2318, 22, 19grpsubid 16446 . . . . . 6  |-  ( ( W  e.  Grp  /\  b  e.  B )  ->  ( b ( -g `  W ) b )  =  .0.  )
2414, 17, 23syl2anc 659 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b
( -g `  W ) b )  =  .0.  )
2511, 12syl 17 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  W  e. oGrp )
26 simplrr 763 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  b  .<  X )
27 archiabllem.t . . . . . . 7  |-  .<  =  ( lt `  W )
2818, 27, 19ogrpsublt 28164 . . . . . 6  |-  ( ( W  e. oGrp  /\  (
b  e.  B  /\  X  e.  B  /\  b  e.  B )  /\  b  .<  X )  ->  ( b (
-g `  W )
b )  .<  ( X ( -g `  W
) b ) )
2925, 17, 16, 17, 26, 28syl131anc 1243 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b
( -g `  W ) b )  .<  ( X ( -g `  W
) b ) )
3024, 29eqbrtrrd 4417 . . . 4  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  .0.  .<  ( X ( -g `  W
) b ) )
31 archiabllem2.1 . . . . . . 7  |-  .+  =  ( +g  `  W )
32 archiabllem2.2 . . . . . . . 8  |-  ( ph  ->  (oppg
`  W )  e. oGrp
)
3311, 32syl 17 . . . . . . 7  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  (oppg
`  W )  e. oGrp
)
3418, 31grpcl 16387 . . . . . . . . . 10  |-  ( ( W  e.  Grp  /\  b  e.  B  /\  b  e.  B )  ->  ( b  .+  b
)  e.  B )
3514, 17, 17, 34syl3anc 1230 . . . . . . . . 9  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b  .+  b )  e.  B
)
36 simpr 459 . . . . . . . . 9  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  X  .<  ( b  .+  b ) )
3718, 27, 19ogrpsublt 28164 . . . . . . . . 9  |-  ( ( W  e. oGrp  /\  ( X  e.  B  /\  ( b  .+  b
)  e.  B  /\  b  e.  B )  /\  X  .<  ( b 
.+  b ) )  ->  ( X (
-g `  W )
b )  .<  (
( b  .+  b
) ( -g `  W
) b ) )
3825, 16, 35, 17, 36, 37syl131anc 1243 . . . . . . . 8  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( X
( -g `  W ) b )  .<  (
( b  .+  b
) ( -g `  W
) b ) )
3918, 31, 19grpaddsubass 16452 . . . . . . . . . 10  |-  ( ( W  e.  Grp  /\  ( b  e.  B  /\  b  e.  B  /\  b  e.  B
) )  ->  (
( b  .+  b
) ( -g `  W
) b )  =  ( b  .+  (
b ( -g `  W
) b ) ) )
4014, 17, 17, 17, 39syl13anc 1232 . . . . . . . . 9  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( (
b  .+  b )
( -g `  W ) b )  =  ( b  .+  ( b ( -g `  W
) b ) ) )
4124oveq2d 6294 . . . . . . . . 9  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b  .+  ( b ( -g `  W ) b ) )  =  ( b 
.+  .0.  ) )
4218, 31, 22grprid 16405 . . . . . . . . . 10  |-  ( ( W  e.  Grp  /\  b  e.  B )  ->  ( b  .+  .0.  )  =  b )
4314, 17, 42syl2anc 659 . . . . . . . . 9  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b  .+  .0.  )  =  b )
4440, 41, 433eqtrd 2447 . . . . . . . 8  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( (
b  .+  b )
( -g `  W ) b )  =  b )
4538, 44breqtrd 4419 . . . . . . 7  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( X
( -g `  W ) b )  .<  b
)
4618, 27, 31, 14, 33, 21, 17, 21, 45ogrpaddltrd 28162 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<  ( ( X (
-g `  W )
b )  .+  b
) )
4718, 31, 19grpnpcan 16454 . . . . . . 7  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  b  e.  B )  ->  ( ( X (
-g `  W )
b )  .+  b
)  =  X )
4814, 16, 17, 47syl3anc 1230 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  b )  =  X )
4946, 48breqtrd 4419 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<  X )
50 ovex 6306 . . . . . . 7  |-  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) )  e.  _V
5150a1i 11 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) )  e.  _V )
52 archiabllem.e . . . . . . 7  |-  .<_  =  ( le `  W )
5352, 27pltle 15915 . . . . . 6  |-  ( ( W  e.  Grp  /\  ( ( X (
-g `  W )
b )  .+  ( X ( -g `  W
) b ) )  e.  _V  /\  X  e.  B )  ->  (
( ( X (
-g `  W )
b )  .+  ( X ( -g `  W
) b ) ) 
.<  X  ->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<_  X ) )
5414, 51, 16, 53syl3anc 1230 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( (
( X ( -g `  W ) b ) 
.+  ( X (
-g `  W )
b ) )  .<  X  ->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<_  X ) )
5549, 54mpd 15 . . . 4  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<_  X )
56 breq2 4399 . . . . . 6  |-  ( c  =  ( X (
-g `  W )
b )  ->  (  .0.  .<  c  <->  .0.  .<  ( X ( -g `  W
) b ) ) )
57 id 22 . . . . . . . 8  |-  ( c  =  ( X (
-g `  W )
b )  ->  c  =  ( X (
-g `  W )
b ) )
5857, 57oveq12d 6296 . . . . . . 7  |-  ( c  =  ( X (
-g `  W )
b )  ->  (
c  .+  c )  =  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) )
5958breq1d 4405 . . . . . 6  |-  ( c  =  ( X (
-g `  W )
b )  ->  (
( c  .+  c
)  .<_  X  <->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<_  X ) )
6056, 59anbi12d 709 . . . . 5  |-  ( c  =  ( X (
-g `  W )
b )  ->  (
(  .0.  .<  c  /\  ( c  .+  c
)  .<_  X )  <->  (  .0.  .< 
( X ( -g `  W ) b )  /\  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<_  X ) ) )
6160rspcev 3160 . . . 4  |-  ( ( ( X ( -g `  W ) b )  e.  B  /\  (  .0.  .<  ( X (
-g `  W )
b )  /\  (
( X ( -g `  W ) b ) 
.+  ( X (
-g `  W )
b ) )  .<_  X ) )  ->  E. c  e.  B  (  .0.  .<  c  /\  ( c  .+  c
)  .<_  X ) )
6221, 30, 55, 61syl12anc 1228 . . 3  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  E. c  e.  B  (  .0.  .< 
c  /\  ( c  .+  c )  .<_  X ) )
6312ad2antrr 724 . . . . 5  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  W  e. oGrp )
64 isogrp 28144 . . . . . 6  |-  ( W  e. oGrp 
<->  ( W  e.  Grp  /\  W  e. oMnd ) )
6564simprbi 462 . . . . 5  |-  ( W  e. oGrp  ->  W  e. oMnd )
66 omndtos 28147 . . . . 5  |-  ( W  e. oMnd  ->  W  e. Toset )
6763, 65, 663syl 18 . . . 4  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  W  e. Toset )
6863, 13syl 17 . . . . 5  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  W  e.  Grp )
69 simplr 754 . . . . 5  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  -> 
b  e.  B )
7068, 69, 69, 34syl3anc 1230 . . . 4  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  -> 
( b  .+  b
)  e.  B )
7115ad2antrr 724 . . . 4  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  X  e.  B )
7218, 52, 27tlt2 28104 . . . 4  |-  ( ( W  e. Toset  /\  (
b  .+  b )  e.  B  /\  X  e.  B )  ->  (
( b  .+  b
)  .<_  X  \/  X  .<  ( b  .+  b
) ) )
7367, 70, 71, 72syl3anc 1230 . . 3  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  -> 
( ( b  .+  b )  .<_  X  \/  X  .<  ( b  .+  b ) ) )
7410, 62, 73mpjaodan 787 . 2  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  E. c  e.  B  (  .0.  .<  c  /\  ( c  .+  c
)  .<_  X ) )
75 archiabllem2.3 . . . . 5  |-  ( (
ph  /\  a  e.  B  /\  .0.  .<  a
)  ->  E. b  e.  B  (  .0.  .< 
b  /\  b  .<  a ) )
76753expia 1199 . . . 4  |-  ( (
ph  /\  a  e.  B )  ->  (  .0.  .<  a  ->  E. b  e.  B  (  .0.  .< 
b  /\  b  .<  a ) ) )
7776ralrimiva 2818 . . 3  |-  ( ph  ->  A. a  e.  B  (  .0.  .<  a  ->  E. b  e.  B  (  .0.  .<  b  /\  b  .<  a ) ) )
78 archiabllem2a.5 . . 3  |-  ( ph  ->  .0.  .<  X )
79 breq2 4399 . . . . 5  |-  ( a  =  X  ->  (  .0.  .<  a  <->  .0.  .<  X ) )
80 breq2 4399 . . . . . . 7  |-  ( a  =  X  ->  (
b  .<  a  <->  b  .<  X ) )
8180anbi2d 702 . . . . . 6  |-  ( a  =  X  ->  (
(  .0.  .<  b  /\  b  .<  a )  <-> 
(  .0.  .<  b  /\  b  .<  X ) ) )
8281rexbidv 2918 . . . . 5  |-  ( a  =  X  ->  ( E. b  e.  B  (  .0.  .<  b  /\  b  .<  a )  <->  E. b  e.  B  (  .0.  .< 
b  /\  b  .<  X ) ) )
8379, 82imbi12d 318 . . . 4  |-  ( a  =  X  ->  (
(  .0.  .<  a  ->  E. b  e.  B  (  .0.  .<  b  /\  b  .<  a ) )  <-> 
(  .0.  .<  X  ->  E. b  e.  B  (  .0.  .<  b  /\  b  .<  X ) ) ) )
8483rspcv 3156 . . 3  |-  ( X  e.  B  ->  ( A. a  e.  B  (  .0.  .<  a  ->  E. b  e.  B  (  .0.  .<  b  /\  b  .<  a ) )  ->  (  .0.  .<  X  ->  E. b  e.  B  (  .0.  .<  b  /\  b  .<  X ) ) ) )
8515, 77, 78, 84syl3c 60 . 2  |-  ( ph  ->  E. b  e.  B  (  .0.  .<  b  /\  b  .<  X ) )
8674, 85r19.29a 2949 1  |-  ( ph  ->  E. c  e.  B  (  .0.  .<  c  /\  ( c  .+  c
)  .<_  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   _Vcvv 3059   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909   lecple 14916   0gc0g 15054   ltcplt 15894  Tosetctos 15987   Grpcgrp 16377   -gcsg 16379  .gcmg 16380  oppgcoppg 16704  oMndcomnd 28139  oGrpcogrp 28140  Archicarchi 28173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-tpos 6958  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-plusg 14922  df-ple 14929  df-0g 15056  df-preset 15881  df-poset 15899  df-plt 15912  df-toset 15988  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381  df-minusg 16382  df-sbg 16383  df-oppg 16705  df-omnd 28141  df-ogrp 28142
This theorem is referenced by:  archiabllem2c  28191
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