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Theorem archiabllem2a 27716
Description: Lemma for archiabl 27720, 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 760 . . . 4  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  ( b 
.+  b )  .<_  X )  ->  b  e.  B )
2 simplrl 761 . . . 4  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  ( b 
.+  b )  .<_  X )  ->  .0.  .< 
b )
3 simpr 461 . . . 4  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  ( b 
.+  b )  .<_  X )  ->  (
b  .+  b )  .<_  X )
4 breq2 4441 . . . . . 6  |-  ( c  =  b  ->  (  .0.  .<  c  <->  .0.  .<  b
) )
5 id 22 . . . . . . . 8  |-  ( c  =  b  ->  c  =  b )
65, 5oveq12d 6299 . . . . . . 7  |-  ( c  =  b  ->  (
c  .+  c )  =  ( b  .+  b ) )
76breq1d 4447 . . . . . 6  |-  ( c  =  b  ->  (
( c  .+  c
)  .<_  X  <->  ( b  .+  b )  .<_  X ) )
84, 7anbi12d 710 . . . . 5  |-  ( c  =  b  ->  (
(  .0.  .<  c  /\  ( c  .+  c
)  .<_  X )  <->  (  .0.  .< 
b  /\  ( b  .+  b )  .<_  X ) ) )
98rspcev 3196 . . . 4  |-  ( ( b  e.  B  /\  (  .0.  .<  b  /\  ( b  .+  b
)  .<_  X ) )  ->  E. c  e.  B  (  .0.  .<  c  /\  ( c  .+  c
)  .<_  X ) )
101, 2, 3, 9syl12anc 1227 . . 3  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  ( b 
.+  b )  .<_  X )  ->  E. c  e.  B  (  .0.  .< 
c  /\  ( c  .+  c )  .<_  X ) )
11 simplll 759 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ph )
12 archiabllem.g . . . . . 6  |-  ( ph  ->  W  e. oGrp )
13 ogrpgrp 27671 . . . . . 6  |-  ( W  e. oGrp  ->  W  e.  Grp )
1411, 12, 133syl 20 . . . . 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 16 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  X  e.  B )
17 simpllr 760 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  b  e.  B )
18 archiabllem.b . . . . . 6  |-  B  =  ( Base `  W
)
19 eqid 2443 . . . . . 6  |-  ( -g `  W )  =  (
-g `  W )
2018, 19grpsubcl 16097 . . . . 5  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  b  e.  B )  ->  ( X ( -g `  W ) b )  e.  B )
2114, 16, 17, 20syl3anc 1229 . . . 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 16101 . . . . . 6  |-  ( ( W  e.  Grp  /\  b  e.  B )  ->  ( b ( -g `  W ) b )  =  .0.  )
2414, 17, 23syl2anc 661 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b
( -g `  W ) b )  =  .0.  )
2511, 12syl 16 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  W  e. oGrp )
26 simplrr 762 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  b  .<  X )
27 archiabllem.t . . . . . . 7  |-  .<  =  ( lt `  W )
2818, 27, 19ogrpsublt 27690 . . . . . 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 1242 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b
( -g `  W ) b )  .<  ( X ( -g `  W
) b ) )
3024, 29eqbrtrrd 4459 . . . 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 16 . . . . . . 7  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  (oppg
`  W )  e. oGrp
)
3418, 31grpcl 16042 . . . . . . . . . 10  |-  ( ( W  e.  Grp  /\  b  e.  B  /\  b  e.  B )  ->  ( b  .+  b
)  e.  B )
3514, 17, 17, 34syl3anc 1229 . . . . . . . . 9  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b  .+  b )  e.  B
)
36 simpr 461 . . . . . . . . 9  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  X  .<  ( b  .+  b ) )
3718, 27, 19ogrpsublt 27690 . . . . . . . . 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 1242 . . . . . . . 8  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( X
( -g `  W ) b )  .<  (
( b  .+  b
) ( -g `  W
) b ) )
3918, 31, 19grpaddsubass 16107 . . . . . . . . . 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 1231 . . . . . . . . 9  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( (
b  .+  b )
( -g `  W ) b )  =  ( b  .+  ( b ( -g `  W
) b ) ) )
4124oveq2d 6297 . . . . . . . . 9  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b  .+  ( b ( -g `  W ) b ) )  =  ( b 
.+  .0.  ) )
4218, 31, 22grprid 16060 . . . . . . . . . 10  |-  ( ( W  e.  Grp  /\  b  e.  B )  ->  ( b  .+  .0.  )  =  b )
4314, 17, 42syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b  .+  .0.  )  =  b )
4440, 41, 433eqtrd 2488 . . . . . . . 8  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( (
b  .+  b )
( -g `  W ) b )  =  b )
4538, 44breqtrd 4461 . . . . . . 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 27688 . . . . . 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 16109 . . . . . . 7  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  b  e.  B )  ->  ( ( X (
-g `  W )
b )  .+  b
)  =  X )
4814, 16, 17, 47syl3anc 1229 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  b )  =  X )
4946, 48breqtrd 4461 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<  X )
50 ovex 6309 . . . . . . 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 15570 . . . . . 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 1229 . . . . 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 4441 . . . . . 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 6299 . . . . . . 7  |-  ( c  =  ( X (
-g `  W )
b )  ->  (
c  .+  c )  =  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) )
5958breq1d 4447 . . . . . 6  |-  ( c  =  ( X (
-g `  W )
b )  ->  (
( c  .+  c
)  .<_  X  <->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<_  X ) )
6056, 59anbi12d 710 . . . . 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 3196 . . . 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 1227 . . 3  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  E. c  e.  B  (  .0.  .< 
c  /\  ( c  .+  c )  .<_  X ) )
6312ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  W  e. oGrp )
64 isogrp 27670 . . . . . 6  |-  ( W  e. oGrp 
<->  ( W  e.  Grp  /\  W  e. oMnd ) )
6564simprbi 464 . . . . 5  |-  ( W  e. oGrp  ->  W  e. oMnd )
66 omndtos 27673 . . . . 5  |-  ( W  e. oMnd  ->  W  e. Toset )
6763, 65, 663syl 20 . . . 4  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  W  e. Toset )
6863, 13syl 16 . . . . 5  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  W  e.  Grp )
69 simplr 755 . . . . 5  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  -> 
b  e.  B )
7068, 69, 69, 34syl3anc 1229 . . . 4  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  -> 
( b  .+  b
)  e.  B )
7115ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  X  e.  B )
7218, 52, 27tlt2 27630 . . . 4  |-  ( ( W  e. Toset  /\  (
b  .+  b )  e.  B  /\  X  e.  B )  ->  (
( b  .+  b
)  .<_  X  \/  X  .<  ( b  .+  b
) ) )
7367, 70, 71, 72syl3anc 1229 . . 3  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  -> 
( ( b  .+  b )  .<_  X  \/  X  .<  ( b  .+  b ) ) )
7410, 62, 73mpjaodan 786 . 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 2857 . . 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 4441 . . . . 5  |-  ( a  =  X  ->  (  .0.  .<  a  <->  .0.  .<  X ) )
80 breq2 4441 . . . . . . 7  |-  ( a  =  X  ->  (
b  .<  a  <->  b  .<  X ) )
8180anbi2d 703 . . . . . 6  |-  ( a  =  X  ->  (
(  .0.  .<  b  /\  b  .<  a )  <-> 
(  .0.  .<  b  /\  b  .<  X ) ) )
8281rexbidv 2954 . . . . 5  |-  ( a  =  X  ->  ( E. b  e.  B  (  .0.  .<  b  /\  b  .<  a )  <->  E. b  e.  B  (  .0.  .< 
b  /\  b  .<  X ) ) )
8379, 82imbi12d 320 . . . 4  |-  ( a  =  X  ->  (
(  .0.  .<  a  ->  E. b  e.  B  (  .0.  .<  b  /\  b  .<  a ) )  <-> 
(  .0.  .<  X  ->  E. b  e.  B  (  .0.  .<  b  /\  b  .<  X ) ) ) )
8483rspcv 3192 . . 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 61 . 2  |-  ( ph  ->  E. b  e.  B  (  .0.  .<  b  /\  b  .<  X ) )
8674, 85r19.29a 2985 1  |-  ( ph  ->  E. c  e.  B  (  .0.  .<  c  /\  ( c  .+  c
)  .<_  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   _Vcvv 3095   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14614   +g cplusg 14679   lecple 14686   0gc0g 14819   ltcplt 15549  Tosetctos 15642   Grpcgrp 16032   -gcsg 16034  .gcmg 16035  oppgcoppg 16359  oMndcomnd 27665  oGrpcogrp 27666  Archicarchi 27699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-plusg 14692  df-ple 14699  df-0g 14821  df-preset 15536  df-poset 15554  df-plt 15567  df-toset 15643  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036  df-minusg 16037  df-sbg 16038  df-oppg 16360  df-omnd 27667  df-ogrp 27668
This theorem is referenced by:  archiabllem2c  27717
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