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Theorem archiabllem2a 26211
Description: Lemma for archiabl 26215, 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 758 . . . 4  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  ( b 
.+  b )  .<_  X )  ->  b  e.  B )
2 simplrl 759 . . . 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 4296 . . . . . 6  |-  ( c  =  b  ->  (  .0.  .<  c  <->  .0.  .<  b
) )
5 id 22 . . . . . . . 8  |-  ( c  =  b  ->  c  =  b )
65, 5oveq12d 6109 . . . . . . 7  |-  ( c  =  b  ->  (
c  .+  c )  =  ( b  .+  b ) )
76breq1d 4302 . . . . . 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 3073 . . . 4  |-  ( ( b  e.  B  /\  (  .0.  .<  b  /\  ( b  .+  b
)  .<_  X ) )  ->  E. c  e.  B  (  .0.  .<  c  /\  ( c  .+  c
)  .<_  X ) )
101, 2, 3, 9syl12anc 1216 . . 3  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  ( b 
.+  b )  .<_  X )  ->  E. c  e.  B  (  .0.  .< 
c  /\  ( c  .+  c )  .<_  X ) )
11 simplll 757 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ph )
12 archiabllem.g . . . . . 6  |-  ( ph  ->  W  e. oGrp )
13 isogrp 26165 . . . . . . 7  |-  ( W  e. oGrp 
<->  ( W  e.  Grp  /\  W  e. oMnd ) )
1413simplbi 460 . . . . . 6  |-  ( W  e. oGrp  ->  W  e.  Grp )
1511, 12, 143syl 20 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  W  e.  Grp )
16 archiabllem2a.4 . . . . . 6  |-  ( ph  ->  X  e.  B )
1711, 16syl 16 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  X  e.  B )
18 simpllr 758 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  b  e.  B )
19 archiabllem.b . . . . . 6  |-  B  =  ( Base `  W
)
20 eqid 2443 . . . . . 6  |-  ( -g `  W )  =  (
-g `  W )
2119, 20grpsubcl 15606 . . . . 5  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  b  e.  B )  ->  ( X ( -g `  W ) b )  e.  B )
2215, 17, 18, 21syl3anc 1218 . . . 4  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( X
( -g `  W ) b )  e.  B
)
23 archiabllem.0 . . . . . . . 8  |-  .0.  =  ( 0g `  W )
2419, 23, 20grpsubid 15610 . . . . . . 7  |-  ( ( W  e.  Grp  /\  b  e.  B )  ->  ( b ( -g `  W ) b )  =  .0.  )
2515, 18, 24syl2anc 661 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b
( -g `  W ) b )  =  .0.  )
2611, 12syl 16 . . . . . . 7  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  W  e. oGrp )
27 simplrr 760 . . . . . . 7  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  b  .<  X )
28 archiabllem.t . . . . . . . 8  |-  .<  =  ( lt `  W )
2919, 28, 20ogrpsublt 26185 . . . . . . 7  |-  ( ( W  e. oGrp  /\  (
b  e.  B  /\  X  e.  B  /\  b  e.  B )  /\  b  .<  X )  ->  ( b (
-g `  W )
b )  .<  ( X ( -g `  W
) b ) )
3026, 18, 17, 18, 27, 29syl131anc 1231 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b
( -g `  W ) b )  .<  ( X ( -g `  W
) b ) )
3125, 30eqbrtrrd 4314 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  .0.  .<  ( X ( -g `  W
) b ) )
32 archiabllem2.2 . . . . . . . . . . 11  |-  ( ph  ->  (oppg
`  W )  e. oGrp
)
3311, 32syl 16 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  (oppg
`  W )  e. oGrp
)
34 archiabllem2.1 . . . . . . . . . . . . . . 15  |-  .+  =  ( +g  `  W )
3519, 34grpcl 15551 . . . . . . . . . . . . . 14  |-  ( ( W  e.  Grp  /\  b  e.  B  /\  b  e.  B )  ->  ( b  .+  b
)  e.  B )
3615, 18, 18, 35syl3anc 1218 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b  .+  b )  e.  B
)
37 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  X  .<  ( b  .+  b ) )
3819, 28, 20ogrpsublt 26185 . . . . . . . . . . . . 13  |-  ( ( 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 ) )
3926, 17, 36, 18, 37, 38syl131anc 1231 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( X
( -g `  W ) b )  .<  (
( b  .+  b
) ( -g `  W
) b ) )
4019, 34, 20grpaddsubass 15615 . . . . . . . . . . . . . 14  |-  ( ( W  e.  Grp  /\  ( b  e.  B  /\  b  e.  B  /\  b  e.  B
) )  ->  (
( b  .+  b
) ( -g `  W
) b )  =  ( b  .+  (
b ( -g `  W
) b ) ) )
4115, 18, 18, 18, 40syl13anc 1220 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( (
b  .+  b )
( -g `  W ) b )  =  ( b  .+  ( b ( -g `  W
) b ) ) )
4225oveq2d 6107 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b  .+  ( b ( -g `  W ) b ) )  =  ( b 
.+  .0.  ) )
4319, 34, 23grprid 15569 . . . . . . . . . . . . . 14  |-  ( ( W  e.  Grp  /\  b  e.  B )  ->  ( b  .+  .0.  )  =  b )
4415, 18, 43syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b  .+  .0.  )  =  b )
4541, 42, 443eqtrd 2479 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( (
b  .+  b )
( -g `  W ) b )  =  b )
4639, 45breqtrd 4316 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( X
( -g `  W ) b )  .<  b
)
47 eqid 2443 . . . . . . . . . . . . . 14  |-  (oppg `  W
)  =  (oppg `  W
)
4847, 28oppglt 26115 . . . . . . . . . . . . 13  |-  ( W  e.  Grp  ->  .<  =  ( lt `  (oppg `  W
) ) )
4915, 48syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  .<  =  ( lt `  (oppg `  W
) ) )
5049breqd 4303 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .< 
b  <->  ( X (
-g `  W )
b ) ( lt
`  (oppg
`  W ) ) b ) )
5146, 50mpbid 210 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( X
( -g `  W ) b ) ( lt
`  (oppg
`  W ) ) b )
5247, 19oppgbas 15866 . . . . . . . . . . 11  |-  B  =  ( Base `  (oppg `  W
) )
53 eqid 2443 . . . . . . . . . . 11  |-  ( lt
`  (oppg
`  W ) )  =  ( lt `  (oppg `  W ) )
54 eqid 2443 . . . . . . . . . . 11  |-  ( +g  `  (oppg
`  W ) )  =  ( +g  `  (oppg `  W
) )
5552, 53, 54ogrpaddlt 26181 . . . . . . . . . 10  |-  ( ( (oppg
`  W )  e. oGrp  /\  ( ( X (
-g `  W )
b )  e.  B  /\  b  e.  B  /\  ( X ( -g `  W ) b )  e.  B )  /\  ( X ( -g `  W
) b ) ( lt `  (oppg `  W
) ) b )  ->  ( ( X ( -g `  W
) b ) ( +g  `  (oppg `  W
) ) ( X ( -g `  W
) b ) ) ( lt `  (oppg `  W
) ) ( b ( +g  `  (oppg `  W
) ) ( X ( -g `  W
) b ) ) )
5633, 22, 18, 22, 51, 55syl131anc 1231 . . . . . . . . 9  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b ) ( +g  `  (oppg `  W
) ) ( X ( -g `  W
) b ) ) ( lt `  (oppg `  W
) ) ( b ( +g  `  (oppg `  W
) ) ( X ( -g `  W
) b ) ) )
5749breqd 4303 . . . . . . . . 9  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( (
( X ( -g `  W ) b ) ( +g  `  (oppg `  W
) ) ( X ( -g `  W
) b ) ) 
.<  ( b ( +g  `  (oppg
`  W ) ) ( X ( -g `  W ) b ) )  <->  ( ( X ( -g `  W
) b ) ( +g  `  (oppg `  W
) ) ( X ( -g `  W
) b ) ) ( lt `  (oppg `  W
) ) ( b ( +g  `  (oppg `  W
) ) ( X ( -g `  W
) b ) ) ) )
5856, 57mpbird 232 . . . . . . . 8  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b ) ( +g  `  (oppg `  W
) ) ( X ( -g `  W
) b ) ) 
.<  ( b ( +g  `  (oppg
`  W ) ) ( X ( -g `  W ) b ) ) )
5934, 47, 54oppgplus 15864 . . . . . . . 8  |-  ( ( X ( -g `  W
) b ) ( +g  `  (oppg `  W
) ) ( X ( -g `  W
) b ) )  =  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) )
6034, 47, 54oppgplus 15864 . . . . . . . 8  |-  ( b ( +g  `  (oppg `  W
) ) ( X ( -g `  W
) b ) )  =  ( ( X ( -g `  W
) b )  .+  b )
6158, 59, 603brtr3g 4323 . . . . . . 7  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<  ( ( X (
-g `  W )
b )  .+  b
) )
6219, 34, 20grpnpcan 15617 . . . . . . . 8  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  b  e.  B )  ->  ( ( X (
-g `  W )
b )  .+  b
)  =  X )
6315, 17, 18, 62syl3anc 1218 . . . . . . 7  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  b )  =  X )
6461, 63breqtrd 4316 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<  X )
65 ovex 6116 . . . . . . . 8  |-  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) )  e.  _V
6665a1i 11 . . . . . . 7  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) )  e.  _V )
67 archiabllem.e . . . . . . . 8  |-  .<_  =  ( le `  W )
6867, 28pltle 15131 . . . . . . 7  |-  ( ( 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 ) )
6915, 66, 17, 68syl3anc 1218 . . . . . 6  |-  ( ( ( ( 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 ) )
7064, 69mpd 15 . . . . 5  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<_  X )
7131, 70jca 532 . . . 4  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  (  .0.  .< 
( X ( -g `  W ) b )  /\  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<_  X ) )
72 breq2 4296 . . . . . 6  |-  ( c  =  ( X (
-g `  W )
b )  ->  (  .0.  .<  c  <->  .0.  .<  ( X ( -g `  W
) b ) ) )
73 id 22 . . . . . . . 8  |-  ( c  =  ( X (
-g `  W )
b )  ->  c  =  ( X (
-g `  W )
b ) )
7473, 73oveq12d 6109 . . . . . . 7  |-  ( c  =  ( X (
-g `  W )
b )  ->  (
c  .+  c )  =  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) )
7574breq1d 4302 . . . . . 6  |-  ( c  =  ( X (
-g `  W )
b )  ->  (
( c  .+  c
)  .<_  X  <->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<_  X ) )
7672, 75anbi12d 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 ) ) )
7776rspcev 3073 . . . 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 ) )
7822, 71, 77syl2anc 661 . . 3  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  E. c  e.  B  (  .0.  .< 
c  /\  ( c  .+  c )  .<_  X ) )
7912ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  W  e. oGrp )
8013simprbi 464 . . . . 5  |-  ( W  e. oGrp  ->  W  e. oMnd )
81 omndtos 26168 . . . . 5  |-  ( W  e. oMnd  ->  W  e. Toset )
8279, 80, 813syl 20 . . . 4  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  W  e. Toset )
8379, 14syl 16 . . . . 5  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  W  e.  Grp )
84 simplr 754 . . . . 5  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  -> 
b  e.  B )
8583, 84, 84, 35syl3anc 1218 . . . 4  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  -> 
( b  .+  b
)  e.  B )
8616ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  X  e.  B )
8719, 67, 28tlt2 26125 . . . 4  |-  ( ( W  e. Toset  /\  (
b  .+  b )  e.  B  /\  X  e.  B )  ->  (
( b  .+  b
)  .<_  X  \/  X  .<  ( b  .+  b
) ) )
8882, 85, 86, 87syl3anc 1218 . . 3  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  -> 
( ( b  .+  b )  .<_  X  \/  X  .<  ( b  .+  b ) ) )
8910, 78, 88mpjaodan 784 . 2  |-  ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  ->  E. c  e.  B  (  .0.  .<  c  /\  ( c  .+  c
)  .<_  X ) )
90 archiabllem2.3 . . . . 5  |-  ( (
ph  /\  a  e.  B  /\  .0.  .<  a
)  ->  E. b  e.  B  (  .0.  .< 
b  /\  b  .<  a ) )
91903expia 1189 . . . 4  |-  ( (
ph  /\  a  e.  B )  ->  (  .0.  .<  a  ->  E. b  e.  B  (  .0.  .< 
b  /\  b  .<  a ) ) )
9291ralrimiva 2799 . . 3  |-  ( ph  ->  A. a  e.  B  (  .0.  .<  a  ->  E. b  e.  B  (  .0.  .<  b  /\  b  .<  a ) ) )
93 archiabllem2a.5 . . 3  |-  ( ph  ->  .0.  .<  X )
94 breq2 4296 . . . . 5  |-  ( a  =  X  ->  (  .0.  .<  a  <->  .0.  .<  X ) )
95 breq2 4296 . . . . . . 7  |-  ( a  =  X  ->  (
b  .<  a  <->  b  .<  X ) )
9695anbi2d 703 . . . . . 6  |-  ( a  =  X  ->  (
(  .0.  .<  b  /\  b  .<  a )  <-> 
(  .0.  .<  b  /\  b  .<  X ) ) )
9796rexbidv 2736 . . . . 5  |-  ( a  =  X  ->  ( E. b  e.  B  (  .0.  .<  b  /\  b  .<  a )  <->  E. b  e.  B  (  .0.  .< 
b  /\  b  .<  X ) ) )
9894, 97imbi12d 320 . . . 4  |-  ( a  =  X  ->  (
(  .0.  .<  a  ->  E. b  e.  B  (  .0.  .<  b  /\  b  .<  a ) )  <-> 
(  .0.  .<  X  ->  E. b  e.  B  (  .0.  .<  b  /\  b  .<  X ) ) ) )
9998rspcv 3069 . . 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 ) ) ) )
10016, 92, 93, 99syl3c 61 . 2  |-  ( ph  ->  E. b  e.  B  (  .0.  .<  b  /\  b  .<  X ) )
10189, 100r19.29a 2862 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 965    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   _Vcvv 2972   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   lecple 14245   0gc0g 14378   ltcplt 15111  Tosetctos 15203   Grpcgrp 15410   -gcsg 15413  .gcmg 15414  oppgcoppg 15860  oMndcomnd 26160  oGrpcogrp 26161  Archicarchi 26194
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-plusg 14251  df-ple 14258  df-0g 14380  df-poset 15116  df-plt 15128  df-toset 15204  df-mnd 15415  df-grp 15545  df-minusg 15546  df-sbg 15547  df-oppg 15861  df-omnd 26162  df-ogrp 26163
This theorem is referenced by:  archiabllem2c  26212
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