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Theorem archiabllem2a 27397
Description: Lemma for archiabl 27401, 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 4451 . . . . . 6  |-  ( c  =  b  ->  (  .0.  .<  c  <->  .0.  .<  b
) )
5 id 22 . . . . . . . 8  |-  ( c  =  b  ->  c  =  b )
65, 5oveq12d 6300 . . . . . . 7  |-  ( c  =  b  ->  (
c  .+  c )  =  ( b  .+  b ) )
76breq1d 4457 . . . . . 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 3214 . . . 4  |-  ( ( b  e.  B  /\  (  .0.  .<  b  /\  ( b  .+  b
)  .<_  X ) )  ->  E. c  e.  B  (  .0.  .<  c  /\  ( c  .+  c
)  .<_  X ) )
101, 2, 3, 9syl12anc 1226 . . 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 27351 . . . . . . 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 2467 . . . . . 6  |-  ( -g `  W )  =  (
-g `  W )
2119, 20grpsubcl 15916 . . . . 5  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  b  e.  B )  ->  ( X ( -g `  W ) b )  e.  B )
2215, 17, 18, 21syl3anc 1228 . . . 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 15920 . . . . . . 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 27371 . . . . . . 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 1241 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b
( -g `  W ) b )  .<  ( X ( -g `  W
) b ) )
3125, 30eqbrtrrd 4469 . . . . 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 15861 . . . . . . . . . . . . . 14  |-  ( ( W  e.  Grp  /\  b  e.  B  /\  b  e.  B )  ->  ( b  .+  b
)  e.  B )
3615, 18, 18, 35syl3anc 1228 . . . . . . . . . . . . 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 27371 . . . . . . . . . . . . 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 1241 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( X
( -g `  W ) b )  .<  (
( b  .+  b
) ( -g `  W
) b ) )
4019, 34, 20grpaddsubass 15926 . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( (
b  .+  b )
( -g `  W ) b )  =  ( b  .+  ( b ( -g `  W
) b ) ) )
4225oveq2d 6298 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( b  .+  ( b ( -g `  W ) b ) )  =  ( b 
.+  .0.  ) )
4319, 34, 23grprid 15879 . . . . . . . . . . . . . 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 2512 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( (
b  .+  b )
( -g `  W ) b )  =  b )
4639, 45breqtrd 4471 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( X
( -g `  W ) b )  .<  b
)
47 eqid 2467 . . . . . . . . . . . . . 14  |-  (oppg `  W
)  =  (oppg `  W
)
4847, 28oppglt 27301 . . . . . . . . . . . . 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 4458 . . . . . . . . . . 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 16178 . . . . . . . . . . 11  |-  B  =  ( Base `  (oppg `  W
) )
53 eqid 2467 . . . . . . . . . . 11  |-  ( lt
`  (oppg
`  W ) )  =  ( lt `  (oppg `  W ) )
54 eqid 2467 . . . . . . . . . . 11  |-  ( +g  `  (oppg
`  W ) )  =  ( +g  `  (oppg `  W
) )
5552, 53, 54ogrpaddlt 27367 . . . . . . . . . 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 1241 . . . . . . . . 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 4458 . . . . . . . . 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 16176 . . . . . . . 8  |-  ( ( X ( -g `  W
) b ) ( +g  `  (oppg `  W
) ) ( X ( -g `  W
) b ) )  =  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) )
6034, 47, 54oppgplus 16176 . . . . . . . 8  |-  ( b ( +g  `  (oppg `  W
) ) ( X ( -g `  W
) b ) )  =  ( ( X ( -g `  W
) b )  .+  b )
6158, 59, 603brtr3g 4478 . . . . . . 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 15928 . . . . . . . 8  |-  ( ( W  e.  Grp  /\  X  e.  B  /\  b  e.  B )  ->  ( ( X (
-g `  W )
b )  .+  b
)  =  X )
6315, 17, 18, 62syl3anc 1228 . . . . . . 7  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  b )  =  X )
6461, 63breqtrd 4471 . . . . . 6  |-  ( ( ( ( ph  /\  b  e.  B )  /\  (  .0.  .<  b  /\  b  .<  X ) )  /\  X  .<  ( b  .+  b ) )  ->  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) 
.<  X )
65 ovex 6307 . . . . . . . 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 15441 . . . . . . 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 1228 . . . . . 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 4451 . . . . . 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 6300 . . . . . . 7  |-  ( c  =  ( X (
-g `  W )
b )  ->  (
c  .+  c )  =  ( ( X ( -g `  W
) b )  .+  ( X ( -g `  W
) b ) ) )
7574breq1d 4457 . . . . . 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 3214 . . . 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 27354 . . . . 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 1228 . . . 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 27311 . . . 4  |-  ( ( W  e. Toset  /\  (
b  .+  b )  e.  B  /\  X  e.  B )  ->  (
( b  .+  b
)  .<_  X  \/  X  .<  ( b  .+  b
) ) )
8882, 85, 86, 87syl3anc 1228 . . 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 1198 . . . 4  |-  ( (
ph  /\  a  e.  B )  ->  (  .0.  .<  a  ->  E. b  e.  B  (  .0.  .< 
b  /\  b  .<  a ) ) )
9291ralrimiva 2878 . . 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 4451 . . . . 5  |-  ( a  =  X  ->  (  .0.  .<  a  <->  .0.  .<  X ) )
95 breq2 4451 . . . . . . 7  |-  ( a  =  X  ->  (
b  .<  a  <->  b  .<  X ) )
9695anbi2d 703 . . . . . 6  |-  ( a  =  X  ->  (
(  .0.  .<  b  /\  b  .<  a )  <-> 
(  .0.  .<  b  /\  b  .<  X ) ) )
9796rexbidv 2973 . . . . 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 3210 . . 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 3003 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14483   +g cplusg 14548   lecple 14555   0gc0g 14688   ltcplt 15421  Tosetctos 15513   Grpcgrp 15720   -gcsg 15723  .gcmg 15724  oppgcoppg 16172  oMndcomnd 27346  oGrpcogrp 27347  Archicarchi 27380
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 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-plusg 14561  df-ple 14568  df-0g 14690  df-poset 15426  df-plt 15438  df-toset 15514  df-mnd 15725  df-grp 15855  df-minusg 15856  df-sbg 15857  df-oppg 16173  df-omnd 27348  df-ogrp 27349
This theorem is referenced by:  archiabllem2c  27398
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