Step |
Hyp |
Ref |
Expression |
1 |
|
archiabllem.b |
|- B = ( Base ` W ) |
2 |
|
archiabllem.0 |
|- .0. = ( 0g ` W ) |
3 |
|
archiabllem.e |
|- .<_ = ( le ` W ) |
4 |
|
archiabllem.t |
|- .< = ( lt ` W ) |
5 |
|
archiabllem.m |
|- .x. = ( .g ` W ) |
6 |
|
archiabllem.g |
|- ( ph -> W e. oGrp ) |
7 |
|
archiabllem.a |
|- ( ph -> W e. Archi ) |
8 |
|
archiabllem2.1 |
|- .+ = ( +g ` W ) |
9 |
|
archiabllem2.2 |
|- ( ph -> ( oppG ` W ) e. oGrp ) |
10 |
|
archiabllem2.3 |
|- ( ( ph /\ a e. B /\ .0. .< a ) -> E. b e. B ( .0. .< b /\ b .< a ) ) |
11 |
|
archiabllem2a.4 |
|- ( ph -> X e. B ) |
12 |
|
archiabllem2a.5 |
|- ( ph -> .0. .< X ) |
13 |
|
simpllr |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ ( b .+ b ) .<_ X ) -> b e. B ) |
14 |
|
simplrl |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ ( b .+ b ) .<_ X ) -> .0. .< b ) |
15 |
|
simpr |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ ( b .+ b ) .<_ X ) -> ( b .+ b ) .<_ X ) |
16 |
|
breq2 |
|- ( c = b -> ( .0. .< c <-> .0. .< b ) ) |
17 |
|
id |
|- ( c = b -> c = b ) |
18 |
17 17
|
oveq12d |
|- ( c = b -> ( c .+ c ) = ( b .+ b ) ) |
19 |
18
|
breq1d |
|- ( c = b -> ( ( c .+ c ) .<_ X <-> ( b .+ b ) .<_ X ) ) |
20 |
16 19
|
anbi12d |
|- ( c = b -> ( ( .0. .< c /\ ( c .+ c ) .<_ X ) <-> ( .0. .< b /\ ( b .+ b ) .<_ X ) ) ) |
21 |
20
|
rspcev |
|- ( ( b e. B /\ ( .0. .< b /\ ( b .+ b ) .<_ X ) ) -> E. c e. B ( .0. .< c /\ ( c .+ c ) .<_ X ) ) |
22 |
13 14 15 21
|
syl12anc |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ ( b .+ b ) .<_ X ) -> E. c e. B ( .0. .< c /\ ( c .+ c ) .<_ X ) ) |
23 |
|
simplll |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ph ) |
24 |
|
ogrpgrp |
|- ( W e. oGrp -> W e. Grp ) |
25 |
23 6 24
|
3syl |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> W e. Grp ) |
26 |
23 11
|
syl |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> X e. B ) |
27 |
|
simpllr |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> b e. B ) |
28 |
|
eqid |
|- ( -g ` W ) = ( -g ` W ) |
29 |
1 28
|
grpsubcl |
|- ( ( W e. Grp /\ X e. B /\ b e. B ) -> ( X ( -g ` W ) b ) e. B ) |
30 |
25 26 27 29
|
syl3anc |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( X ( -g ` W ) b ) e. B ) |
31 |
1 2 28
|
grpsubid |
|- ( ( W e. Grp /\ b e. B ) -> ( b ( -g ` W ) b ) = .0. ) |
32 |
25 27 31
|
syl2anc |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( b ( -g ` W ) b ) = .0. ) |
33 |
23 6
|
syl |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> W e. oGrp ) |
34 |
|
simplrr |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> b .< X ) |
35 |
1 4 28
|
ogrpsublt |
|- ( ( W e. oGrp /\ ( b e. B /\ X e. B /\ b e. B ) /\ b .< X ) -> ( b ( -g ` W ) b ) .< ( X ( -g ` W ) b ) ) |
36 |
33 27 26 27 34 35
|
syl131anc |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( b ( -g ` W ) b ) .< ( X ( -g ` W ) b ) ) |
37 |
32 36
|
eqbrtrrd |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> .0. .< ( X ( -g ` W ) b ) ) |
38 |
23 9
|
syl |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( oppG ` W ) e. oGrp ) |
39 |
1 8
|
grpcl |
|- ( ( W e. Grp /\ b e. B /\ b e. B ) -> ( b .+ b ) e. B ) |
40 |
25 27 27 39
|
syl3anc |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( b .+ b ) e. B ) |
41 |
|
simpr |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> X .< ( b .+ b ) ) |
42 |
1 4 28
|
ogrpsublt |
|- ( ( 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 ) ) |
43 |
33 26 40 27 41 42
|
syl131anc |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( X ( -g ` W ) b ) .< ( ( b .+ b ) ( -g ` W ) b ) ) |
44 |
1 8 28
|
grpaddsubass |
|- ( ( W e. Grp /\ ( b e. B /\ b e. B /\ b e. B ) ) -> ( ( b .+ b ) ( -g ` W ) b ) = ( b .+ ( b ( -g ` W ) b ) ) ) |
45 |
25 27 27 27 44
|
syl13anc |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( b .+ b ) ( -g ` W ) b ) = ( b .+ ( b ( -g ` W ) b ) ) ) |
46 |
32
|
oveq2d |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( b .+ ( b ( -g ` W ) b ) ) = ( b .+ .0. ) ) |
47 |
1 8 2
|
grprid |
|- ( ( W e. Grp /\ b e. B ) -> ( b .+ .0. ) = b ) |
48 |
25 27 47
|
syl2anc |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( b .+ .0. ) = b ) |
49 |
45 46 48
|
3eqtrd |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( b .+ b ) ( -g ` W ) b ) = b ) |
50 |
43 49
|
breqtrd |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( X ( -g ` W ) b ) .< b ) |
51 |
1 4 8 25 38 30 27 30 50
|
ogrpaddltrd |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .< ( ( X ( -g ` W ) b ) .+ b ) ) |
52 |
1 8 28
|
grpnpcan |
|- ( ( W e. Grp /\ X e. B /\ b e. B ) -> ( ( X ( -g ` W ) b ) .+ b ) = X ) |
53 |
25 26 27 52
|
syl3anc |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( X ( -g ` W ) b ) .+ b ) = X ) |
54 |
51 53
|
breqtrd |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .< X ) |
55 |
|
ovexd |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) e. _V ) |
56 |
3 4
|
pltle |
|- ( ( 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 ) ) |
57 |
25 55 26 56
|
syl3anc |
|- ( ( ( ( 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 ) ) |
58 |
54 57
|
mpd |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .<_ X ) |
59 |
|
breq2 |
|- ( c = ( X ( -g ` W ) b ) -> ( .0. .< c <-> .0. .< ( X ( -g ` W ) b ) ) ) |
60 |
|
id |
|- ( c = ( X ( -g ` W ) b ) -> c = ( X ( -g ` W ) b ) ) |
61 |
60 60
|
oveq12d |
|- ( c = ( X ( -g ` W ) b ) -> ( c .+ c ) = ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) ) |
62 |
61
|
breq1d |
|- ( c = ( X ( -g ` W ) b ) -> ( ( c .+ c ) .<_ X <-> ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .<_ X ) ) |
63 |
59 62
|
anbi12d |
|- ( c = ( X ( -g ` W ) b ) -> ( ( .0. .< c /\ ( c .+ c ) .<_ X ) <-> ( .0. .< ( X ( -g ` W ) b ) /\ ( ( X ( -g ` W ) b ) .+ ( X ( -g ` W ) b ) ) .<_ X ) ) ) |
64 |
63
|
rspcev |
|- ( ( ( 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 ) ) |
65 |
30 37 58 64
|
syl12anc |
|- ( ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) /\ X .< ( b .+ b ) ) -> E. c e. B ( .0. .< c /\ ( c .+ c ) .<_ X ) ) |
66 |
6
|
ad2antrr |
|- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> W e. oGrp ) |
67 |
|
isogrp |
|- ( W e. oGrp <-> ( W e. Grp /\ W e. oMnd ) ) |
68 |
67
|
simprbi |
|- ( W e. oGrp -> W e. oMnd ) |
69 |
|
omndtos |
|- ( W e. oMnd -> W e. Toset ) |
70 |
66 68 69
|
3syl |
|- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> W e. Toset ) |
71 |
66 24
|
syl |
|- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> W e. Grp ) |
72 |
|
simplr |
|- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> b e. B ) |
73 |
71 72 72 39
|
syl3anc |
|- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> ( b .+ b ) e. B ) |
74 |
11
|
ad2antrr |
|- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> X e. B ) |
75 |
1 3 4
|
tlt2 |
|- ( ( W e. Toset /\ ( b .+ b ) e. B /\ X e. B ) -> ( ( b .+ b ) .<_ X \/ X .< ( b .+ b ) ) ) |
76 |
70 73 74 75
|
syl3anc |
|- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> ( ( b .+ b ) .<_ X \/ X .< ( b .+ b ) ) ) |
77 |
22 65 76
|
mpjaodan |
|- ( ( ( ph /\ b e. B ) /\ ( .0. .< b /\ b .< X ) ) -> E. c e. B ( .0. .< c /\ ( c .+ c ) .<_ X ) ) |
78 |
10
|
3expia |
|- ( ( ph /\ a e. B ) -> ( .0. .< a -> E. b e. B ( .0. .< b /\ b .< a ) ) ) |
79 |
78
|
ralrimiva |
|- ( ph -> A. a e. B ( .0. .< a -> E. b e. B ( .0. .< b /\ b .< a ) ) ) |
80 |
|
breq2 |
|- ( a = X -> ( .0. .< a <-> .0. .< X ) ) |
81 |
|
breq2 |
|- ( a = X -> ( b .< a <-> b .< X ) ) |
82 |
81
|
anbi2d |
|- ( a = X -> ( ( .0. .< b /\ b .< a ) <-> ( .0. .< b /\ b .< X ) ) ) |
83 |
82
|
rexbidv |
|- ( a = X -> ( E. b e. B ( .0. .< b /\ b .< a ) <-> E. b e. B ( .0. .< b /\ b .< X ) ) ) |
84 |
80 83
|
imbi12d |
|- ( a = X -> ( ( .0. .< a -> E. b e. B ( .0. .< b /\ b .< a ) ) <-> ( .0. .< X -> E. b e. B ( .0. .< b /\ b .< X ) ) ) ) |
85 |
84
|
rspcv |
|- ( 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 ) ) ) ) |
86 |
11 79 12 85
|
syl3c |
|- ( ph -> E. b e. B ( .0. .< b /\ b .< X ) ) |
87 |
77 86
|
r19.29a |
|- ( ph -> E. c e. B ( .0. .< c /\ ( c .+ c ) .<_ X ) ) |