Step |
Hyp |
Ref |
Expression |
1 |
|
archirng.b |
|- B = ( Base ` W ) |
2 |
|
archirng.0 |
|- .0. = ( 0g ` W ) |
3 |
|
archirng.i |
|- .< = ( lt ` W ) |
4 |
|
archirng.l |
|- .<_ = ( le ` W ) |
5 |
|
archirng.x |
|- .x. = ( .g ` W ) |
6 |
|
archirng.1 |
|- ( ph -> W e. oGrp ) |
7 |
|
archirng.2 |
|- ( ph -> W e. Archi ) |
8 |
|
archirng.3 |
|- ( ph -> X e. B ) |
9 |
|
archirng.4 |
|- ( ph -> Y e. B ) |
10 |
|
archirng.5 |
|- ( ph -> .0. .< X ) |
11 |
|
archirng.6 |
|- ( ph -> .0. .< Y ) |
12 |
|
oveq1 |
|- ( m = 0 -> ( m .x. X ) = ( 0 .x. X ) ) |
13 |
12
|
breq2d |
|- ( m = 0 -> ( Y .<_ ( m .x. X ) <-> Y .<_ ( 0 .x. X ) ) ) |
14 |
|
oveq1 |
|- ( m = n -> ( m .x. X ) = ( n .x. X ) ) |
15 |
14
|
breq2d |
|- ( m = n -> ( Y .<_ ( m .x. X ) <-> Y .<_ ( n .x. X ) ) ) |
16 |
|
oveq1 |
|- ( m = ( n + 1 ) -> ( m .x. X ) = ( ( n + 1 ) .x. X ) ) |
17 |
16
|
breq2d |
|- ( m = ( n + 1 ) -> ( Y .<_ ( m .x. X ) <-> Y .<_ ( ( n + 1 ) .x. X ) ) ) |
18 |
|
isogrp |
|- ( W e. oGrp <-> ( W e. Grp /\ W e. oMnd ) ) |
19 |
18
|
simprbi |
|- ( W e. oGrp -> W e. oMnd ) |
20 |
|
omndtos |
|- ( W e. oMnd -> W e. Toset ) |
21 |
6 19 20
|
3syl |
|- ( ph -> W e. Toset ) |
22 |
|
ogrpgrp |
|- ( W e. oGrp -> W e. Grp ) |
23 |
6 22
|
syl |
|- ( ph -> W e. Grp ) |
24 |
1 2
|
grpidcl |
|- ( W e. Grp -> .0. e. B ) |
25 |
23 24
|
syl |
|- ( ph -> .0. e. B ) |
26 |
1 4 3
|
tltnle |
|- ( ( W e. Toset /\ .0. e. B /\ Y e. B ) -> ( .0. .< Y <-> -. Y .<_ .0. ) ) |
27 |
21 25 9 26
|
syl3anc |
|- ( ph -> ( .0. .< Y <-> -. Y .<_ .0. ) ) |
28 |
11 27
|
mpbid |
|- ( ph -> -. Y .<_ .0. ) |
29 |
1 2 5
|
mulg0 |
|- ( X e. B -> ( 0 .x. X ) = .0. ) |
30 |
8 29
|
syl |
|- ( ph -> ( 0 .x. X ) = .0. ) |
31 |
30
|
breq2d |
|- ( ph -> ( Y .<_ ( 0 .x. X ) <-> Y .<_ .0. ) ) |
32 |
28 31
|
mtbird |
|- ( ph -> -. Y .<_ ( 0 .x. X ) ) |
33 |
8 9
|
jca |
|- ( ph -> ( X e. B /\ Y e. B ) ) |
34 |
|
omndmnd |
|- ( W e. oMnd -> W e. Mnd ) |
35 |
6 19 34
|
3syl |
|- ( ph -> W e. Mnd ) |
36 |
1 2 5 4 3
|
isarchi2 |
|- ( ( W e. Toset /\ W e. Mnd ) -> ( W e. Archi <-> A. x e. B A. y e. B ( .0. .< x -> E. m e. NN y .<_ ( m .x. x ) ) ) ) |
37 |
36
|
biimpa |
|- ( ( ( W e. Toset /\ W e. Mnd ) /\ W e. Archi ) -> A. x e. B A. y e. B ( .0. .< x -> E. m e. NN y .<_ ( m .x. x ) ) ) |
38 |
21 35 7 37
|
syl21anc |
|- ( ph -> A. x e. B A. y e. B ( .0. .< x -> E. m e. NN y .<_ ( m .x. x ) ) ) |
39 |
|
breq2 |
|- ( x = X -> ( .0. .< x <-> .0. .< X ) ) |
40 |
|
oveq2 |
|- ( x = X -> ( m .x. x ) = ( m .x. X ) ) |
41 |
40
|
breq2d |
|- ( x = X -> ( y .<_ ( m .x. x ) <-> y .<_ ( m .x. X ) ) ) |
42 |
41
|
rexbidv |
|- ( x = X -> ( E. m e. NN y .<_ ( m .x. x ) <-> E. m e. NN y .<_ ( m .x. X ) ) ) |
43 |
39 42
|
imbi12d |
|- ( x = X -> ( ( .0. .< x -> E. m e. NN y .<_ ( m .x. x ) ) <-> ( .0. .< X -> E. m e. NN y .<_ ( m .x. X ) ) ) ) |
44 |
|
breq1 |
|- ( y = Y -> ( y .<_ ( m .x. X ) <-> Y .<_ ( m .x. X ) ) ) |
45 |
44
|
rexbidv |
|- ( y = Y -> ( E. m e. NN y .<_ ( m .x. X ) <-> E. m e. NN Y .<_ ( m .x. X ) ) ) |
46 |
45
|
imbi2d |
|- ( y = Y -> ( ( .0. .< X -> E. m e. NN y .<_ ( m .x. X ) ) <-> ( .0. .< X -> E. m e. NN Y .<_ ( m .x. X ) ) ) ) |
47 |
43 46
|
rspc2v |
|- ( ( X e. B /\ Y e. B ) -> ( A. x e. B A. y e. B ( .0. .< x -> E. m e. NN y .<_ ( m .x. x ) ) -> ( .0. .< X -> E. m e. NN Y .<_ ( m .x. X ) ) ) ) |
48 |
33 38 10 47
|
syl3c |
|- ( ph -> E. m e. NN Y .<_ ( m .x. X ) ) |
49 |
13 15 17 32 48
|
nn0min |
|- ( ph -> E. n e. NN0 ( -. Y .<_ ( n .x. X ) /\ Y .<_ ( ( n + 1 ) .x. X ) ) ) |
50 |
21
|
adantr |
|- ( ( ph /\ n e. NN0 ) -> W e. Toset ) |
51 |
23
|
adantr |
|- ( ( ph /\ n e. NN0 ) -> W e. Grp ) |
52 |
|
simpr |
|- ( ( ph /\ n e. NN0 ) -> n e. NN0 ) |
53 |
52
|
nn0zd |
|- ( ( ph /\ n e. NN0 ) -> n e. ZZ ) |
54 |
8
|
adantr |
|- ( ( ph /\ n e. NN0 ) -> X e. B ) |
55 |
1 5
|
mulgcl |
|- ( ( W e. Grp /\ n e. ZZ /\ X e. B ) -> ( n .x. X ) e. B ) |
56 |
51 53 54 55
|
syl3anc |
|- ( ( ph /\ n e. NN0 ) -> ( n .x. X ) e. B ) |
57 |
9
|
adantr |
|- ( ( ph /\ n e. NN0 ) -> Y e. B ) |
58 |
1 4 3
|
tltnle |
|- ( ( W e. Toset /\ ( n .x. X ) e. B /\ Y e. B ) -> ( ( n .x. X ) .< Y <-> -. Y .<_ ( n .x. X ) ) ) |
59 |
50 56 57 58
|
syl3anc |
|- ( ( ph /\ n e. NN0 ) -> ( ( n .x. X ) .< Y <-> -. Y .<_ ( n .x. X ) ) ) |
60 |
59
|
anbi1d |
|- ( ( ph /\ n e. NN0 ) -> ( ( ( n .x. X ) .< Y /\ Y .<_ ( ( n + 1 ) .x. X ) ) <-> ( -. Y .<_ ( n .x. X ) /\ Y .<_ ( ( n + 1 ) .x. X ) ) ) ) |
61 |
60
|
rexbidva |
|- ( ph -> ( E. n e. NN0 ( ( n .x. X ) .< Y /\ Y .<_ ( ( n + 1 ) .x. X ) ) <-> E. n e. NN0 ( -. Y .<_ ( n .x. X ) /\ Y .<_ ( ( n + 1 ) .x. X ) ) ) ) |
62 |
49 61
|
mpbird |
|- ( ph -> E. n e. NN0 ( ( n .x. X ) .< Y /\ Y .<_ ( ( n + 1 ) .x. X ) ) ) |