Step |
Hyp |
Ref |
Expression |
1 |
|
iccpartgtprec.m |
|- ( ph -> M e. NN ) |
2 |
|
iccpartgtprec.p |
|- ( ph -> P e. ( RePart ` M ) ) |
3 |
|
nnnn0 |
|- ( M e. NN -> M e. NN0 ) |
4 |
|
elnn0uz |
|- ( M e. NN0 <-> M e. ( ZZ>= ` 0 ) ) |
5 |
3 4
|
sylib |
|- ( M e. NN -> M e. ( ZZ>= ` 0 ) ) |
6 |
1 5
|
syl |
|- ( ph -> M e. ( ZZ>= ` 0 ) ) |
7 |
|
fzisfzounsn |
|- ( M e. ( ZZ>= ` 0 ) -> ( 0 ... M ) = ( ( 0 ..^ M ) u. { M } ) ) |
8 |
6 7
|
syl |
|- ( ph -> ( 0 ... M ) = ( ( 0 ..^ M ) u. { M } ) ) |
9 |
8
|
eleq2d |
|- ( ph -> ( i e. ( 0 ... M ) <-> i e. ( ( 0 ..^ M ) u. { M } ) ) ) |
10 |
|
elun |
|- ( i e. ( ( 0 ..^ M ) u. { M } ) <-> ( i e. ( 0 ..^ M ) \/ i e. { M } ) ) |
11 |
10
|
a1i |
|- ( ph -> ( i e. ( ( 0 ..^ M ) u. { M } ) <-> ( i e. ( 0 ..^ M ) \/ i e. { M } ) ) ) |
12 |
|
velsn |
|- ( i e. { M } <-> i = M ) |
13 |
12
|
a1i |
|- ( ph -> ( i e. { M } <-> i = M ) ) |
14 |
13
|
orbi2d |
|- ( ph -> ( ( i e. ( 0 ..^ M ) \/ i e. { M } ) <-> ( i e. ( 0 ..^ M ) \/ i = M ) ) ) |
15 |
9 11 14
|
3bitrd |
|- ( ph -> ( i e. ( 0 ... M ) <-> ( i e. ( 0 ..^ M ) \/ i = M ) ) ) |
16 |
1
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> M e. NN ) |
17 |
2
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> P e. ( RePart ` M ) ) |
18 |
|
fzossfz |
|- ( 0 ..^ M ) C_ ( 0 ... M ) |
19 |
18
|
a1i |
|- ( ph -> ( 0 ..^ M ) C_ ( 0 ... M ) ) |
20 |
19
|
sselda |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> i e. ( 0 ... M ) ) |
21 |
16 17 20
|
iccpartxr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( P ` i ) e. RR* ) |
22 |
|
nn0fz0 |
|- ( M e. NN0 <-> M e. ( 0 ... M ) ) |
23 |
3 22
|
sylib |
|- ( M e. NN -> M e. ( 0 ... M ) ) |
24 |
1 23
|
syl |
|- ( ph -> M e. ( 0 ... M ) ) |
25 |
1 2 24
|
iccpartxr |
|- ( ph -> ( P ` M ) e. RR* ) |
26 |
25
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( P ` M ) e. RR* ) |
27 |
1 2
|
iccpartltu |
|- ( ph -> A. k e. ( 0 ..^ M ) ( P ` k ) < ( P ` M ) ) |
28 |
|
fveq2 |
|- ( k = i -> ( P ` k ) = ( P ` i ) ) |
29 |
28
|
breq1d |
|- ( k = i -> ( ( P ` k ) < ( P ` M ) <-> ( P ` i ) < ( P ` M ) ) ) |
30 |
29
|
rspccv |
|- ( A. k e. ( 0 ..^ M ) ( P ` k ) < ( P ` M ) -> ( i e. ( 0 ..^ M ) -> ( P ` i ) < ( P ` M ) ) ) |
31 |
27 30
|
syl |
|- ( ph -> ( i e. ( 0 ..^ M ) -> ( P ` i ) < ( P ` M ) ) ) |
32 |
31
|
imp |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( P ` i ) < ( P ` M ) ) |
33 |
21 26 32
|
xrltled |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( P ` i ) <_ ( P ` M ) ) |
34 |
33
|
expcom |
|- ( i e. ( 0 ..^ M ) -> ( ph -> ( P ` i ) <_ ( P ` M ) ) ) |
35 |
|
fveq2 |
|- ( i = M -> ( P ` i ) = ( P ` M ) ) |
36 |
35
|
adantr |
|- ( ( i = M /\ ph ) -> ( P ` i ) = ( P ` M ) ) |
37 |
25
|
xrleidd |
|- ( ph -> ( P ` M ) <_ ( P ` M ) ) |
38 |
37
|
adantl |
|- ( ( i = M /\ ph ) -> ( P ` M ) <_ ( P ` M ) ) |
39 |
36 38
|
eqbrtrd |
|- ( ( i = M /\ ph ) -> ( P ` i ) <_ ( P ` M ) ) |
40 |
39
|
ex |
|- ( i = M -> ( ph -> ( P ` i ) <_ ( P ` M ) ) ) |
41 |
34 40
|
jaoi |
|- ( ( i e. ( 0 ..^ M ) \/ i = M ) -> ( ph -> ( P ` i ) <_ ( P ` M ) ) ) |
42 |
41
|
com12 |
|- ( ph -> ( ( i e. ( 0 ..^ M ) \/ i = M ) -> ( P ` i ) <_ ( P ` M ) ) ) |
43 |
15 42
|
sylbid |
|- ( ph -> ( i e. ( 0 ... M ) -> ( P ` i ) <_ ( P ` M ) ) ) |
44 |
43
|
ralrimiv |
|- ( ph -> A. i e. ( 0 ... M ) ( P ` i ) <_ ( P ` M ) ) |