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Theorem vdwlem1 14867
Description: Lemma for vdw 14880. (Contributed by Mario Carneiro, 12-Sep-2014.)
Hypotheses
Ref Expression
vdwlem1.r  |-  ( ph  ->  R  e.  Fin )
vdwlem1.k  |-  ( ph  ->  K  e.  NN )
vdwlem1.w  |-  ( ph  ->  W  e.  NN )
vdwlem1.f  |-  ( ph  ->  F : ( 1 ... W ) --> R )
vdwlem1.a  |-  ( ph  ->  A  e.  NN )
vdwlem1.m  |-  ( ph  ->  M  e.  NN )
vdwlem1.d  |-  ( ph  ->  D : ( 1 ... M ) --> NN )
vdwlem1.s  |-  ( ph  ->  A. i  e.  ( 1 ... M ) ( ( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  C_  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } ) )
vdwlem1.i  |-  ( ph  ->  I  e.  ( 1 ... M ) )
vdwlem1.e  |-  ( ph  ->  ( F `  A
)  =  ( F `
 ( A  +  ( D `  I ) ) ) )
Assertion
Ref Expression
vdwlem1  |-  ( ph  ->  ( K  +  1 ) MonoAP  F )
Distinct variable groups:    A, i    D, i    i, I    i, K    i, F    i, M    ph, i    R, i    i, W

Proof of Theorem vdwlem1
Dummy variables  a 
c  d  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdwlem1.a . . . 4  |-  ( ph  ->  A  e.  NN )
2 vdwlem1.d . . . . 5  |-  ( ph  ->  D : ( 1 ... M ) --> NN )
3 vdwlem1.i . . . . 5  |-  ( ph  ->  I  e.  ( 1 ... M ) )
42, 3ffvelrnd 5975 . . . 4  |-  ( ph  ->  ( D `  I
)  e.  NN )
5 vdwlem1.k . . . . . . 7  |-  ( ph  ->  K  e.  NN )
65nnnn0d 10869 . . . . . 6  |-  ( ph  ->  K  e.  NN0 )
7 vdwapun 14860 . . . . . 6  |-  ( ( K  e.  NN0  /\  A  e.  NN  /\  ( D `  I )  e.  NN )  ->  ( A (AP `  ( K  +  1 ) ) ( D `  I
) )  =  ( { A }  u.  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) ) ) )
86, 1, 4, 7syl3anc 1264 . . . . 5  |-  ( ph  ->  ( A (AP `  ( K  +  1
) ) ( D `
 I ) )  =  ( { A }  u.  ( ( A  +  ( D `  I ) ) (AP
`  K ) ( D `  I ) ) ) )
91nnred 10568 . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR )
10 vdwlem1.m . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  NN )
11 nnuz 11138 . . . . . . . . . . . . . . 15  |-  NN  =  ( ZZ>= `  1 )
1210, 11syl6eleq 2510 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
13 eluzfz1 11750 . . . . . . . . . . . . . 14  |-  ( M  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... M
) )
1412, 13syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  ( 1 ... M ) )
152, 14ffvelrnd 5975 . . . . . . . . . . . 12  |-  ( ph  ->  ( D `  1
)  e.  NN )
161, 15nnaddcld 10600 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  ( D `  1 ) )  e.  NN )
1716nnred 10568 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  ( D `  1 ) )  e.  RR )
18 vdwlem1.w . . . . . . . . . . 11  |-  ( ph  ->  W  e.  NN )
1918nnred 10568 . . . . . . . . . 10  |-  ( ph  ->  W  e.  RR )
2015nnrpd 11283 . . . . . . . . . . . 12  |-  ( ph  ->  ( D `  1
)  e.  RR+ )
219, 20ltaddrpd 11315 . . . . . . . . . . 11  |-  ( ph  ->  A  <  ( A  +  ( D ` 
1 ) ) )
229, 17, 21ltled 9727 . . . . . . . . . 10  |-  ( ph  ->  A  <_  ( A  +  ( D ` 
1 ) ) )
23 vdwlem1.s . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. i  e.  ( 1 ... M ) ( ( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  C_  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } ) )
2423r19.21bi 2728 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  C_  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } ) )
25 cnvimass 5143 . . . . . . . . . . . . . . . . 17  |-  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  C_  dom  F
26 vdwlem1.f . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F : ( 1 ... W ) --> R )
27 fdm 5686 . . . . . . . . . . . . . . . . . 18  |-  ( F : ( 1 ... W ) --> R  ->  dom  F  =  ( 1 ... W ) )
2826, 27syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  dom  F  =  ( 1 ... W ) )
2925, 28syl5sseq 3448 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( `' F " { ( F `  ( A  +  ( D `  i )
) ) } ) 
C_  ( 1 ... W ) )
3029adantr 466 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  C_  (
1 ... W ) )
3124, 30sstrd 3410 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  C_  ( 1 ... W
) )
32 nnm1nn0 10855 . . . . . . . . . . . . . . . . . . . 20  |-  ( K  e.  NN  ->  ( K  -  1 )  e.  NN0 )
335, 32syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( K  -  1 )  e.  NN0 )
34 nn0uz 11137 . . . . . . . . . . . . . . . . . . 19  |-  NN0  =  ( ZZ>= `  0 )
3533, 34syl6eleq 2510 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( K  -  1 )  e.  ( ZZ>= ` 
0 ) )
36 eluzfz1 11750 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  -  1 )  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... ( K  -  1 ) ) )
3735, 36syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  e.  ( 0 ... ( K  - 
1 ) ) )
3837adantr 466 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  0  e.  ( 0 ... ( K  -  1 ) ) )
392ffvelrnda 5974 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( D `  i )  e.  NN )
4039nncnd 10569 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( D `  i )  e.  CC )
4140mul02d 9775 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
0  x.  ( D `
 i ) )  =  0 )
4241oveq2d 6258 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i ) ) )  =  ( ( A  +  ( D `  i ) )  +  0 ) )
431adantr 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  A  e.  NN )
4443, 39nnaddcld 10600 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  NN )
4544nncnd 10569 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  CC )
4645addid1d 9777 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) )  +  0 )  =  ( A  +  ( D `  i ) ) )
4742, 46eqtr2d 2457 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i ) ) ) )
48 oveq1 6249 . . . . . . . . . . . . . . . . . . 19  |-  ( m  =  0  ->  (
m  x.  ( D `
 i ) )  =  ( 0  x.  ( D `  i
) ) )
4948oveq2d 6258 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  0  ->  (
( A  +  ( D `  i ) )  +  ( m  x.  ( D `  i ) ) )  =  ( ( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i )
) ) )
5049eqeq2d 2432 . . . . . . . . . . . . . . . . 17  |-  ( m  =  0  ->  (
( A  +  ( D `  i ) )  =  ( ( A  +  ( D `
 i ) )  +  ( m  x.  ( D `  i
) ) )  <->  ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i ) ) ) ) )
5150rspcev 3118 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  ( 0 ... ( K  - 
1 ) )  /\  ( A  +  ( D `  i )
)  =  ( ( A  +  ( D `
 i ) )  +  ( 0  x.  ( D `  i
) ) ) )  ->  E. m  e.  ( 0 ... ( K  -  1 ) ) ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `
 i ) )  +  ( m  x.  ( D `  i
) ) ) )
5238, 47, 51syl2anc 665 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  E. m  e.  ( 0 ... ( K  -  1 ) ) ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `
 i ) )  +  ( m  x.  ( D `  i
) ) ) )
535adantr 466 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  K  e.  NN )
5453nnnn0d 10869 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  K  e.  NN0 )
55 vdwapval 14859 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  NN0  /\  ( A  +  ( D `  i )
)  e.  NN  /\  ( D `  i )  e.  NN )  -> 
( ( A  +  ( D `  i ) )  e.  ( ( A  +  ( D `
 i ) ) (AP `  K ) ( D `  i
) )  <->  E. m  e.  ( 0 ... ( K  -  1 ) ) ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `
 i ) )  +  ( m  x.  ( D `  i
) ) ) ) )
5654, 44, 39, 55syl3anc 1264 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) )  e.  ( ( A  +  ( D `
 i ) ) (AP `  K ) ( D `  i
) )  <->  E. m  e.  ( 0 ... ( K  -  1 ) ) ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `
 i ) )  +  ( m  x.  ( D `  i
) ) ) ) )
5752, 56mpbird 235 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  ( ( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) ) )
5831, 57sseldd 3401 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  ( 1 ... W
) )
5958ralrimiva 2773 . . . . . . . . . . . 12  |-  ( ph  ->  A. i  e.  ( 1 ... M ) ( A  +  ( D `  i ) )  e.  ( 1 ... W ) )
60 fveq2 5818 . . . . . . . . . . . . . . 15  |-  ( i  =  1  ->  ( D `  i )  =  ( D ` 
1 ) )
6160oveq2d 6258 . . . . . . . . . . . . . 14  |-  ( i  =  1  ->  ( A  +  ( D `  i ) )  =  ( A  +  ( D `  1 ) ) )
6261eleq1d 2484 . . . . . . . . . . . . 13  |-  ( i  =  1  ->  (
( A  +  ( D `  i ) )  e.  ( 1 ... W )  <->  ( A  +  ( D ` 
1 ) )  e.  ( 1 ... W
) ) )
6362rspcv 3114 . . . . . . . . . . . 12  |-  ( 1  e.  ( 1 ... M )  ->  ( A. i  e.  (
1 ... M ) ( A  +  ( D `
 i ) )  e.  ( 1 ... W )  ->  ( A  +  ( D `  1 ) )  e.  ( 1 ... W ) ) )
6414, 59, 63sylc 62 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  ( D `  1 ) )  e.  ( 1 ... W ) )
65 elfzle2 11747 . . . . . . . . . . 11  |-  ( ( A  +  ( D `
 1 ) )  e.  ( 1 ... W )  ->  ( A  +  ( D `  1 ) )  <_  W )
6664, 65syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  ( D `  1 ) )  <_  W )
679, 17, 19, 22, 66letrd 9736 . . . . . . . . 9  |-  ( ph  ->  A  <_  W )
681, 11syl6eleq 2510 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( ZZ>= ` 
1 ) )
6918nnzd 10983 . . . . . . . . . 10  |-  ( ph  ->  W  e.  ZZ )
70 elfz5 11736 . . . . . . . . . 10  |-  ( ( A  e.  ( ZZ>= ` 
1 )  /\  W  e.  ZZ )  ->  ( A  e.  ( 1 ... W )  <->  A  <_  W ) )
7168, 69, 70syl2anc 665 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  ( 1 ... W )  <-> 
A  <_  W )
)
7267, 71mpbird 235 . . . . . . . 8  |-  ( ph  ->  A  e.  ( 1 ... W ) )
73 eqidd 2423 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  =  ( F `
 A ) )
74 ffn 5682 . . . . . . . . 9  |-  ( F : ( 1 ... W ) --> R  ->  F  Fn  ( 1 ... W ) )
75 fniniseg 5955 . . . . . . . . 9  |-  ( F  Fn  ( 1 ... W )  ->  ( A  e.  ( `' F " { ( F `
 A ) } )  <->  ( A  e.  ( 1 ... W
)  /\  ( F `  A )  =  ( F `  A ) ) ) )
7626, 74, 753syl 18 . . . . . . . 8  |-  ( ph  ->  ( A  e.  ( `' F " { ( F `  A ) } )  <->  ( A  e.  ( 1 ... W
)  /\  ( F `  A )  =  ( F `  A ) ) ) )
7772, 73, 76mpbir2and 930 . . . . . . 7  |-  ( ph  ->  A  e.  ( `' F " { ( F `  A ) } ) )
7877snssd 4081 . . . . . 6  |-  ( ph  ->  { A }  C_  ( `' F " { ( F `  A ) } ) )
79 fveq2 5818 . . . . . . . . . . . 12  |-  ( i  =  I  ->  ( D `  i )  =  ( D `  I ) )
8079oveq2d 6258 . . . . . . . . . . 11  |-  ( i  =  I  ->  ( A  +  ( D `  i ) )  =  ( A  +  ( D `  I ) ) )
8180, 79oveq12d 6260 . . . . . . . . . 10  |-  ( i  =  I  ->  (
( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  =  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) ) )
8280fveq2d 5822 . . . . . . . . . . . 12  |-  ( i  =  I  ->  ( F `  ( A  +  ( D `  i ) ) )  =  ( F `  ( A  +  ( D `  I )
) ) )
8382sneqd 3946 . . . . . . . . . . 11  |-  ( i  =  I  ->  { ( F `  ( A  +  ( D `  i ) ) ) }  =  { ( F `  ( A  +  ( D `  I ) ) ) } )
8483imaeq2d 5123 . . . . . . . . . 10  |-  ( i  =  I  ->  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  =  ( `' F " { ( F `  ( A  +  ( D `  I ) ) ) } ) )
8581, 84sseq12d 3429 . . . . . . . . 9  |-  ( i  =  I  ->  (
( ( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  C_  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  <->  ( ( A  +  ( D `  I ) ) (AP
`  K ) ( D `  I ) )  C_  ( `' F " { ( F `
 ( A  +  ( D `  I ) ) ) } ) ) )
8685rspcv 3114 . . . . . . . 8  |-  ( I  e.  ( 1 ... M )  ->  ( A. i  e.  (
1 ... M ) ( ( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  C_  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  ->  (
( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) )  C_  ( `' F " { ( F `  ( A  +  ( D `  I ) ) ) } ) ) )
873, 23, 86sylc 62 . . . . . . 7  |-  ( ph  ->  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) )  C_  ( `' F " { ( F `  ( A  +  ( D `  I ) ) ) } ) )
88 vdwlem1.e . . . . . . . . 9  |-  ( ph  ->  ( F `  A
)  =  ( F `
 ( A  +  ( D `  I ) ) ) )
8988sneqd 3946 . . . . . . . 8  |-  ( ph  ->  { ( F `  A ) }  =  { ( F `  ( A  +  ( D `  I )
) ) } )
9089imaeq2d 5123 . . . . . . 7  |-  ( ph  ->  ( `' F " { ( F `  A ) } )  =  ( `' F " { ( F `  ( A  +  ( D `  I )
) ) } ) )
9187, 90sseqtr4d 3437 . . . . . 6  |-  ( ph  ->  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) )  C_  ( `' F " { ( F `  A ) } ) )
9278, 91unssd 3578 . . . . 5  |-  ( ph  ->  ( { A }  u.  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) ) ) 
C_  ( `' F " { ( F `  A ) } ) )
938, 92eqsstrd 3434 . . . 4  |-  ( ph  ->  ( A (AP `  ( K  +  1
) ) ( D `
 I ) ) 
C_  ( `' F " { ( F `  A ) } ) )
94 oveq1 6249 . . . . . 6  |-  ( a  =  A  ->  (
a (AP `  ( K  +  1 ) ) d )  =  ( A (AP `  ( K  +  1
) ) d ) )
9594sseq1d 3427 . . . . 5  |-  ( a  =  A  ->  (
( a (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { ( F `  A ) } )  <-> 
( A (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { ( F `  A ) } ) ) )
96 oveq2 6250 . . . . . 6  |-  ( d  =  ( D `  I )  ->  ( A (AP `  ( K  +  1 ) ) d )  =  ( A (AP `  ( K  +  1 ) ) ( D `  I ) ) )
9796sseq1d 3427 . . . . 5  |-  ( d  =  ( D `  I )  ->  (
( A (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { ( F `  A ) } )  <-> 
( A (AP `  ( K  +  1
) ) ( D `
 I ) ) 
C_  ( `' F " { ( F `  A ) } ) ) )
9895, 97rspc2ev 3129 . . . 4  |-  ( ( A  e.  NN  /\  ( D `  I )  e.  NN  /\  ( A (AP `  ( K  +  1 ) ) ( D `  I
) )  C_  ( `' F " { ( F `  A ) } ) )  ->  E. a  e.  NN  E. d  e.  NN  (
a (AP `  ( K  +  1 ) ) d )  C_  ( `' F " { ( F `  A ) } ) )
991, 4, 93, 98syl3anc 1264 . . 3  |-  ( ph  ->  E. a  e.  NN  E. d  e.  NN  (
a (AP `  ( K  +  1 ) ) d )  C_  ( `' F " { ( F `  A ) } ) )
100 fvex 5828 . . . 4  |-  ( F `
 A )  e. 
_V
101 sneq 3944 . . . . . . 7  |-  ( c  =  ( F `  A )  ->  { c }  =  { ( F `  A ) } )
102101imaeq2d 5123 . . . . . 6  |-  ( c  =  ( F `  A )  ->  ( `' F " { c } )  =  ( `' F " { ( F `  A ) } ) )
103102sseq2d 3428 . . . . 5  |-  ( c  =  ( F `  A )  ->  (
( a (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { c } )  <-> 
( a (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { ( F `  A ) } ) ) )
1041032rexbidv 2879 . . . 4  |-  ( c  =  ( F `  A )  ->  ( E. a  e.  NN  E. d  e.  NN  (
a (AP `  ( K  +  1 ) ) d )  C_  ( `' F " { c } )  <->  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  ( K  + 
1 ) ) d )  C_  ( `' F " { ( F `
 A ) } ) ) )
105100, 104spcev 3109 . . 3  |-  ( E. a  e.  NN  E. d  e.  NN  (
a (AP `  ( K  +  1 ) ) d )  C_  ( `' F " { ( F `  A ) } )  ->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  ( K  +  1 ) ) d )  C_  ( `' F " { c } ) )
10699, 105syl 17 . 2  |-  ( ph  ->  E. c E. a  e.  NN  E. d  e.  NN  ( a (AP
`  ( K  + 
1 ) ) d )  C_  ( `' F " { c } ) )
107 ovex 6270 . . 3  |-  ( 1 ... W )  e. 
_V
108 peano2nn0 10854 . . . 4  |-  ( K  e.  NN0  ->  ( K  +  1 )  e. 
NN0 )
1096, 108syl 17 . . 3  |-  ( ph  ->  ( K  +  1 )  e.  NN0 )
110107, 109, 26vdwmc 14864 . 2  |-  ( ph  ->  ( ( K  + 
1 ) MonoAP  F  <->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  ( K  +  1 ) ) d )  C_  ( `' F " { c } ) ) )
111106, 110mpbird 235 1  |-  ( ph  ->  ( K  +  1 ) MonoAP  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   A.wral 2708   E.wrex 2709    u. cun 3370    C_ wss 3372   {csn 3934   class class class wbr 4359   `'ccnv 4788   dom cdm 4789   "cima 4792    Fn wfn 5532   -->wf 5533   ` cfv 5537  (class class class)co 6242   Fincfn 7517   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488    <_ cle 9620    - cmin 9804   NNcn 10553   NN0cn0 10813   ZZcz 10881   ZZ>=cuz 11103   ...cfz 11728  APcvdwa 14851   MonoAP cvdwm 14852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-rep 4472  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-nel 2596  df-ral 2713  df-rex 2714  df-reu 2715  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-tp 3939  df-op 3941  df-uni 4156  df-iun 4237  df-br 4360  df-opab 4419  df-mpt 4420  df-tr 4455  df-eprel 4700  df-id 4704  df-po 4710  df-so 4711  df-fr 4748  df-we 4750  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-pred 5335  df-ord 5381  df-on 5382  df-lim 5383  df-suc 5384  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-riota 6204  df-ov 6245  df-oprab 6246  df-mpt2 6247  df-om 6644  df-1st 6744  df-2nd 6745  df-wrecs 6976  df-recs 7038  df-rdg 7076  df-er 7311  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9806  df-neg 9807  df-nn 10554  df-n0 10814  df-z 10882  df-uz 11104  df-rp 11247  df-fz 11729  df-vdwap 14854  df-vdwmc 14855
This theorem is referenced by:  vdwlem6  14872
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