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Theorem vdwlem1 14358
Description: Lemma for vdw 14371. (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 6022 . . . 4  |-  ( ph  ->  ( D `  I
)  e.  NN )
5 vdwlem1.k . . . . . . 7  |-  ( ph  ->  K  e.  NN )
65nnnn0d 10852 . . . . . 6  |-  ( ph  ->  K  e.  NN0 )
7 vdwapun 14351 . . . . . 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 1228 . . . . 5  |-  ( ph  ->  ( A (AP `  ( K  +  1
) ) ( D `
 I ) )  =  ( { A }  u.  ( ( A  +  ( D `  I ) ) (AP
`  K ) ( D `  I ) ) ) )
91nnred 10551 . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR )
10 vdwlem1.m . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  NN )
11 nnuz 11117 . . . . . . . . . . . . . . 15  |-  NN  =  ( ZZ>= `  1 )
1210, 11syl6eleq 2565 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
13 eluzfz1 11693 . . . . . . . . . . . . . 14  |-  ( M  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... M
) )
1412, 13syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  ( 1 ... M ) )
152, 14ffvelrnd 6022 . . . . . . . . . . . 12  |-  ( ph  ->  ( D `  1
)  e.  NN )
161, 15nnaddcld 10582 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  ( D `  1 ) )  e.  NN )
1716nnred 10551 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  ( D `  1 ) )  e.  RR )
18 vdwlem1.w . . . . . . . . . . 11  |-  ( ph  ->  W  e.  NN )
1918nnred 10551 . . . . . . . . . 10  |-  ( ph  ->  W  e.  RR )
2015nnrpd 11255 . . . . . . . . . . . 12  |-  ( ph  ->  ( D `  1
)  e.  RR+ )
219, 20ltaddrpd 11285 . . . . . . . . . . 11  |-  ( ph  ->  A  <  ( A  +  ( D ` 
1 ) ) )
229, 17, 21ltled 9732 . . . . . . . . . 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 2833 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  C_  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } ) )
25 cnvimass 5357 . . . . . . . . . . . . . . . . 17  |-  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  C_  dom  F
26 vdwlem1.f . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F : ( 1 ... W ) --> R )
27 fdm 5735 . . . . . . . . . . . . . . . . . 18  |-  ( F : ( 1 ... W ) --> R  ->  dom  F  =  ( 1 ... W ) )
2826, 27syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  dom  F  =  ( 1 ... W ) )
2925, 28syl5sseq 3552 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( `' F " { ( F `  ( A  +  ( D `  i )
) ) } ) 
C_  ( 1 ... W ) )
3029adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  C_  (
1 ... W ) )
3124, 30sstrd 3514 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  C_  ( 1 ... W
) )
32 nnm1nn0 10837 . . . . . . . . . . . . . . . . . . . 20  |-  ( K  e.  NN  ->  ( K  -  1 )  e.  NN0 )
335, 32syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( K  -  1 )  e.  NN0 )
34 nn0uz 11116 . . . . . . . . . . . . . . . . . . 19  |-  NN0  =  ( ZZ>= `  0 )
3533, 34syl6eleq 2565 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( K  -  1 )  e.  ( ZZ>= ` 
0 ) )
36 eluzfz1 11693 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  -  1 )  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... ( K  -  1 ) ) )
3735, 36syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  e.  ( 0 ... ( K  - 
1 ) ) )
3837adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  0  e.  ( 0 ... ( K  -  1 ) ) )
392ffvelrnda 6021 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( D `  i )  e.  NN )
4039nncnd 10552 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( D `  i )  e.  CC )
4140mul02d 9777 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
0  x.  ( D `
 i ) )  =  0 )
4241oveq2d 6300 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i ) ) )  =  ( ( A  +  ( D `  i ) )  +  0 ) )
431adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  A  e.  NN )
4443, 39nnaddcld 10582 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  NN )
4544nncnd 10552 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  CC )
4645addid1d 9779 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) )  +  0 )  =  ( A  +  ( D `  i ) ) )
4742, 46eqtr2d 2509 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i ) ) ) )
48 oveq1 6291 . . . . . . . . . . . . . . . . . . 19  |-  ( m  =  0  ->  (
m  x.  ( D `
 i ) )  =  ( 0  x.  ( D `  i
) ) )
4948oveq2d 6300 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  0  ->  (
( A  +  ( D `  i ) )  +  ( m  x.  ( D `  i ) ) )  =  ( ( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i )
) ) )
5049eqeq2d 2481 . . . . . . . . . . . . . . . . 17  |-  ( m  =  0  ->  (
( A  +  ( D `  i ) )  =  ( ( A  +  ( D `
 i ) )  +  ( m  x.  ( D `  i
) ) )  <->  ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i ) ) ) ) )
5150rspcev 3214 . . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  E. m  e.  ( 0 ... ( K  -  1 ) ) ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `
 i ) )  +  ( m  x.  ( D `  i
) ) ) )
535adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  K  e.  NN )
5453nnnn0d 10852 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  K  e.  NN0 )
55 vdwapval 14350 . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  ( ( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) ) )
5831, 57sseldd 3505 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  ( 1 ... W
) )
5958ralrimiva 2878 . . . . . . . . . . . 12  |-  ( ph  ->  A. i  e.  ( 1 ... M ) ( A  +  ( D `  i ) )  e.  ( 1 ... W ) )
60 fveq2 5866 . . . . . . . . . . . . . . 15  |-  ( i  =  1  ->  ( D `  i )  =  ( D ` 
1 ) )
6160oveq2d 6300 . . . . . . . . . . . . . 14  |-  ( i  =  1  ->  ( A  +  ( D `  i ) )  =  ( A  +  ( D `  1 ) ) )
6261eleq1d 2536 . . . . . . . . . . . . 13  |-  ( i  =  1  ->  (
( A  +  ( D `  i ) )  e.  ( 1 ... W )  <->  ( A  +  ( D ` 
1 ) )  e.  ( 1 ... W
) ) )
6362rspcv 3210 . . . . . . . . . . . 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 60 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  ( D `  1 ) )  e.  ( 1 ... W ) )
65 elfzle2 11690 . . . . . . . . . . 11  |-  ( ( A  +  ( D `
 1 ) )  e.  ( 1 ... W )  ->  ( A  +  ( D `  1 ) )  <_  W )
6664, 65syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  ( D `  1 ) )  <_  W )
679, 17, 19, 22, 66letrd 9738 . . . . . . . . 9  |-  ( ph  ->  A  <_  W )
681, 11syl6eleq 2565 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( ZZ>= ` 
1 ) )
6918nnzd 10965 . . . . . . . . . 10  |-  ( ph  ->  W  e.  ZZ )
70 elfz5 11680 . . . . . . . . . 10  |-  ( ( A  e.  ( ZZ>= ` 
1 )  /\  W  e.  ZZ )  ->  ( A  e.  ( 1 ... W )  <->  A  <_  W ) )
7168, 69, 70syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  ( 1 ... W )  <-> 
A  <_  W )
)
7267, 71mpbird 232 . . . . . . . 8  |-  ( ph  ->  A  e.  ( 1 ... W ) )
73 eqidd 2468 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  =  ( F `
 A ) )
74 ffn 5731 . . . . . . . . 9  |-  ( F : ( 1 ... W ) --> R  ->  F  Fn  ( 1 ... W ) )
75 fniniseg 6002 . . . . . . . . 9  |-  ( F  Fn  ( 1 ... W )  ->  ( A  e.  ( `' F " { ( F `
 A ) } )  <->  ( A  e.  ( 1 ... W
)  /\  ( F `  A )  =  ( F `  A ) ) ) )
7626, 74, 753syl 20 . . . . . . . 8  |-  ( ph  ->  ( A  e.  ( `' F " { ( F `  A ) } )  <->  ( A  e.  ( 1 ... W
)  /\  ( F `  A )  =  ( F `  A ) ) ) )
7772, 73, 76mpbir2and 920 . . . . . . 7  |-  ( ph  ->  A  e.  ( `' F " { ( F `  A ) } ) )
7877snssd 4172 . . . . . 6  |-  ( ph  ->  { A }  C_  ( `' F " { ( F `  A ) } ) )
79 fveq2 5866 . . . . . . . . . . . 12  |-  ( i  =  I  ->  ( D `  i )  =  ( D `  I ) )
8079oveq2d 6300 . . . . . . . . . . 11  |-  ( i  =  I  ->  ( A  +  ( D `  i ) )  =  ( A  +  ( D `  I ) ) )
8180, 79oveq12d 6302 . . . . . . . . . 10  |-  ( i  =  I  ->  (
( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  =  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) ) )
8280fveq2d 5870 . . . . . . . . . . . 12  |-  ( i  =  I  ->  ( F `  ( A  +  ( D `  i ) ) )  =  ( F `  ( A  +  ( D `  I )
) ) )
8382sneqd 4039 . . . . . . . . . . 11  |-  ( i  =  I  ->  { ( F `  ( A  +  ( D `  i ) ) ) }  =  { ( F `  ( A  +  ( D `  I ) ) ) } )
8483imaeq2d 5337 . . . . . . . . . 10  |-  ( i  =  I  ->  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  =  ( `' F " { ( F `  ( A  +  ( D `  I ) ) ) } ) )
8581, 84sseq12d 3533 . . . . . . . . 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 3210 . . . . . . . 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 60 . . . . . . 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 4039 . . . . . . . 8  |-  ( ph  ->  { ( F `  A ) }  =  { ( F `  ( A  +  ( D `  I )
) ) } )
9089imaeq2d 5337 . . . . . . 7  |-  ( ph  ->  ( `' F " { ( F `  A ) } )  =  ( `' F " { ( F `  ( A  +  ( D `  I )
) ) } ) )
9187, 90sseqtr4d 3541 . . . . . 6  |-  ( ph  ->  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) )  C_  ( `' F " { ( F `  A ) } ) )
9278, 91unssd 3680 . . . . 5  |-  ( ph  ->  ( { A }  u.  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) ) ) 
C_  ( `' F " { ( F `  A ) } ) )
938, 92eqsstrd 3538 . . . 4  |-  ( ph  ->  ( A (AP `  ( K  +  1
) ) ( D `
 I ) ) 
C_  ( `' F " { ( F `  A ) } ) )
94 oveq1 6291 . . . . . 6  |-  ( a  =  A  ->  (
a (AP `  ( K  +  1 ) ) d )  =  ( A (AP `  ( K  +  1
) ) d ) )
9594sseq1d 3531 . . . . 5  |-  ( a  =  A  ->  (
( a (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { ( F `  A ) } )  <-> 
( A (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { ( F `  A ) } ) ) )
96 oveq2 6292 . . . . . 6  |-  ( d  =  ( D `  I )  ->  ( A (AP `  ( K  +  1 ) ) d )  =  ( A (AP `  ( K  +  1 ) ) ( D `  I ) ) )
9796sseq1d 3531 . . . . 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 3225 . . . 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 1228 . . 3  |-  ( ph  ->  E. a  e.  NN  E. d  e.  NN  (
a (AP `  ( K  +  1 ) ) d )  C_  ( `' F " { ( F `  A ) } ) )
100 fvex 5876 . . . 4  |-  ( F `
 A )  e. 
_V
101 sneq 4037 . . . . . . 7  |-  ( c  =  ( F `  A )  ->  { c }  =  { ( F `  A ) } )
102101imaeq2d 5337 . . . . . 6  |-  ( c  =  ( F `  A )  ->  ( `' F " { c } )  =  ( `' F " { ( F `  A ) } ) )
103102sseq2d 3532 . . . . 5  |-  ( c  =  ( F `  A )  ->  (
( a (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { c } )  <-> 
( a (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { ( F `  A ) } ) ) )
1041032rexbidv 2980 . . . 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 3205 . . 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 16 . 2  |-  ( ph  ->  E. c E. a  e.  NN  E. d  e.  NN  ( a (AP
`  ( K  + 
1 ) ) d )  C_  ( `' F " { c } ) )
107 ovex 6309 . . 3  |-  ( 1 ... W )  e. 
_V
108 peano2nn0 10836 . . . 4  |-  ( K  e.  NN0  ->  ( K  +  1 )  e. 
NN0 )
1096, 108syl 16 . . 3  |-  ( ph  ->  ( K  +  1 )  e.  NN0 )
110107, 109, 26vdwmc 14355 . 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 232 1  |-  ( ph  ->  ( K  +  1 ) MonoAP  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2814   E.wrex 2815    u. cun 3474    C_ wss 3476   {csn 4027   class class class wbr 4447   `'ccnv 4998   dom cdm 4999   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   Fincfn 7516   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    <_ cle 9629    - cmin 9805   NNcn 10536   NN0cn0 10795   ZZcz 10864   ZZ>=cuz 11082   ...cfz 11672  APcvdwa 14342   MonoAP cvdwm 14343
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 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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-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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-vdwap 14345  df-vdwmc 14346
This theorem is referenced by:  vdwlem6  14363
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