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Theorem vdwlem1 13304
Description: Lemma for vdw 13317. (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 5830 . . . 4  |-  ( ph  ->  ( D `  I
)  e.  NN )
5 vdwlem1.k . . . . . . 7  |-  ( ph  ->  K  e.  NN )
65nnnn0d 10230 . . . . . 6  |-  ( ph  ->  K  e.  NN0 )
7 vdwapun 13297 . . . . . 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 1184 . . . . 5  |-  ( ph  ->  ( A (AP `  ( K  +  1
) ) ( D `
 I ) )  =  ( { A }  u.  ( ( A  +  ( D `  I ) ) (AP
`  K ) ( D `  I ) ) ) )
91nnred 9971 . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR )
10 vdwlem1.m . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  NN )
11 nnuz 10477 . . . . . . . . . . . . . . 15  |-  NN  =  ( ZZ>= `  1 )
1210, 11syl6eleq 2494 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
13 eluzfz1 11020 . . . . . . . . . . . . . 14  |-  ( M  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... M
) )
1412, 13syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  ( 1 ... M ) )
152, 14ffvelrnd 5830 . . . . . . . . . . . 12  |-  ( ph  ->  ( D `  1
)  e.  NN )
161, 15nnaddcld 10002 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  ( D `  1 ) )  e.  NN )
1716nnred 9971 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  ( D `  1 ) )  e.  RR )
18 vdwlem1.w . . . . . . . . . . 11  |-  ( ph  ->  W  e.  NN )
1918nnred 9971 . . . . . . . . . 10  |-  ( ph  ->  W  e.  RR )
2015nnrpd 10603 . . . . . . . . . . . 12  |-  ( ph  ->  ( D `  1
)  e.  RR+ )
219, 20ltaddrpd 10633 . . . . . . . . . . 11  |-  ( ph  ->  A  <  ( A  +  ( D ` 
1 ) ) )
229, 17, 21ltled 9177 . . . . . . . . . 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 2764 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  C_  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } ) )
25 cnvimass 5183 . . . . . . . . . . . . . . . . 17  |-  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  C_  dom  F
26 vdwlem1.f . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F : ( 1 ... W ) --> R )
27 fdm 5554 . . . . . . . . . . . . . . . . . 18  |-  ( F : ( 1 ... W ) --> R  ->  dom  F  =  ( 1 ... W ) )
2826, 27syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  dom  F  =  ( 1 ... W ) )
2925, 28syl5sseq 3356 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( `' F " { ( F `  ( A  +  ( D `  i )
) ) } ) 
C_  ( 1 ... W ) )
3029adantr 452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  C_  (
1 ... W ) )
3124, 30sstrd 3318 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  C_  ( 1 ... W
) )
32 nnm1nn0 10217 . . . . . . . . . . . . . . . . . . . 20  |-  ( K  e.  NN  ->  ( K  -  1 )  e.  NN0 )
335, 32syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( K  -  1 )  e.  NN0 )
34 nn0uz 10476 . . . . . . . . . . . . . . . . . . 19  |-  NN0  =  ( ZZ>= `  0 )
3533, 34syl6eleq 2494 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( K  -  1 )  e.  ( ZZ>= ` 
0 ) )
36 eluzfz1 11020 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  -  1 )  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... ( K  -  1 ) ) )
3735, 36syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  e.  ( 0 ... ( K  - 
1 ) ) )
3837adantr 452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  0  e.  ( 0 ... ( K  -  1 ) ) )
392ffvelrnda 5829 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( D `  i )  e.  NN )
4039nncnd 9972 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( D `  i )  e.  CC )
4140mul02d 9220 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
0  x.  ( D `
 i ) )  =  0 )
4241oveq2d 6056 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i ) ) )  =  ( ( A  +  ( D `  i ) )  +  0 ) )
431adantr 452 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  A  e.  NN )
4443, 39nnaddcld 10002 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  NN )
4544nncnd 9972 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  CC )
4645addid1d 9222 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) )  +  0 )  =  ( A  +  ( D `  i ) ) )
4742, 46eqtr2d 2437 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i ) ) ) )
48 oveq1 6047 . . . . . . . . . . . . . . . . . . 19  |-  ( m  =  0  ->  (
m  x.  ( D `
 i ) )  =  ( 0  x.  ( D `  i
) ) )
4948oveq2d 6056 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  0  ->  (
( A  +  ( D `  i ) )  +  ( m  x.  ( D `  i ) ) )  =  ( ( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i )
) ) )
5049eqeq2d 2415 . . . . . . . . . . . . . . . . 17  |-  ( m  =  0  ->  (
( A  +  ( D `  i ) )  =  ( ( A  +  ( D `
 i ) )  +  ( m  x.  ( D `  i
) ) )  <->  ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i ) ) ) ) )
5150rspcev 3012 . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  E. m  e.  ( 0 ... ( K  -  1 ) ) ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `
 i ) )  +  ( m  x.  ( D `  i
) ) ) )
535adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  K  e.  NN )
5453nnnn0d 10230 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  K  e.  NN0 )
55 vdwapval 13296 . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  ( ( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) ) )
5831, 57sseldd 3309 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  ( 1 ... W
) )
5958ralrimiva 2749 . . . . . . . . . . . 12  |-  ( ph  ->  A. i  e.  ( 1 ... M ) ( A  +  ( D `  i ) )  e.  ( 1 ... W ) )
60 fveq2 5687 . . . . . . . . . . . . . . 15  |-  ( i  =  1  ->  ( D `  i )  =  ( D ` 
1 ) )
6160oveq2d 6056 . . . . . . . . . . . . . 14  |-  ( i  =  1  ->  ( A  +  ( D `  i ) )  =  ( A  +  ( D `  1 ) ) )
6261eleq1d 2470 . . . . . . . . . . . . 13  |-  ( i  =  1  ->  (
( A  +  ( D `  i ) )  e.  ( 1 ... W )  <->  ( A  +  ( D ` 
1 ) )  e.  ( 1 ... W
) ) )
6362rspcv 3008 . . . . . . . . . . . 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 58 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  ( D `  1 ) )  e.  ( 1 ... W ) )
65 elfzle2 11017 . . . . . . . . . . 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 9183 . . . . . . . . 9  |-  ( ph  ->  A  <_  W )
681, 11syl6eleq 2494 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( ZZ>= ` 
1 ) )
6918nnzd 10330 . . . . . . . . . 10  |-  ( ph  ->  W  e.  ZZ )
70 elfz5 11007 . . . . . . . . . 10  |-  ( ( A  e.  ( ZZ>= ` 
1 )  /\  W  e.  ZZ )  ->  ( A  e.  ( 1 ... W )  <->  A  <_  W ) )
7168, 69, 70syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  ( 1 ... W )  <-> 
A  <_  W )
)
7267, 71mpbird 224 . . . . . . . 8  |-  ( ph  ->  A  e.  ( 1 ... W ) )
73 eqidd 2405 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  =  ( F `
 A ) )
74 ffn 5550 . . . . . . . . 9  |-  ( F : ( 1 ... W ) --> R  ->  F  Fn  ( 1 ... W ) )
75 fniniseg 5810 . . . . . . . . 9  |-  ( F  Fn  ( 1 ... W )  ->  ( A  e.  ( `' F " { ( F `
 A ) } )  <->  ( A  e.  ( 1 ... W
)  /\  ( F `  A )  =  ( F `  A ) ) ) )
7626, 74, 753syl 19 . . . . . . . 8  |-  ( ph  ->  ( A  e.  ( `' F " { ( F `  A ) } )  <->  ( A  e.  ( 1 ... W
)  /\  ( F `  A )  =  ( F `  A ) ) ) )
7772, 73, 76mpbir2and 889 . . . . . . 7  |-  ( ph  ->  A  e.  ( `' F " { ( F `  A ) } ) )
7877snssd 3903 . . . . . 6  |-  ( ph  ->  { A }  C_  ( `' F " { ( F `  A ) } ) )
79 fveq2 5687 . . . . . . . . . . . 12  |-  ( i  =  I  ->  ( D `  i )  =  ( D `  I ) )
8079oveq2d 6056 . . . . . . . . . . 11  |-  ( i  =  I  ->  ( A  +  ( D `  i ) )  =  ( A  +  ( D `  I ) ) )
8180, 79oveq12d 6058 . . . . . . . . . 10  |-  ( i  =  I  ->  (
( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  =  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) ) )
8280fveq2d 5691 . . . . . . . . . . . 12  |-  ( i  =  I  ->  ( F `  ( A  +  ( D `  i ) ) )  =  ( F `  ( A  +  ( D `  I )
) ) )
8382sneqd 3787 . . . . . . . . . . 11  |-  ( i  =  I  ->  { ( F `  ( A  +  ( D `  i ) ) ) }  =  { ( F `  ( A  +  ( D `  I ) ) ) } )
8483imaeq2d 5162 . . . . . . . . . 10  |-  ( i  =  I  ->  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  =  ( `' F " { ( F `  ( A  +  ( D `  I ) ) ) } ) )
8581, 84sseq12d 3337 . . . . . . . . 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 3008 . . . . . . . 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 58 . . . . . . 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 3787 . . . . . . . 8  |-  ( ph  ->  { ( F `  A ) }  =  { ( F `  ( A  +  ( D `  I )
) ) } )
9089imaeq2d 5162 . . . . . . 7  |-  ( ph  ->  ( `' F " { ( F `  A ) } )  =  ( `' F " { ( F `  ( A  +  ( D `  I )
) ) } ) )
9187, 90sseqtr4d 3345 . . . . . 6  |-  ( ph  ->  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) )  C_  ( `' F " { ( F `  A ) } ) )
9278, 91unssd 3483 . . . . 5  |-  ( ph  ->  ( { A }  u.  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) ) ) 
C_  ( `' F " { ( F `  A ) } ) )
938, 92eqsstrd 3342 . . . 4  |-  ( ph  ->  ( A (AP `  ( K  +  1
) ) ( D `
 I ) ) 
C_  ( `' F " { ( F `  A ) } ) )
94 oveq1 6047 . . . . . 6  |-  ( a  =  A  ->  (
a (AP `  ( K  +  1 ) ) d )  =  ( A (AP `  ( K  +  1
) ) d ) )
9594sseq1d 3335 . . . . 5  |-  ( a  =  A  ->  (
( a (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { ( F `  A ) } )  <-> 
( A (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { ( F `  A ) } ) ) )
96 oveq2 6048 . . . . . 6  |-  ( d  =  ( D `  I )  ->  ( A (AP `  ( K  +  1 ) ) d )  =  ( A (AP `  ( K  +  1 ) ) ( D `  I ) ) )
9796sseq1d 3335 . . . . 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 3020 . . . 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 1184 . . 3  |-  ( ph  ->  E. a  e.  NN  E. d  e.  NN  (
a (AP `  ( K  +  1 ) ) d )  C_  ( `' F " { ( F `  A ) } ) )
100 fvex 5701 . . . 4  |-  ( F `
 A )  e. 
_V
101 sneq 3785 . . . . . . 7  |-  ( c  =  ( F `  A )  ->  { c }  =  { ( F `  A ) } )
102101imaeq2d 5162 . . . . . 6  |-  ( c  =  ( F `  A )  ->  ( `' F " { c } )  =  ( `' F " { ( F `  A ) } ) )
103102sseq2d 3336 . . . . 5  |-  ( c  =  ( F `  A )  ->  (
( a (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { c } )  <-> 
( a (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { ( F `  A ) } ) ) )
1041032rexbidv 2709 . . . 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 3003 . . 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 6065 . . 3  |-  ( 1 ... W )  e. 
_V
108 peano2nn0 10216 . . . 4  |-  ( K  e.  NN0  ->  ( K  +  1 )  e. 
NN0 )
1096, 108syl 16 . . 3  |-  ( ph  ->  ( K  +  1 )  e.  NN0 )
110107, 109, 26vdwmc 13301 . 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 224 1  |-  ( ph  ->  ( K  +  1 ) MonoAP  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667    u. cun 3278    C_ wss 3280   {csn 3774   class class class wbr 4172   `'ccnv 4836   dom cdm 4837   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    <_ cle 9077    - cmin 9247   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999  APcvdwa 13288   MonoAP cvdwm 13289
This theorem is referenced by:  vdwlem6  13309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-vdwap 13291  df-vdwmc 13292
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