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Theorem vdwlem1 13351
Description: Lemma for vdw 13364. (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 5873 . . . 4  |-  ( ph  ->  ( D `  I
)  e.  NN )
5 vdwlem1.k . . . . . . 7  |-  ( ph  ->  K  e.  NN )
65nnnn0d 10276 . . . . . 6  |-  ( ph  ->  K  e.  NN0 )
7 vdwapun 13344 . . . . . 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 1185 . . . . 5  |-  ( ph  ->  ( A (AP `  ( K  +  1
) ) ( D `
 I ) )  =  ( { A }  u.  ( ( A  +  ( D `  I ) ) (AP
`  K ) ( D `  I ) ) ) )
91nnred 10017 . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR )
10 vdwlem1.m . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  NN )
11 nnuz 10523 . . . . . . . . . . . . . . 15  |-  NN  =  ( ZZ>= `  1 )
1210, 11syl6eleq 2528 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
13 eluzfz1 11066 . . . . . . . . . . . . . 14  |-  ( M  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... M
) )
1412, 13syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  ( 1 ... M ) )
152, 14ffvelrnd 5873 . . . . . . . . . . . 12  |-  ( ph  ->  ( D `  1
)  e.  NN )
161, 15nnaddcld 10048 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  ( D `  1 ) )  e.  NN )
1716nnred 10017 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  ( D `  1 ) )  e.  RR )
18 vdwlem1.w . . . . . . . . . . 11  |-  ( ph  ->  W  e.  NN )
1918nnred 10017 . . . . . . . . . 10  |-  ( ph  ->  W  e.  RR )
2015nnrpd 10649 . . . . . . . . . . . 12  |-  ( ph  ->  ( D `  1
)  e.  RR+ )
219, 20ltaddrpd 10679 . . . . . . . . . . 11  |-  ( ph  ->  A  <  ( A  +  ( D ` 
1 ) ) )
229, 17, 21ltled 9223 . . . . . . . . . 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 2806 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  C_  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } ) )
25 cnvimass 5226 . . . . . . . . . . . . . . . . 17  |-  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  C_  dom  F
26 vdwlem1.f . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F : ( 1 ... W ) --> R )
27 fdm 5597 . . . . . . . . . . . . . . . . . 18  |-  ( F : ( 1 ... W ) --> R  ->  dom  F  =  ( 1 ... W ) )
2826, 27syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  dom  F  =  ( 1 ... W ) )
2925, 28syl5sseq 3398 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( `' F " { ( F `  ( A  +  ( D `  i )
) ) } ) 
C_  ( 1 ... W ) )
3029adantr 453 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  C_  (
1 ... W ) )
3124, 30sstrd 3360 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  C_  ( 1 ... W
) )
32 nnm1nn0 10263 . . . . . . . . . . . . . . . . . . . 20  |-  ( K  e.  NN  ->  ( K  -  1 )  e.  NN0 )
335, 32syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( K  -  1 )  e.  NN0 )
34 nn0uz 10522 . . . . . . . . . . . . . . . . . . 19  |-  NN0  =  ( ZZ>= `  0 )
3533, 34syl6eleq 2528 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( K  -  1 )  e.  ( ZZ>= ` 
0 ) )
36 eluzfz1 11066 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  -  1 )  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... ( K  -  1 ) ) )
3735, 36syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  e.  ( 0 ... ( K  - 
1 ) ) )
3837adantr 453 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  0  e.  ( 0 ... ( K  -  1 ) ) )
392ffvelrnda 5872 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( D `  i )  e.  NN )
4039nncnd 10018 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( D `  i )  e.  CC )
4140mul02d 9266 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
0  x.  ( D `
 i ) )  =  0 )
4241oveq2d 6099 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i ) ) )  =  ( ( A  +  ( D `  i ) )  +  0 ) )
431adantr 453 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  A  e.  NN )
4443, 39nnaddcld 10048 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  NN )
4544nncnd 10018 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  CC )
4645addid1d 9268 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( A  +  ( D `  i ) )  +  0 )  =  ( A  +  ( D `  i ) ) )
4742, 46eqtr2d 2471 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i ) ) ) )
48 oveq1 6090 . . . . . . . . . . . . . . . . . . 19  |-  ( m  =  0  ->  (
m  x.  ( D `
 i ) )  =  ( 0  x.  ( D `  i
) ) )
4948oveq2d 6099 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  0  ->  (
( A  +  ( D `  i ) )  +  ( m  x.  ( D `  i ) ) )  =  ( ( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i )
) ) )
5049eqeq2d 2449 . . . . . . . . . . . . . . . . 17  |-  ( m  =  0  ->  (
( A  +  ( D `  i ) )  =  ( ( A  +  ( D `
 i ) )  +  ( m  x.  ( D `  i
) ) )  <->  ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `  i ) )  +  ( 0  x.  ( D `  i ) ) ) ) )
5150rspcev 3054 . . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  E. m  e.  ( 0 ... ( K  -  1 ) ) ( A  +  ( D `  i ) )  =  ( ( A  +  ( D `
 i ) )  +  ( m  x.  ( D `  i
) ) ) )
535adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  K  e.  NN )
5453nnnn0d 10276 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  K  e.  NN0 )
55 vdwapval 13343 . . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  ( ( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) ) )
5831, 57sseldd 3351 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( A  +  ( D `  i ) )  e.  ( 1 ... W
) )
5958ralrimiva 2791 . . . . . . . . . . . 12  |-  ( ph  ->  A. i  e.  ( 1 ... M ) ( A  +  ( D `  i ) )  e.  ( 1 ... W ) )
60 fveq2 5730 . . . . . . . . . . . . . . 15  |-  ( i  =  1  ->  ( D `  i )  =  ( D ` 
1 ) )
6160oveq2d 6099 . . . . . . . . . . . . . 14  |-  ( i  =  1  ->  ( A  +  ( D `  i ) )  =  ( A  +  ( D `  1 ) ) )
6261eleq1d 2504 . . . . . . . . . . . . 13  |-  ( i  =  1  ->  (
( A  +  ( D `  i ) )  e.  ( 1 ... W )  <->  ( A  +  ( D ` 
1 ) )  e.  ( 1 ... W
) ) )
6362rspcv 3050 . . . . . . . . . . . 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 59 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  ( D `  1 ) )  e.  ( 1 ... W ) )
65 elfzle2 11063 . . . . . . . . . . 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 9229 . . . . . . . . 9  |-  ( ph  ->  A  <_  W )
681, 11syl6eleq 2528 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( ZZ>= ` 
1 ) )
6918nnzd 10376 . . . . . . . . . 10  |-  ( ph  ->  W  e.  ZZ )
70 elfz5 11053 . . . . . . . . . 10  |-  ( ( A  e.  ( ZZ>= ` 
1 )  /\  W  e.  ZZ )  ->  ( A  e.  ( 1 ... W )  <->  A  <_  W ) )
7168, 69, 70syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  ( 1 ... W )  <-> 
A  <_  W )
)
7267, 71mpbird 225 . . . . . . . 8  |-  ( ph  ->  A  e.  ( 1 ... W ) )
73 eqidd 2439 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  =  ( F `
 A ) )
74 ffn 5593 . . . . . . . . 9  |-  ( F : ( 1 ... W ) --> R  ->  F  Fn  ( 1 ... W ) )
75 fniniseg 5853 . . . . . . . . 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 890 . . . . . . 7  |-  ( ph  ->  A  e.  ( `' F " { ( F `  A ) } ) )
7877snssd 3945 . . . . . 6  |-  ( ph  ->  { A }  C_  ( `' F " { ( F `  A ) } ) )
79 fveq2 5730 . . . . . . . . . . . 12  |-  ( i  =  I  ->  ( D `  i )  =  ( D `  I ) )
8079oveq2d 6099 . . . . . . . . . . 11  |-  ( i  =  I  ->  ( A  +  ( D `  i ) )  =  ( A  +  ( D `  I ) ) )
8180, 79oveq12d 6101 . . . . . . . . . 10  |-  ( i  =  I  ->  (
( A  +  ( D `  i ) ) (AP `  K
) ( D `  i ) )  =  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) ) )
8280fveq2d 5734 . . . . . . . . . . . 12  |-  ( i  =  I  ->  ( F `  ( A  +  ( D `  i ) ) )  =  ( F `  ( A  +  ( D `  I )
) ) )
8382sneqd 3829 . . . . . . . . . . 11  |-  ( i  =  I  ->  { ( F `  ( A  +  ( D `  i ) ) ) }  =  { ( F `  ( A  +  ( D `  I ) ) ) } )
8483imaeq2d 5205 . . . . . . . . . 10  |-  ( i  =  I  ->  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )  =  ( `' F " { ( F `  ( A  +  ( D `  I ) ) ) } ) )
8581, 84sseq12d 3379 . . . . . . . . 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 3050 . . . . . . . 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 59 . . . . . . 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 3829 . . . . . . . 8  |-  ( ph  ->  { ( F `  A ) }  =  { ( F `  ( A  +  ( D `  I )
) ) } )
9089imaeq2d 5205 . . . . . . 7  |-  ( ph  ->  ( `' F " { ( F `  A ) } )  =  ( `' F " { ( F `  ( A  +  ( D `  I )
) ) } ) )
9187, 90sseqtr4d 3387 . . . . . 6  |-  ( ph  ->  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) )  C_  ( `' F " { ( F `  A ) } ) )
9278, 91unssd 3525 . . . . 5  |-  ( ph  ->  ( { A }  u.  ( ( A  +  ( D `  I ) ) (AP `  K
) ( D `  I ) ) ) 
C_  ( `' F " { ( F `  A ) } ) )
938, 92eqsstrd 3384 . . . 4  |-  ( ph  ->  ( A (AP `  ( K  +  1
) ) ( D `
 I ) ) 
C_  ( `' F " { ( F `  A ) } ) )
94 oveq1 6090 . . . . . 6  |-  ( a  =  A  ->  (
a (AP `  ( K  +  1 ) ) d )  =  ( A (AP `  ( K  +  1
) ) d ) )
9594sseq1d 3377 . . . . 5  |-  ( a  =  A  ->  (
( a (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { ( F `  A ) } )  <-> 
( A (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { ( F `  A ) } ) ) )
96 oveq2 6091 . . . . . 6  |-  ( d  =  ( D `  I )  ->  ( A (AP `  ( K  +  1 ) ) d )  =  ( A (AP `  ( K  +  1 ) ) ( D `  I ) ) )
9796sseq1d 3377 . . . . 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 3062 . . . 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 1185 . . 3  |-  ( ph  ->  E. a  e.  NN  E. d  e.  NN  (
a (AP `  ( K  +  1 ) ) d )  C_  ( `' F " { ( F `  A ) } ) )
100 fvex 5744 . . . 4  |-  ( F `
 A )  e. 
_V
101 sneq 3827 . . . . . . 7  |-  ( c  =  ( F `  A )  ->  { c }  =  { ( F `  A ) } )
102101imaeq2d 5205 . . . . . 6  |-  ( c  =  ( F `  A )  ->  ( `' F " { c } )  =  ( `' F " { ( F `  A ) } ) )
103102sseq2d 3378 . . . . 5  |-  ( c  =  ( F `  A )  ->  (
( a (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { c } )  <-> 
( a (AP `  ( K  +  1
) ) d ) 
C_  ( `' F " { ( F `  A ) } ) ) )
1041032rexbidv 2750 . . . 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 3045 . . 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 6108 . . 3  |-  ( 1 ... W )  e. 
_V
108 peano2nn0 10262 . . . 4  |-  ( K  e.  NN0  ->  ( K  +  1 )  e. 
NN0 )
1096, 108syl 16 . . 3  |-  ( ph  ->  ( K  +  1 )  e.  NN0 )
110107, 109, 26vdwmc 13348 . 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 225 1  |-  ( ph  ->  ( K  +  1 ) MonoAP  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    u. cun 3320    C_ wss 3322   {csn 3816   class class class wbr 4214   `'ccnv 4879   dom cdm 4880   "cima 4883    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   Fincfn 7111   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997    <_ cle 9123    - cmin 9293   NNcn 10002   NN0cn0 10223   ZZcz 10284   ZZ>=cuz 10490   ...cfz 11045  APcvdwa 13335   MonoAP cvdwm 13336
This theorem is referenced by:  vdwlem6  13356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-vdwap 13338  df-vdwmc 13339
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