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Theorem dmgmseqn0 23711
Description: If  A is not a nonpositive integer, then  prod_ m  e.  ( 1 ... N
) A  +  m is nonzero for any  N. (Contributed by Mario Carneiro, 12-Jul-2014.)
Hypothesis
Ref Expression
gamcvg.1  |-  S  =  seq  0 (  x.  ,  ( m  e. 
_V  |->  ( A  +  m ) ) )
Assertion
Ref Expression
dmgmseqn0  |-  ( ( A  e.  ( CC 
\  ( ZZ  \  NN ) )  /\  N  e.  NN0 )  ->  ( S `  N )  =/=  0 )
Distinct variable group:    A, m
Allowed substitution hints:    S( m)    N( m)

Proof of Theorem dmgmseqn0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gamcvg.1 . . . 4  |-  S  =  seq  0 (  x.  ,  ( m  e. 
_V  |->  ( A  +  m ) ) )
21fveq1i 5542 . . 3  |-  ( S `
 N )  =  (  seq  0 (  x.  ,  ( m  e.  _V  |->  ( A  +  m ) ) ) `  N )
3 simpr 447 . . . . 5  |-  ( ( A  e.  ( CC 
\  ( ZZ  \  NN ) )  /\  N  e.  NN0 )  ->  N  e.  NN0 )
4 nn0uz 10278 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
53, 4syl6eleq 2386 . . . 4  |-  ( ( A  e.  ( CC 
\  ( ZZ  \  NN ) )  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= `  0 )
)
6 vex 2804 . . . . . 6  |-  x  e. 
_V
7 oveq2 5882 . . . . . . 7  |-  ( m  =  x  ->  ( A  +  m )  =  ( A  +  x ) )
8 eqid 2296 . . . . . . 7  |-  ( m  e.  _V  |->  ( A  +  m ) )  =  ( m  e. 
_V  |->  ( A  +  m ) )
9 ovex 5899 . . . . . . 7  |-  ( A  +  x )  e. 
_V
107, 8, 9fvmpt 5618 . . . . . 6  |-  ( x  e.  _V  ->  (
( m  e.  _V  |->  ( A  +  m
) ) `  x
)  =  ( A  +  x ) )
116, 10ax-mp 8 . . . . 5  |-  ( ( m  e.  _V  |->  ( A  +  m ) ) `  x )  =  ( A  +  x )
12 eldifi 3311 . . . . . . . 8  |-  ( A  e.  ( CC  \ 
( ZZ  \  NN ) )  ->  A  e.  CC )
1312adantr 451 . . . . . . 7  |-  ( ( A  e.  ( CC 
\  ( ZZ  \  NN ) )  /\  N  e.  NN0 )  ->  A  e.  CC )
14 elfznn0 10838 . . . . . . . 8  |-  ( x  e.  ( 0 ... N )  ->  x  e.  NN0 )
1514nn0cnd 10036 . . . . . . 7  |-  ( x  e.  ( 0 ... N )  ->  x  e.  CC )
16 addcl 8835 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  +  x
)  e.  CC )
1713, 15, 16syl2an 463 . . . . . 6  |-  ( ( ( A  e.  ( CC  \  ( ZZ 
\  NN ) )  /\  N  e.  NN0 )  /\  x  e.  ( 0 ... N ) )  ->  ( A  +  x )  e.  CC )
18 simpl 443 . . . . . . 7  |-  ( ( A  e.  ( CC 
\  ( ZZ  \  NN ) )  /\  N  e.  NN0 )  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )
19 dmgmaddn0 23710 . . . . . . 7  |-  ( ( A  e.  ( CC 
\  ( ZZ  \  NN ) )  /\  x  e.  NN0 )  ->  ( A  +  x )  =/=  0 )
2018, 14, 19syl2an 463 . . . . . 6  |-  ( ( ( A  e.  ( CC  \  ( ZZ 
\  NN ) )  /\  N  e.  NN0 )  /\  x  e.  ( 0 ... N ) )  ->  ( A  +  x )  =/=  0
)
21 eldifsn 3762 . . . . . 6  |-  ( ( A  +  x )  e.  ( CC  \  { 0 } )  <-> 
( ( A  +  x )  e.  CC  /\  ( A  +  x
)  =/=  0 ) )
2217, 20, 21sylanbrc 645 . . . . 5  |-  ( ( ( A  e.  ( CC  \  ( ZZ 
\  NN ) )  /\  N  e.  NN0 )  /\  x  e.  ( 0 ... N ) )  ->  ( A  +  x )  e.  ( CC  \  { 0 } ) )
2311, 22syl5eqel 2380 . . . 4  |-  ( ( ( A  e.  ( CC  \  ( ZZ 
\  NN ) )  /\  N  e.  NN0 )  /\  x  e.  ( 0 ... N ) )  ->  ( (
m  e.  _V  |->  ( A  +  m ) ) `  x )  e.  ( CC  \  { 0 } ) )
24 eldifsn 3762 . . . . . 6  |-  ( x  e.  ( CC  \  { 0 } )  <-> 
( x  e.  CC  /\  x  =/=  0 ) )
25 eldifsn 3762 . . . . . 6  |-  ( y  e.  ( CC  \  { 0 } )  <-> 
( y  e.  CC  /\  y  =/=  0 ) )
26 mulcl 8837 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
2726ad2ant2r 727 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  e.  CC )
28 mulne0 9426 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  =/=  0 )
29 eldifsn 3762 . . . . . . 7  |-  ( ( x  x.  y )  e.  ( CC  \  { 0 } )  <-> 
( ( x  x.  y )  e.  CC  /\  ( x  x.  y
)  =/=  0 ) )
3027, 28, 29sylanbrc 645 . . . . . 6  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  e.  ( CC 
\  { 0 } ) )
3124, 25, 30syl2anb 465 . . . . 5  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x  x.  y )  e.  ( CC  \  { 0 } ) )
3231adantl 452 . . . 4  |-  ( ( ( A  e.  ( CC  \  ( ZZ 
\  NN ) )  /\  N  e.  NN0 )  /\  ( x  e.  ( CC  \  {
0 } )  /\  y  e.  ( CC  \  { 0 } ) ) )  ->  (
x  x.  y )  e.  ( CC  \  { 0 } ) )
335, 23, 32seqcl 11082 . . 3  |-  ( ( A  e.  ( CC 
\  ( ZZ  \  NN ) )  /\  N  e.  NN0 )  ->  (  seq  0 (  x.  , 
( m  e.  _V  |->  ( A  +  m
) ) ) `  N )  e.  ( CC  \  { 0 } ) )
342, 33syl5eqel 2380 . 2  |-  ( ( A  e.  ( CC 
\  ( ZZ  \  NN ) )  /\  N  e.  NN0 )  ->  ( S `  N )  e.  ( CC  \  {
0 } ) )
35 eldifsni 3763 . 2  |-  ( ( S `  N )  e.  ( CC  \  { 0 } )  ->  ( S `  N )  =/=  0
)
3634, 35syl 15 1  |-  ( ( A  e.  ( CC 
\  ( ZZ  \  NN ) )  /\  N  e.  NN0 )  ->  ( S `  N )  =/=  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    \ cdif 3162   {csn 3653    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753    + caddc 8756    x. cmul 8758   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063
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