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Theorem fvmptnn04ifb 19479
Description: The function value of a mapping from the nonnegative integers with four distinct cases for the second case. (Contributed by AV, 10-Nov-2019.)
Hypotheses
Ref Expression
fvmptnn04if.g  |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) ) )
fvmptnn04if.s  |-  ( ph  ->  S  e.  NN )
fvmptnn04if.n  |-  ( ph  ->  N  e.  NN0 )
Assertion
Ref Expression
fvmptnn04ifb  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  ->  ( G `  N )  =  [_ N  /  n ]_ B )
Distinct variable groups:    n, N    S, n    A, n    n, V
Allowed substitution hints:    ph( n)    B( n)    C( n)    D( n)    G( n)

Proof of Theorem fvmptnn04ifb
StepHypRef Expression
1 fvmptnn04if.g . 2  |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) ) )
2 fvmptnn04if.s . . 3  |-  ( ph  ->  S  e.  NN )
323ad2ant1 1017 . 2  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  ->  S  e.  NN )
4 fvmptnn04if.n . . 3  |-  ( ph  ->  N  e.  NN0 )
543ad2ant1 1017 . 2  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  ->  N  e.  NN0 )
6 simp3 998 . 2  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  ->  [_ N  /  n ]_ B  e.  V )
7 nn0re 10825 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  RR )
8 nn0ge0 10842 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  0  <_  N )
97, 8jca 532 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  e.  RR  /\  0  <_  N ) )
10 ne0gt0 9706 . . . . . . . . 9  |-  ( ( N  e.  RR  /\  0  <_  N )  -> 
( N  =/=  0  <->  0  <  N ) )
114, 9, 103syl 20 . . . . . . . 8  |-  ( ph  ->  ( N  =/=  0  <->  0  <  N ) )
1211biimprcd 225 . . . . . . 7  |-  ( 0  <  N  ->  ( ph  ->  N  =/=  0
) )
1312adantr 465 . . . . . 6  |-  ( ( 0  <  N  /\  N  <  S )  -> 
( ph  ->  N  =/=  0 ) )
1413impcom 430 . . . . 5  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S ) )  ->  N  =/=  0 )
15143adant3 1016 . . . 4  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  ->  N  =/=  0 )
16 df-ne 2654 . . . . . 6  |-  ( N  =/=  0  <->  -.  N  =  0 )
1716biimpi 194 . . . . 5  |-  ( N  =/=  0  ->  -.  N  =  0 )
1817pm2.21d 106 . . . 4  |-  ( N  =/=  0  ->  ( N  =  0  ->  [_ N  /  n ]_ B  =  [_ N  /  n ]_ A ) )
1915, 18syl 16 . . 3  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  ->  ( N  =  0  ->  [_ N  /  n ]_ B  =  [_ N  /  n ]_ A ) )
2019imp 429 . 2  |-  ( ( ( ph  /\  (
0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  /\  N  =  0
)  ->  [_ N  /  n ]_ B  =  [_ N  /  n ]_ A
)
21 eqidd 2458 . 2  |-  ( ( ( ph  /\  (
0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  /\  0  <  N  /\  N  <  S )  ->  [_ N  /  n ]_ B  =  [_ N  /  n ]_ B )
224, 7syl 16 . . . . . . . . 9  |-  ( ph  ->  N  e.  RR )
2322adantr 465 . . . . . . . 8  |-  ( (
ph  /\  N  <  S )  ->  N  e.  RR )
24 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  N  <  S )  ->  N  <  S )
2523, 24ltned 9738 . . . . . . 7  |-  ( (
ph  /\  N  <  S )  ->  N  =/=  S )
2625neneqd 2659 . . . . . 6  |-  ( (
ph  /\  N  <  S )  ->  -.  N  =  S )
2726adantrl 715 . . . . 5  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S ) )  ->  -.  N  =  S
)
28273adant3 1016 . . . 4  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  ->  -.  N  =  S )
2928pm2.21d 106 . . 3  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  ->  ( N  =  S  ->  [_ N  /  n ]_ B  =  [_ N  /  n ]_ C ) )
3029imp 429 . 2  |-  ( ( ( ph  /\  (
0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  /\  N  =  S
)  ->  [_ N  /  n ]_ B  =  [_ N  /  n ]_ C
)
312nnred 10571 . . . . . . . . 9  |-  ( ph  ->  S  e.  RR )
32 ltnsym 9700 . . . . . . . . 9  |-  ( ( N  e.  RR  /\  S  e.  RR )  ->  ( N  <  S  ->  -.  S  <  N
) )
3322, 31, 32syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( N  <  S  ->  -.  S  <  N
) )
3433com12 31 . . . . . . 7  |-  ( N  <  S  ->  ( ph  ->  -.  S  <  N ) )
3534adantl 466 . . . . . 6  |-  ( ( 0  <  N  /\  N  <  S )  -> 
( ph  ->  -.  S  <  N ) )
3635impcom 430 . . . . 5  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S ) )  ->  -.  S  <  N )
37363adant3 1016 . . . 4  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  ->  -.  S  <  N )
3837pm2.21d 106 . . 3  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  ->  ( S  <  N  ->  [_ N  /  n ]_ B  = 
[_ N  /  n ]_ D ) )
3938imp 429 . 2  |-  ( ( ( ph  /\  (
0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  /\  S  <  N )  ->  [_ N  /  n ]_ B  =  [_ N  /  n ]_ D )
401, 3, 5, 6, 20, 21, 30, 39fvmptnn04if 19477 1  |-  ( (
ph  /\  ( 0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  ->  ( G `  N )  =  [_ N  /  n ]_ B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   [_csb 3430   ifcif 3944   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594   RRcr 9508   0cc0 9509    < clt 9645    <_ cle 9646   NNcn 10556   NN0cn0 10816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817
This theorem is referenced by: (None)
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