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Theorem fzsplit1nn0 25981
Description: Split a finite 1-based set of integers in the middle, allowing either end to be empty ( ( 1 ... 0 )). (Contributed by Stefan O'Rear, 8-Oct-2014.)
Assertion
Ref Expression
fzsplit1nn0  |-  ( ( A  e.  NN0  /\  B  e.  NN0  /\  A  <_  B )  ->  (
1 ... B )  =  ( ( 1 ... A )  u.  (
( A  +  1 ) ... B ) ) )

Proof of Theorem fzsplit1nn0
StepHypRef Expression
1 elnn0 10014 . . 3  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
2 nnge1 9817 . . . . . . . 8  |-  ( A  e.  NN  ->  1  <_  A )
32adantr 451 . . . . . . 7  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  1  <_  A )
4 simprr 733 . . . . . . 7  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  A  <_  B )
5 nnz 10092 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  ZZ )
65adantr 451 . . . . . . . 8  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  A  e.  ZZ )
7 1z 10100 . . . . . . . . 9  |-  1  e.  ZZ
87a1i 10 . . . . . . . 8  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  1  e.  ZZ )
9 nn0z 10093 . . . . . . . . 9  |-  ( B  e.  NN0  ->  B  e.  ZZ )
109ad2antrl 708 . . . . . . . 8  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  B  e.  ZZ )
11 elfz 10835 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  1  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  e.  ( 1 ... B )  <->  ( 1  <_  A  /\  A  <_  B ) ) )
126, 8, 10, 11syl3anc 1182 . . . . . . 7  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( A  e.  ( 1 ... B
)  <->  ( 1  <_  A  /\  A  <_  B
) ) )
133, 4, 12mpbir2and 888 . . . . . 6  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  A  e.  ( 1 ... B
) )
14 fzsplit 10863 . . . . . 6  |-  ( A  e.  ( 1 ... B )  ->  (
1 ... B )  =  ( ( 1 ... A )  u.  (
( A  +  1 ) ... B ) ) )
1513, 14syl 15 . . . . 5  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( 1 ... B )  =  ( ( 1 ... A )  u.  (
( A  +  1 ) ... B ) ) )
16 uncom 3353 . . . . . 6  |-  ( ( 1 ... A )  u.  ( ( A  +  1 ) ... B ) )  =  ( ( ( A  +  1 ) ... B )  u.  (
1 ... A ) )
17 oveq1 5907 . . . . . . . . . . 11  |-  ( A  =  0  ->  ( A  +  1 )  =  ( 0  +  1 ) )
1817adantr 451 . . . . . . . . . 10  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( A  +  1 )  =  ( 0  +  1 ) )
19 0p1e1 9884 . . . . . . . . . 10  |-  ( 0  +  1 )  =  1
2018, 19syl6eq 2364 . . . . . . . . 9  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( A  +  1 )  =  1 )
2120oveq1d 5915 . . . . . . . 8  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( ( A  +  1 ) ... B )  =  ( 1 ... B
) )
22 oveq2 5908 . . . . . . . . . 10  |-  ( A  =  0  ->  (
1 ... A )  =  ( 1 ... 0
) )
2322adantr 451 . . . . . . . . 9  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( 1 ... A )  =  ( 1 ... 0
) )
24 fz10 10861 . . . . . . . . 9  |-  ( 1 ... 0 )  =  (/)
2523, 24syl6eq 2364 . . . . . . . 8  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( 1 ... A )  =  (/) )
2621, 25uneq12d 3364 . . . . . . 7  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( (
( A  +  1 ) ... B )  u.  ( 1 ... A ) )  =  ( ( 1 ... B )  u.  (/) ) )
27 un0 3513 . . . . . . 7  |-  ( ( 1 ... B )  u.  (/) )  =  ( 1 ... B )
2826, 27syl6eq 2364 . . . . . 6  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( (
( A  +  1 ) ... B )  u.  ( 1 ... A ) )  =  ( 1 ... B
) )
2916, 28syl5req 2361 . . . . 5  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( 1 ... B )  =  ( ( 1 ... A )  u.  (
( A  +  1 ) ... B ) ) )
3015, 29jaoian 759 . . . 4  |-  ( ( ( A  e.  NN  \/  A  =  0
)  /\  ( B  e.  NN0  /\  A  <_  B ) )  -> 
( 1 ... B
)  =  ( ( 1 ... A )  u.  ( ( A  +  1 ) ... B ) ) )
3130ex 423 . . 3  |-  ( ( A  e.  NN  \/  A  =  0 )  ->  ( ( B  e.  NN0  /\  A  <_  B )  ->  (
1 ... B )  =  ( ( 1 ... A )  u.  (
( A  +  1 ) ... B ) ) ) )
321, 31sylbi 187 . 2  |-  ( A  e.  NN0  ->  ( ( B  e.  NN0  /\  A  <_  B )  -> 
( 1 ... B
)  =  ( ( 1 ... A )  u.  ( ( A  +  1 ) ... B ) ) ) )
33323impib 1149 1  |-  ( ( A  e.  NN0  /\  B  e.  NN0  /\  A  <_  B )  ->  (
1 ... B )  =  ( ( 1 ... A )  u.  (
( A  +  1 ) ... B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    u. cun 3184   (/)c0 3489   class class class wbr 4060  (class class class)co 5900   0cc0 8782   1c1 8783    + caddc 8785    <_ cle 8913   NNcn 9791   NN0cn0 10012   ZZcz 10071   ...cfz 10829
This theorem is referenced by:  eldioph2lem1  25987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830
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