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Theorem fzsplit1nn0 26813
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 10224 . . 3  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
2 nnge1 10027 . . . . . . . 8  |-  ( A  e.  NN  ->  1  <_  A )
32adantr 453 . . . . . . 7  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  1  <_  A )
4 simprr 735 . . . . . . 7  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  A  <_  B )
5 nnz 10304 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  ZZ )
65adantr 453 . . . . . . . 8  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  A  e.  ZZ )
7 1z 10312 . . . . . . . . 9  |-  1  e.  ZZ
87a1i 11 . . . . . . . 8  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  1  e.  ZZ )
9 nn0z 10305 . . . . . . . . 9  |-  ( B  e.  NN0  ->  B  e.  ZZ )
109ad2antrl 710 . . . . . . . 8  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  B  e.  ZZ )
11 elfz 11050 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  1  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  e.  ( 1 ... B )  <->  ( 1  <_  A  /\  A  <_  B ) ) )
126, 8, 10, 11syl3anc 1185 . . . . . . 7  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( A  e.  ( 1 ... B
)  <->  ( 1  <_  A  /\  A  <_  B
) ) )
133, 4, 12mpbir2and 890 . . . . . 6  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  A  e.  ( 1 ... B
) )
14 fzsplit 11078 . . . . . 6  |-  ( A  e.  ( 1 ... B )  ->  (
1 ... B )  =  ( ( 1 ... A )  u.  (
( A  +  1 ) ... B ) ) )
1513, 14syl 16 . . . . 5  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( 1 ... B )  =  ( ( 1 ... A )  u.  (
( A  +  1 ) ... B ) ) )
16 uncom 3492 . . . . . 6  |-  ( ( 1 ... A )  u.  ( ( A  +  1 ) ... B ) )  =  ( ( ( A  +  1 ) ... B )  u.  (
1 ... A ) )
17 oveq1 6089 . . . . . . . . . . 11  |-  ( A  =  0  ->  ( A  +  1 )  =  ( 0  +  1 ) )
1817adantr 453 . . . . . . . . . 10  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( A  +  1 )  =  ( 0  +  1 ) )
19 0p1e1 10094 . . . . . . . . . 10  |-  ( 0  +  1 )  =  1
2018, 19syl6eq 2485 . . . . . . . . 9  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( A  +  1 )  =  1 )
2120oveq1d 6097 . . . . . . . 8  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( ( A  +  1 ) ... B )  =  ( 1 ... B
) )
22 oveq2 6090 . . . . . . . . . 10  |-  ( A  =  0  ->  (
1 ... A )  =  ( 1 ... 0
) )
2322adantr 453 . . . . . . . . 9  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( 1 ... A )  =  ( 1 ... 0
) )
24 fz10 11076 . . . . . . . . 9  |-  ( 1 ... 0 )  =  (/)
2523, 24syl6eq 2485 . . . . . . . 8  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( 1 ... A )  =  (/) )
2621, 25uneq12d 3503 . . . . . . 7  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( (
( A  +  1 ) ... B )  u.  ( 1 ... A ) )  =  ( ( 1 ... B )  u.  (/) ) )
27 un0 3653 . . . . . . 7  |-  ( ( 1 ... B )  u.  (/) )  =  ( 1 ... B )
2826, 27syl6eq 2485 . . . . . 6  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( (
( A  +  1 ) ... B )  u.  ( 1 ... A ) )  =  ( 1 ... B
) )
2916, 28syl5req 2482 . . . . 5  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( 1 ... B )  =  ( ( 1 ... A )  u.  (
( A  +  1 ) ... B ) ) )
3015, 29jaoian 761 . . . 4  |-  ( ( ( A  e.  NN  \/  A  =  0
)  /\  ( B  e.  NN0  /\  A  <_  B ) )  -> 
( 1 ... B
)  =  ( ( 1 ... A )  u.  ( ( A  +  1 ) ... B ) ) )
3130ex 425 . . 3  |-  ( ( A  e.  NN  \/  A  =  0 )  ->  ( ( B  e.  NN0  /\  A  <_  B )  ->  (
1 ... B )  =  ( ( 1 ... A )  u.  (
( A  +  1 ) ... B ) ) ) )
321, 31sylbi 189 . 2  |-  ( A  e.  NN0  ->  ( ( B  e.  NN0  /\  A  <_  B )  -> 
( 1 ... B
)  =  ( ( 1 ... A )  u.  ( ( A  +  1 ) ... B ) ) ) )
33323impib 1152 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 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    u. cun 3319   (/)c0 3629   class class class wbr 4213  (class class class)co 6082   0cc0 8991   1c1 8992    + caddc 8994    <_ cle 9122   NNcn 10001   NN0cn0 10222   ZZcz 10283   ...cfz 11044
This theorem is referenced by:  eldioph2lem1  26819
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 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-n0 10223  df-z 10284  df-uz 10490  df-fz 11045
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