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Theorem fzsplit1nn0 29235
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 10687 . . 3  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
2 nnge1 10454 . . . . . . . 8  |-  ( A  e.  NN  ->  1  <_  A )
32adantr 465 . . . . . . 7  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  1  <_  A )
4 simprr 756 . . . . . . 7  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  A  <_  B )
5 nnz 10774 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  ZZ )
65adantr 465 . . . . . . . 8  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  A  e.  ZZ )
7 1z 10782 . . . . . . . . 9  |-  1  e.  ZZ
87a1i 11 . . . . . . . 8  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  1  e.  ZZ )
9 nn0z 10775 . . . . . . . . 9  |-  ( B  e.  NN0  ->  B  e.  ZZ )
109ad2antrl 727 . . . . . . . 8  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  B  e.  ZZ )
11 elfz 11555 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  1  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  e.  ( 1 ... B )  <->  ( 1  <_  A  /\  A  <_  B ) ) )
126, 8, 10, 11syl3anc 1219 . . . . . . 7  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( A  e.  ( 1 ... B
)  <->  ( 1  <_  A  /\  A  <_  B
) ) )
133, 4, 12mpbir2and 913 . . . . . 6  |-  ( ( A  e.  NN  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  A  e.  ( 1 ... B
) )
14 fzsplit 11587 . . . . . 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 3603 . . . . . 6  |-  ( ( 1 ... A )  u.  ( ( A  +  1 ) ... B ) )  =  ( ( ( A  +  1 ) ... B )  u.  (
1 ... A ) )
17 oveq1 6202 . . . . . . . . . . 11  |-  ( A  =  0  ->  ( A  +  1 )  =  ( 0  +  1 ) )
1817adantr 465 . . . . . . . . . 10  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( A  +  1 )  =  ( 0  +  1 ) )
19 0p1e1 10539 . . . . . . . . . 10  |-  ( 0  +  1 )  =  1
2018, 19syl6eq 2509 . . . . . . . . 9  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( A  +  1 )  =  1 )
2120oveq1d 6210 . . . . . . . 8  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( ( A  +  1 ) ... B )  =  ( 1 ... B
) )
22 oveq2 6203 . . . . . . . . . 10  |-  ( A  =  0  ->  (
1 ... A )  =  ( 1 ... 0
) )
2322adantr 465 . . . . . . . . 9  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( 1 ... A )  =  ( 1 ... 0
) )
24 fz10 11582 . . . . . . . . 9  |-  ( 1 ... 0 )  =  (/)
2523, 24syl6eq 2509 . . . . . . . 8  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( 1 ... A )  =  (/) )
2621, 25uneq12d 3614 . . . . . . 7  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( (
( A  +  1 ) ... B )  u.  ( 1 ... A ) )  =  ( ( 1 ... B )  u.  (/) ) )
27 un0 3765 . . . . . . 7  |-  ( ( 1 ... B )  u.  (/) )  =  ( 1 ... B )
2826, 27syl6eq 2509 . . . . . 6  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( (
( A  +  1 ) ... B )  u.  ( 1 ... A ) )  =  ( 1 ... B
) )
2916, 28syl5req 2506 . . . . 5  |-  ( ( A  =  0  /\  ( B  e.  NN0  /\  A  <_  B )
)  ->  ( 1 ... B )  =  ( ( 1 ... A )  u.  (
( A  +  1 ) ... B ) ) )
3015, 29jaoian 782 . . . 4  |-  ( ( ( A  e.  NN  \/  A  =  0
)  /\  ( B  e.  NN0  /\  A  <_  B ) )  -> 
( 1 ... B
)  =  ( ( 1 ... A )  u.  ( ( A  +  1 ) ... B ) ) )
3130ex 434 . . 3  |-  ( ( A  e.  NN  \/  A  =  0 )  ->  ( ( B  e.  NN0  /\  A  <_  B )  ->  (
1 ... B )  =  ( ( 1 ... A )  u.  (
( A  +  1 ) ... B ) ) ) )
321, 31sylbi 195 . 2  |-  ( A  e.  NN0  ->  ( ( B  e.  NN0  /\  A  <_  B )  -> 
( 1 ... B
)  =  ( ( 1 ... A )  u.  ( ( A  +  1 ) ... B ) ) ) )
33323impib 1186 1  |-  ( ( A  e.  NN0  /\  B  e.  NN0  /\  A  <_  B )  ->  (
1 ... B )  =  ( ( 1 ... A )  u.  (
( A  +  1 ) ... B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    u. cun 3429   (/)c0 3740   class class class wbr 4395  (class class class)co 6195   0cc0 9388   1c1 9389    + caddc 9391    <_ cle 9525   NNcn 10428   NN0cn0 10685   ZZcz 10752   ...cfz 11549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550
This theorem is referenced by:  eldioph2lem1  29241
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