MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sadass Structured version   Unicode version

Theorem sadass 14222
Description: Sequence addition is associative. (Contributed by Mario Carneiro, 9-Sep-2016.)
Assertion
Ref Expression
sadass  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B
) sadd  C )  =  ( A sadd  ( B sadd  C
) ) )

Proof of Theorem sadass
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 sadcl 14213 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0 )  ->  ( A sadd  B )  C_  NN0 )
2 sadcl 14213 . . . . 5  |-  ( ( ( A sadd  B ) 
C_  NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B ) sadd  C ) 
C_  NN0 )
31, 2stoic3 1630 . . . 4  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B
) sadd  C )  C_  NN0 )
43sseld 3440 . . 3  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( k  e.  ( ( A sadd  B ) sadd 
C )  ->  k  e.  NN0 ) )
5 simp1 997 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  A  C_  NN0 )
6 sadcl 14213 . . . . . 6  |-  ( ( B  C_  NN0  /\  C  C_ 
NN0 )  ->  ( B sadd  C )  C_  NN0 )
763adant1 1015 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( B sadd  C ) 
C_  NN0 )
8 sadcl 14213 . . . . 5  |-  ( ( A  C_  NN0  /\  ( B sadd  C )  C_  NN0 )  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
95, 7, 8syl2anc 659 . . . 4  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
109sseld 3440 . . 3  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( k  e.  ( A sadd  ( B sadd  C
) )  ->  k  e.  NN0 ) )
11 simpl1 1000 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  A  C_ 
NN0 )
12 simpl2 1001 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  B  C_ 
NN0 )
13 simpl3 1002 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  C  C_ 
NN0 )
14 simpr 459 . . . . . . . . 9  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  NN0 )
15 1nn0 10772 . . . . . . . . . 10  |-  1  e.  NN0
1615a1i 11 . . . . . . . . 9  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  1  e.  NN0 )
1714, 16nn0addcld 10817 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
k  +  1 )  e.  NN0 )
1811, 12, 13, 17sadasslem 14221 . . . . . . 7  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ ( k  +  1 ) ) )  =  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ ( k  +  1 ) ) ) )
1918eleq2d 2472 . . . . . 6  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
k  e.  ( ( ( A sadd  B ) sadd 
C )  i^i  (
0..^ ( k  +  1 ) ) )  <-> 
k  e.  ( ( A sadd  ( B sadd  C
) )  i^i  (
0..^ ( k  +  1 ) ) ) ) )
20 elin 3625 . . . . . 6  |-  ( k  e.  ( ( ( A sadd  B ) sadd  C
)  i^i  ( 0..^ ( k  +  1 ) ) )  <->  ( k  e.  ( ( A sadd  B
) sadd  C )  /\  k  e.  ( 0..^ ( k  +  1 ) ) ) )
21 elin 3625 . . . . . 6  |-  ( k  e.  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ ( k  +  1 ) ) )  <->  ( k  e.  ( A sadd  ( B sadd 
C ) )  /\  k  e.  ( 0..^ ( k  +  1 ) ) ) )
2219, 20, 213bitr3g 287 . . . . 5  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
( k  e.  ( ( A sadd  B ) sadd 
C )  /\  k  e.  ( 0..^ ( k  +  1 ) ) )  <->  ( k  e.  ( A sadd  ( B sadd 
C ) )  /\  k  e.  ( 0..^ ( k  +  1 ) ) ) ) )
23 nn0uz 11079 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
2414, 23syl6eleq 2500 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ( ZZ>= `  0 )
)
25 eluzfz2 11665 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  ( 0 ... k
) )
2624, 25syl 17 . . . . . . 7  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ( 0 ... k
) )
2714nn0zd 10926 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ZZ )
28 fzval3 11834 . . . . . . . 8  |-  ( k  e.  ZZ  ->  (
0 ... k )  =  ( 0..^ ( k  +  1 ) ) )
2927, 28syl 17 . . . . . . 7  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
0 ... k )  =  ( 0..^ ( k  +  1 ) ) )
3026, 29eleqtrd 2492 . . . . . 6  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ( 0..^ ( k  +  1 ) ) )
3130biantrud 505 . . . . 5  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
k  e.  ( ( A sadd  B ) sadd  C
)  <->  ( k  e.  ( ( A sadd  B
) sadd  C )  /\  k  e.  ( 0..^ ( k  +  1 ) ) ) ) )
3230biantrud 505 . . . . 5  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
k  e.  ( A sadd  ( B sadd  C ) )  <->  ( k  e.  ( A sadd  ( B sadd 
C ) )  /\  k  e.  ( 0..^ ( k  +  1 ) ) ) ) )
3322, 31, 323bitr4d 285 . . . 4  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
k  e.  ( ( A sadd  B ) sadd  C
)  <->  k  e.  ( A sadd  ( B sadd  C
) ) ) )
3433ex 432 . . 3  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( k  e.  NN0  ->  ( k  e.  ( ( A sadd  B ) sadd 
C )  <->  k  e.  ( A sadd  ( B sadd  C ) ) ) ) )
354, 10, 34pm5.21ndd 352 . 2  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( k  e.  ( ( A sadd  B ) sadd 
C )  <->  k  e.  ( A sadd  ( B sadd  C ) ) ) )
3635eqrdv 2399 1  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B
) sadd  C )  =  ( A sadd  ( B sadd  C
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    i^i cin 3412    C_ wss 3413   ` cfv 5525  (class class class)co 6234   0cc0 9442   1c1 9443    + caddc 9445   NN0cn0 10756   ZZcz 10825   ZZ>=cuz 11045   ...cfz 11643  ..^cfzo 11767   sadd csad 14171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-xor 1367  df-tru 1408  df-fal 1411  df-had 1461  df-cad 1462  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-disj 4366  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-2o 7088  df-oadd 7091  df-er 7268  df-map 7379  df-pm 7380  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-sup 7855  df-oi 7889  df-card 8272  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-rp 11184  df-fz 11644  df-fzo 11768  df-fl 11879  df-mod 11948  df-seq 12062  df-exp 12121  df-hash 12360  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-clim 13367  df-sum 13565  df-dvds 14088  df-bits 14173  df-sad 14202
This theorem is referenced by:  bitsres  14224
  Copyright terms: Public domain W3C validator