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Theorem sadass 13665
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 13656 . . . . . 6  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0 )  ->  ( A sadd  B )  C_  NN0 )
213adant3 1008 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( A sadd  B ) 
C_  NN0 )
3 simp3 990 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  C  C_  NN0 )
4 sadcl 13656 . . . . 5  |-  ( ( ( A sadd  B ) 
C_  NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B ) sadd  C ) 
C_  NN0 )
52, 3, 4syl2anc 661 . . . 4  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B
) sadd  C )  C_  NN0 )
65sseld 3353 . . 3  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( k  e.  ( ( A sadd  B ) sadd 
C )  ->  k  e.  NN0 ) )
7 simp1 988 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  A  C_  NN0 )
8 sadcl 13656 . . . . . 6  |-  ( ( B  C_  NN0  /\  C  C_ 
NN0 )  ->  ( B sadd  C )  C_  NN0 )
983adant1 1006 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( B sadd  C ) 
C_  NN0 )
10 sadcl 13656 . . . . 5  |-  ( ( A  C_  NN0  /\  ( B sadd  C )  C_  NN0 )  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
117, 9, 10syl2anc 661 . . . 4  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
1211sseld 3353 . . 3  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( k  e.  ( A sadd  ( B sadd  C
) )  ->  k  e.  NN0 ) )
137adantr 465 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  A  C_ 
NN0 )
14 simpl2 992 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  B  C_ 
NN0 )
153adantr 465 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  C  C_ 
NN0 )
16 simpr 461 . . . . . . . . 9  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  NN0 )
17 1nn0 10593 . . . . . . . . . 10  |-  1  e.  NN0
1817a1i 11 . . . . . . . . 9  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  1  e.  NN0 )
1916, 18nn0addcld 10638 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
k  +  1 )  e.  NN0 )
2013, 14, 15, 19sadasslem 13664 . . . . . . 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 ) ) ) )
2120eleq2d 2508 . . . . . 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 ) ) ) ) )
22 elin 3537 . . . . . 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 ) ) ) )
23 elin 3537 . . . . . 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 ) ) ) )
2421, 22, 233bitr3g 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 ) ) ) ) )
25 nn0uz 10893 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
2616, 25syl6eleq 2531 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ( ZZ>= `  0 )
)
27 eluzfz2 11457 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  ( 0 ... k
) )
2826, 27syl 16 . . . . . . 7  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ( 0 ... k
) )
2916nn0zd 10743 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ZZ )
30 fzval3 11603 . . . . . . . 8  |-  ( k  e.  ZZ  ->  (
0 ... k )  =  ( 0..^ ( k  +  1 ) ) )
3129, 30syl 16 . . . . . . 7  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
0 ... k )  =  ( 0..^ ( k  +  1 ) ) )
3228, 31eleqtrd 2517 . . . . . 6  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ( 0..^ ( k  +  1 ) ) )
3332biantrud 507 . . . . 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 ) ) ) ) )
3432biantrud 507 . . . . 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 ) ) ) ) )
3524, 33, 343bitr4d 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
) ) ) )
3635ex 434 . . 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 ) ) ) ) )
376, 12, 36pm5.21ndd 354 . 2  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( k  e.  ( ( A sadd  B ) sadd 
C )  <->  k  e.  ( A sadd  ( B sadd  C ) ) ) )
3837eqrdv 2439 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3325    C_ wss 3326   ` cfv 5416  (class class class)co 6089   0cc0 9280   1c1 9281    + caddc 9283   NN0cn0 10577   ZZcz 10644   ZZ>=cuz 10859   ...cfz 11435  ..^cfzo 11546   sadd csad 13614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  df-tru 1372  df-fal 1375  df-had 1421  df-cad 1422  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-disj 4261  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-oi 7722  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-fz 11436  df-fzo 11547  df-fl 11640  df-mod 11707  df-seq 11805  df-exp 11864  df-hash 12102  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-clim 12964  df-sum 13162  df-dvds 13534  df-bits 13616  df-sad 13645
This theorem is referenced by:  bitsres  13667
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