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Theorem sadass 12983
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 12974 . . . . . 6  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0 )  ->  ( A sadd  B )  C_  NN0 )
213adant3 977 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( A sadd  B ) 
C_  NN0 )
3 simp3 959 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  C  C_  NN0 )
4 sadcl 12974 . . . . 5  |-  ( ( ( A sadd  B ) 
C_  NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B ) sadd  C ) 
C_  NN0 )
52, 3, 4syl2anc 643 . . . 4  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B
) sadd  C )  C_  NN0 )
65sseld 3347 . . 3  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( k  e.  ( ( A sadd  B ) sadd 
C )  ->  k  e.  NN0 ) )
7 simp1 957 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  A  C_  NN0 )
8 sadcl 12974 . . . . . 6  |-  ( ( B  C_  NN0  /\  C  C_ 
NN0 )  ->  ( B sadd  C )  C_  NN0 )
983adant1 975 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( B sadd  C ) 
C_  NN0 )
10 sadcl 12974 . . . . 5  |-  ( ( A  C_  NN0  /\  ( B sadd  C )  C_  NN0 )  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
117, 9, 10syl2anc 643 . . . 4  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
1211sseld 3347 . . 3  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( k  e.  ( A sadd  ( B sadd  C
) )  ->  k  e.  NN0 ) )
137adantr 452 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  A  C_ 
NN0 )
14 simpl2 961 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  B  C_ 
NN0 )
153adantr 452 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  C  C_ 
NN0 )
16 simpr 448 . . . . . . . . 9  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  NN0 )
17 1nn0 10237 . . . . . . . . . 10  |-  1  e.  NN0
1817a1i 11 . . . . . . . . 9  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  1  e.  NN0 )
1916, 18nn0addcld 10278 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
k  +  1 )  e.  NN0 )
2013, 14, 15, 19sadasslem 12982 . . . . . . 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 2503 . . . . . 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 3530 . . . . . 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 3530 . . . . . 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 279 . . . . 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 10520 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
2616, 25syl6eleq 2526 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ( ZZ>= `  0 )
)
27 eluzfz2 11065 . . . . . . . 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 10373 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ZZ )
30 fzval3 11180 . . . . . . . 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 2512 . . . . . 6  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ( 0..^ ( k  +  1 ) ) )
3332biantrud 494 . . . . 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 494 . . . . 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 277 . . . 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 424 . . 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 344 . 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 2434 1  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B
) sadd  C )  =  ( A sadd  ( B sadd  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991    + caddc 8993   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043  ..^cfzo 11135   sadd csad 12932
This theorem is referenced by:  bitsres  12985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1314  df-tru 1328  df-had 1389  df-cad 1390  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-disj 4183  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-dvds 12853  df-bits 12934  df-sad 12963
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