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Theorem sadass 12678
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 12669 . . . . . 6  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0 )  ->  ( A sadd  B )  C_  NN0 )
213adant3 975 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( A sadd  B ) 
C_  NN0 )
3 simp3 957 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  C  C_  NN0 )
4 sadcl 12669 . . . . 5  |-  ( ( ( A sadd  B ) 
C_  NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B ) sadd  C ) 
C_  NN0 )
52, 3, 4syl2anc 642 . . . 4  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B
) sadd  C )  C_  NN0 )
65sseld 3192 . . 3  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( k  e.  ( ( A sadd  B ) sadd 
C )  ->  k  e.  NN0 ) )
7 simp1 955 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  A  C_  NN0 )
8 sadcl 12669 . . . . . 6  |-  ( ( B  C_  NN0  /\  C  C_ 
NN0 )  ->  ( B sadd  C )  C_  NN0 )
983adant1 973 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( B sadd  C ) 
C_  NN0 )
10 sadcl 12669 . . . . 5  |-  ( ( A  C_  NN0  /\  ( B sadd  C )  C_  NN0 )  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
117, 9, 10syl2anc 642 . . . 4  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
1211sseld 3192 . . 3  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( k  e.  ( A sadd  ( B sadd  C
) )  ->  k  e.  NN0 ) )
137adantr 451 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  A  C_ 
NN0 )
14 simpl2 959 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  B  C_ 
NN0 )
153adantr 451 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  C  C_ 
NN0 )
16 simpr 447 . . . . . . . . 9  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  NN0 )
17 1nn0 9997 . . . . . . . . . 10  |-  1  e.  NN0
1817a1i 10 . . . . . . . . 9  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  1  e.  NN0 )
1916, 18nn0addcld 10038 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
k  +  1 )  e.  NN0 )
2013, 14, 15, 19sadasslem 12677 . . . . . . 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 2363 . . . . . 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 3371 . . . . . 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 3371 . . . . . 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 278 . . . . 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 10278 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
2616, 25syl6eleq 2386 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ( ZZ>= `  0 )
)
27 eluzfz2 10820 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  ( 0 ... k
) )
2826, 27syl 15 . . . . . . 7  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ( 0 ... k
) )
2916nn0zd 10131 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ZZ )
30 fzval3 10927 . . . . . . . 8  |-  ( k  e.  ZZ  ->  (
0 ... k )  =  ( 0..^ ( k  +  1 ) ) )
3129, 30syl 15 . . . . . . 7  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
0 ... k )  =  ( 0..^ ( k  +  1 ) ) )
3228, 31eleqtrd 2372 . . . . . 6  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ( 0..^ ( k  +  1 ) ) )
3332biantrud 493 . . . . 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 493 . . . . 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 276 . . . 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 423 . . 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 343 . 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 2294 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    + caddc 8756   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798  ..^cfzo 10886   sadd csad 12627
This theorem is referenced by:  bitsres  12680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-xor 1296  df-tru 1310  df-had 1370  df-cad 1371  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-bits 12629  df-sad 12658
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