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Theorem sadass 13969
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 13960 . . . . . 6  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0 )  ->  ( A sadd  B )  C_  NN0 )
213adant3 1011 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( A sadd  B ) 
C_  NN0 )
3 simp3 993 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  C  C_  NN0 )
4 sadcl 13960 . . . . 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 3496 . . 3  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( k  e.  ( ( A sadd  B ) sadd 
C )  ->  k  e.  NN0 ) )
7 simp1 991 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  A  C_  NN0 )
8 sadcl 13960 . . . . . 6  |-  ( ( B  C_  NN0  /\  C  C_ 
NN0 )  ->  ( B sadd  C )  C_  NN0 )
983adant1 1009 . . . . 5  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0  /\  C  C_  NN0 )  ->  ( B sadd  C ) 
C_  NN0 )
10 sadcl 13960 . . . . 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 3496 . . 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 995 . . . . . . . 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 10800 . . . . . . . . . 10  |-  1  e.  NN0
1817a1i 11 . . . . . . . . 9  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  1  e.  NN0 )
1916, 18nn0addcld 10845 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  (
k  +  1 )  e.  NN0 )
2013, 14, 15, 19sadasslem 13968 . . . . . . 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 2530 . . . . . 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 3680 . . . . . 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 3680 . . . . . 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 11105 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
2616, 25syl6eleq 2558 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ( ZZ>= `  0 )
)
27 eluzfz2 11683 . . . . . . . 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 10953 . . . . . . . 8  |-  ( ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  ZZ )
30 fzval3 11842 . . . . . . . 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 2550 . . . . . 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 2457 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 968    = wceq 1374    e. wcel 1762    i^i cin 3468    C_ wss 3469   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482    + caddc 9484   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   ...cfz 11661  ..^cfzo 11781   sadd csad 13918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-xor 1356  df-tru 1377  df-fal 1380  df-had 1426  df-cad 1427  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-disj 4411  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-dvds 13837  df-bits 13920  df-sad 13949
This theorem is referenced by:  bitsres  13971
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