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Theorem saddisjlem 12968
Description: Lemma for sadadd 12971. (Contributed by Mario Carneiro, 9-Sep-2016.)
Hypotheses
Ref Expression
saddisj.1  |-  ( ph  ->  A  C_  NN0 )
saddisj.2  |-  ( ph  ->  B  C_  NN0 )
saddisj.3  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
saddisjlem.c  |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) )
saddisjlem.3  |-  ( ph  ->  N  e.  NN0 )
Assertion
Ref Expression
saddisjlem  |-  ( ph  ->  ( N  e.  ( A sadd  B )  <->  N  e.  ( A  u.  B
) ) )
Distinct variable groups:    m, c, n    A, c, m    B, c, m    n, N
Allowed substitution hints:    ph( m, n, c)    A( n)    B( n)    C( m, n, c)    N( m, c)

Proof of Theorem saddisjlem
Dummy variables  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 saddisj.1 . . 3  |-  ( ph  ->  A  C_  NN0 )
2 saddisj.2 . . 3  |-  ( ph  ->  B  C_  NN0 )
3 saddisjlem.c . . 3  |-  C  =  seq  0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) )
4 saddisjlem.3 . . 3  |-  ( ph  ->  N  e.  NN0 )
51, 2, 3, 4sadval 12960 . 2  |-  ( ph  ->  ( N  e.  ( A sadd  B )  <-> hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `  N ) ) ) )
6 fveq2 5720 . . . . . . . 8  |-  ( x  =  0  ->  ( C `  x )  =  ( C ` 
0 ) )
76eleq2d 2502 . . . . . . 7  |-  ( x  =  0  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  0 ) ) )
87notbid 286 . . . . . 6  |-  ( x  =  0  ->  ( -.  (/)  e.  ( C `
 x )  <->  -.  (/)  e.  ( C `  0 ) ) )
98imbi2d 308 . . . . 5  |-  ( x  =  0  ->  (
( ph  ->  -.  (/)  e.  ( C `  x ) )  <->  ( ph  ->  -.  (/)  e.  ( C ` 
0 ) ) ) )
10 fveq2 5720 . . . . . . . 8  |-  ( x  =  k  ->  ( C `  x )  =  ( C `  k ) )
1110eleq2d 2502 . . . . . . 7  |-  ( x  =  k  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  k ) ) )
1211notbid 286 . . . . . 6  |-  ( x  =  k  ->  ( -.  (/)  e.  ( C `
 x )  <->  -.  (/)  e.  ( C `  k ) ) )
1312imbi2d 308 . . . . 5  |-  ( x  =  k  ->  (
( ph  ->  -.  (/)  e.  ( C `  x ) )  <->  ( ph  ->  -.  (/)  e.  ( C `  k ) ) ) )
14 fveq2 5720 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( C `  x )  =  ( C `  ( k  +  1 ) ) )
1514eleq2d 2502 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  ( k  +  1 ) ) ) )
1615notbid 286 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( -.  (/)  e.  ( C `
 x )  <->  -.  (/)  e.  ( C `  ( k  +  1 ) ) ) )
1716imbi2d 308 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
( ph  ->  -.  (/)  e.  ( C `  x ) )  <->  ( ph  ->  -.  (/)  e.  ( C `  ( k  +  1 ) ) ) ) )
18 fveq2 5720 . . . . . . . 8  |-  ( x  =  N  ->  ( C `  x )  =  ( C `  N ) )
1918eleq2d 2502 . . . . . . 7  |-  ( x  =  N  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  N ) ) )
2019notbid 286 . . . . . 6  |-  ( x  =  N  ->  ( -.  (/)  e.  ( C `
 x )  <->  -.  (/)  e.  ( C `  N ) ) )
2120imbi2d 308 . . . . 5  |-  ( x  =  N  ->  (
( ph  ->  -.  (/)  e.  ( C `  x ) )  <->  ( ph  ->  -.  (/)  e.  ( C `  N ) ) ) )
221, 2, 3sadc0 12958 . . . . 5  |-  ( ph  ->  -.  (/)  e.  ( C `
 0 ) )
23 noel 3624 . . . . . . . . 9  |-  -.  k  e.  (/)
241ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  ->  A  C_  NN0 )
252ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  ->  B  C_  NN0 )
26 simplr 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  -> 
k  e.  NN0 )
2724, 25, 3, 26sadcp1 12959 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  -> 
( (/)  e.  ( C `
 ( k  +  1 ) )  <-> cadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `  k ) ) ) )
28 cad0 1409 . . . . . . . . . . 11  |-  ( -.  (/)  e.  ( C `  k )  ->  (cadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `
 k ) )  <-> 
( k  e.  A  /\  k  e.  B
) ) )
2928adantl 453 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  -> 
(cadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `  k ) )  <->  ( k  e.  A  /\  k  e.  B ) ) )
30 elin 3522 . . . . . . . . . . 11  |-  ( k  e.  ( A  i^i  B )  <->  ( k  e.  A  /\  k  e.  B ) )
31 saddisj.3 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
3231ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  -> 
( A  i^i  B
)  =  (/) )
3332eleq2d 2502 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  -> 
( k  e.  ( A  i^i  B )  <-> 
k  e.  (/) ) )
3430, 33syl5bbr 251 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  -> 
( ( k  e.  A  /\  k  e.  B )  <->  k  e.  (/) ) )
3527, 29, 343bitrd 271 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  -> 
( (/)  e.  ( C `
 ( k  +  1 ) )  <->  k  e.  (/) ) )
3623, 35mtbiri 295 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  ->  -.  (/)  e.  ( C `
 ( k  +  1 ) ) )
3736ex 424 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( -.  (/) 
e.  ( C `  k )  ->  -.  (/) 
e.  ( C `  ( k  +  1 ) ) ) )
3837expcom 425 . . . . . 6  |-  ( k  e.  NN0  ->  ( ph  ->  ( -.  (/)  e.  ( C `  k )  ->  -.  (/)  e.  ( C `  ( k  +  1 ) ) ) ) )
3938a2d 24 . . . . 5  |-  ( k  e.  NN0  ->  ( (
ph  ->  -.  (/)  e.  ( C `  k ) )  ->  ( ph  ->  -.  (/)  e.  ( C `
 ( k  +  1 ) ) ) ) )
409, 13, 17, 21, 22, 39nn0ind 10358 . . . 4  |-  ( N  e.  NN0  ->  ( ph  ->  -.  (/)  e.  ( C `
 N ) ) )
414, 40mpcom 34 . . 3  |-  ( ph  ->  -.  (/)  e.  ( C `
 N ) )
42 hadrot 1399 . . . 4  |-  (hadd (
(/)  e.  ( C `  N ) ,  N  e.  A ,  N  e.  B )  <-> hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `  N ) ) )
43 had0 1412 . . . 4  |-  ( -.  (/)  e.  ( C `  N )  ->  (hadd ( (/)  e.  ( C `
 N ) ,  N  e.  A ,  N  e.  B )  <->  ( N  e.  A  \/_  N  e.  B )
) )
4442, 43syl5bbr 251 . . 3  |-  ( -.  (/)  e.  ( C `  N )  ->  (hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
 N ) )  <-> 
( N  e.  A  \/_  N  e.  B ) ) )
4541, 44syl 16 . 2  |-  ( ph  ->  (hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `  N ) )  <->  ( N  e.  A  \/_  N  e.  B ) ) )
46 noel 3624 . . . . 5  |-  -.  N  e.  (/)
47 elin 3522 . . . . . 6  |-  ( N  e.  ( A  i^i  B )  <->  ( N  e.  A  /\  N  e.  B ) )
4831eleq2d 2502 . . . . . 6  |-  ( ph  ->  ( N  e.  ( A  i^i  B )  <-> 
N  e.  (/) ) )
4947, 48syl5bbr 251 . . . . 5  |-  ( ph  ->  ( ( N  e.  A  /\  N  e.  B )  <->  N  e.  (/) ) )
5046, 49mtbiri 295 . . . 4  |-  ( ph  ->  -.  ( N  e.  A  /\  N  e.  B ) )
51 xor2 1319 . . . . 5  |-  ( ( N  e.  A  \/_  N  e.  B )  <->  ( ( N  e.  A  \/  N  e.  B
)  /\  -.  ( N  e.  A  /\  N  e.  B )
) )
5251rbaib 874 . . . 4  |-  ( -.  ( N  e.  A  /\  N  e.  B
)  ->  ( ( N  e.  A  \/_  N  e.  B )  <->  ( N  e.  A  \/  N  e.  B )
) )
5350, 52syl 16 . . 3  |-  ( ph  ->  ( ( N  e.  A  \/_  N  e.  B )  <->  ( N  e.  A  \/  N  e.  B ) ) )
54 elun 3480 . . 3  |-  ( N  e.  ( A  u.  B )  <->  ( N  e.  A  \/  N  e.  B ) )
5553, 54syl6bbr 255 . 2  |-  ( ph  ->  ( ( N  e.  A  \/_  N  e.  B )  <->  N  e.  ( A  u.  B
) ) )
565, 45, 553bitrd 271 1  |-  ( ph  ->  ( N  e.  ( A sadd  B )  <->  N  e.  ( A  u.  B
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    \/_ wxo 1313  haddwhad 1387  caddwcad 1388    = wceq 1652    e. wcel 1725    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   ifcif 3731    e. cmpt 4258   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1oc1o 6709   2oc2o 6710   0cc0 8982   1c1 8983    + caddc 8985    - cmin 9283   NN0cn0 10213    seq cseq 11315   sadd csad 12924
This theorem is referenced by:  saddisj  12969
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-seq 11316  df-sad 12955
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