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Theorem saddisjlem 12655
Description: Lemma for sadadd 12658. (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 12647 . 2  |-  ( ph  ->  ( N  e.  ( A sadd  B )  <-> hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `  N ) ) ) )
6 fveq2 5525 . . . . . . . 8  |-  ( x  =  0  ->  ( C `  x )  =  ( C ` 
0 ) )
76eleq2d 2350 . . . . . . 7  |-  ( x  =  0  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  0 ) ) )
87notbid 285 . . . . . 6  |-  ( x  =  0  ->  ( -.  (/)  e.  ( C `
 x )  <->  -.  (/)  e.  ( C `  0 ) ) )
98imbi2d 307 . . . . 5  |-  ( x  =  0  ->  (
( ph  ->  -.  (/)  e.  ( C `  x ) )  <->  ( ph  ->  -.  (/)  e.  ( C ` 
0 ) ) ) )
10 fveq2 5525 . . . . . . . 8  |-  ( x  =  k  ->  ( C `  x )  =  ( C `  k ) )
1110eleq2d 2350 . . . . . . 7  |-  ( x  =  k  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  k ) ) )
1211notbid 285 . . . . . 6  |-  ( x  =  k  ->  ( -.  (/)  e.  ( C `
 x )  <->  -.  (/)  e.  ( C `  k ) ) )
1312imbi2d 307 . . . . 5  |-  ( x  =  k  ->  (
( ph  ->  -.  (/)  e.  ( C `  x ) )  <->  ( ph  ->  -.  (/)  e.  ( C `  k ) ) ) )
14 fveq2 5525 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( C `  x )  =  ( C `  ( k  +  1 ) ) )
1514eleq2d 2350 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  ( k  +  1 ) ) ) )
1615notbid 285 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( -.  (/)  e.  ( C `
 x )  <->  -.  (/)  e.  ( C `  ( k  +  1 ) ) ) )
1716imbi2d 307 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
( ph  ->  -.  (/)  e.  ( C `  x ) )  <->  ( ph  ->  -.  (/)  e.  ( C `  ( k  +  1 ) ) ) ) )
18 fveq2 5525 . . . . . . . 8  |-  ( x  =  N  ->  ( C `  x )  =  ( C `  N ) )
1918eleq2d 2350 . . . . . . 7  |-  ( x  =  N  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  N ) ) )
2019notbid 285 . . . . . 6  |-  ( x  =  N  ->  ( -.  (/)  e.  ( C `
 x )  <->  -.  (/)  e.  ( C `  N ) ) )
2120imbi2d 307 . . . . 5  |-  ( x  =  N  ->  (
( ph  ->  -.  (/)  e.  ( C `  x ) )  <->  ( ph  ->  -.  (/)  e.  ( C `  N ) ) ) )
221, 2, 3sadc0 12645 . . . . 5  |-  ( ph  ->  -.  (/)  e.  ( C `
 0 ) )
23 noel 3459 . . . . . . . . 9  |-  -.  k  e.  (/)
241ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  ->  A  C_  NN0 )
252ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  ->  B  C_  NN0 )
26 simplr 731 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  -> 
k  e.  NN0 )
2724, 25, 3, 26sadcp1 12646 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  -> 
( (/)  e.  ( C `
 ( k  +  1 ) )  <-> cadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `  k ) ) ) )
28 cad0 1390 . . . . . . . . . . 11  |-  ( -.  (/)  e.  ( C `  k )  ->  (cadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `
 k ) )  <-> 
( k  e.  A  /\  k  e.  B
) ) )
2928adantl 452 . . . . . . . . . 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 3358 . . . . . . . . . . 11  |-  ( k  e.  ( A  i^i  B )  <->  ( k  e.  A  /\  k  e.  B ) )
31 saddisj.3 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
3231ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  -> 
( A  i^i  B
)  =  (/) )
3332eleq2d 2350 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  -> 
( k  e.  ( A  i^i  B )  <-> 
k  e.  (/) ) )
3430, 33syl5bbr 250 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  -> 
( ( k  e.  A  /\  k  e.  B )  <->  k  e.  (/) ) )
3527, 29, 343bitrd 270 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  -> 
( (/)  e.  ( C `
 ( k  +  1 ) )  <->  k  e.  (/) ) )
3623, 35mtbiri 294 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  (/) 
e.  ( C `  k ) )  ->  -.  (/)  e.  ( C `
 ( k  +  1 ) ) )
3736ex 423 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( -.  (/) 
e.  ( C `  k )  ->  -.  (/) 
e.  ( C `  ( k  +  1 ) ) ) )
3837expcom 424 . . . . . 6  |-  ( k  e.  NN0  ->  ( ph  ->  ( -.  (/)  e.  ( C `  k )  ->  -.  (/)  e.  ( C `  ( k  +  1 ) ) ) ) )
3938a2d 23 . . . . 5  |-  ( k  e.  NN0  ->  ( (
ph  ->  -.  (/)  e.  ( C `  k ) )  ->  ( ph  ->  -.  (/)  e.  ( C `
 ( k  +  1 ) ) ) ) )
409, 13, 17, 21, 22, 39nn0ind 10108 . . . 4  |-  ( N  e.  NN0  ->  ( ph  ->  -.  (/)  e.  ( C `
 N ) ) )
414, 40mpcom 32 . . 3  |-  ( ph  ->  -.  (/)  e.  ( C `
 N ) )
42 hadrot 1380 . . . 4  |-  (hadd (
(/)  e.  ( C `  N ) ,  N  e.  A ,  N  e.  B )  <-> hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `  N ) ) )
43 had0 1393 . . . 4  |-  ( -.  (/)  e.  ( C `  N )  ->  (hadd ( (/)  e.  ( C `
 N ) ,  N  e.  A ,  N  e.  B )  <->  ( N  e.  A \/_ N  e.  B )
) )
4442, 43syl5bbr 250 . . 3  |-  ( -.  (/)  e.  ( C `  N )  ->  (hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
 N ) )  <-> 
( N  e.  A \/_ N  e.  B ) ) )
4541, 44syl 15 . 2  |-  ( ph  ->  (hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `  N ) )  <->  ( N  e.  A \/_ N  e.  B ) ) )
46 noel 3459 . . . . 5  |-  -.  N  e.  (/)
47 elin 3358 . . . . . 6  |-  ( N  e.  ( A  i^i  B )  <->  ( N  e.  A  /\  N  e.  B ) )
4831eleq2d 2350 . . . . . 6  |-  ( ph  ->  ( N  e.  ( A  i^i  B )  <-> 
N  e.  (/) ) )
4947, 48syl5bbr 250 . . . . 5  |-  ( ph  ->  ( ( N  e.  A  /\  N  e.  B )  <->  N  e.  (/) ) )
5046, 49mtbiri 294 . . . 4  |-  ( ph  ->  -.  ( N  e.  A  /\  N  e.  B ) )
51 xor2 1301 . . . . 5  |-  ( ( N  e.  A \/_ N  e.  B )  <->  ( ( N  e.  A  \/  N  e.  B
)  /\  -.  ( N  e.  A  /\  N  e.  B )
) )
5251rbaib 873 . . . 4  |-  ( -.  ( N  e.  A  /\  N  e.  B
)  ->  ( ( N  e.  A \/_ N  e.  B )  <->  ( N  e.  A  \/  N  e.  B )
) )
5350, 52syl 15 . . 3  |-  ( ph  ->  ( ( N  e.  A \/_ N  e.  B )  <->  ( N  e.  A  \/  N  e.  B ) ) )
54 elun 3316 . . 3  |-  ( N  e.  ( A  u.  B )  <->  ( N  e.  A  \/  N  e.  B ) )
5553, 54syl6bbr 254 . 2  |-  ( ph  ->  ( ( N  e.  A \/_ N  e.  B )  <->  N  e.  ( A  u.  B
) ) )
565, 45, 553bitrd 270 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 176    \/ wo 357    /\ wa 358   \/_wxo 1295  haddwhad 1368  caddwcad 1369    = wceq 1623    e. wcel 1684    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ifcif 3565    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   0cc0 8737   1c1 8738    + caddc 8740    - cmin 9037   NN0cn0 9965    seq cseq 11046   sadd csad 12611
This theorem is referenced by:  saddisj  12656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-sad 12642
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