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Theorem omopthlem2 7317
Description: Lemma for omopthi 7318. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
Hypotheses
Ref Expression
omopthlem2.1  |-  A  e. 
om
omopthlem2.2  |-  B  e. 
om
omopthlem2.3  |-  C  e. 
om
omopthlem2.4  |-  D  e. 
om
Assertion
Ref Expression
omopthlem2  |-  ( ( A  +o  B )  e.  C  ->  -.  ( ( C  .o  C )  +o  D
)  =  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B ) )

Proof of Theorem omopthlem2
StepHypRef Expression
1 omopthlem2.3 . . . . . . 7  |-  C  e. 
om
21, 1nnmcli 7276 . . . . . 6  |-  ( C  .o  C )  e. 
om
3 omopthlem2.4 . . . . . 6  |-  D  e. 
om
42, 3nnacli 7275 . . . . 5  |-  ( ( C  .o  C )  +o  D )  e. 
om
54nnoni 6702 . . . 4  |-  ( ( C  .o  C )  +o  D )  e.  On
65onirri 4990 . . 3  |-  -.  (
( C  .o  C
)  +o  D )  e.  ( ( C  .o  C )  +o  D )
7 eleq1 2539 . . 3  |-  ( ( ( C  .o  C
)  +o  D )  =  ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  ->  (
( ( C  .o  C )  +o  D
)  e.  ( ( C  .o  C )  +o  D )  <->  ( (
( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  e.  ( ( C  .o  C )  +o  D
) ) )
86, 7mtbii 302 . 2  |-  ( ( ( C  .o  C
)  +o  D )  =  ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  ->  -.  ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  B
)  e.  ( ( C  .o  C )  +o  D ) )
9 nnaword1 7290 . . . 4  |-  ( ( ( C  .o  C
)  e.  om  /\  D  e.  om )  ->  ( C  .o  C
)  C_  ( ( C  .o  C )  +o  D ) )
102, 3, 9mp2an 672 . . 3  |-  ( C  .o  C )  C_  ( ( C  .o  C )  +o  D
)
11 omopthlem2.2 . . . . . . . . 9  |-  B  e. 
om
12 omopthlem2.1 . . . . . . . . . . 11  |-  A  e. 
om
1312, 11nnacli 7275 . . . . . . . . . 10  |-  ( A  +o  B )  e. 
om
1413, 12nnacli 7275 . . . . . . . . 9  |-  ( ( A  +o  B )  +o  A )  e. 
om
15 nnaword1 7290 . . . . . . . . 9  |-  ( ( B  e.  om  /\  ( ( A  +o  B )  +o  A
)  e.  om )  ->  B  C_  ( B  +o  ( ( A  +o  B )  +o  A
) ) )
1611, 14, 15mp2an 672 . . . . . . . 8  |-  B  C_  ( B  +o  (
( A  +o  B
)  +o  A ) )
17 nnacom 7278 . . . . . . . . 9  |-  ( ( B  e.  om  /\  ( ( A  +o  B )  +o  A
)  e.  om )  ->  ( B  +o  (
( A  +o  B
)  +o  A ) )  =  ( ( ( A  +o  B
)  +o  A )  +o  B ) )
1811, 14, 17mp2an 672 . . . . . . . 8  |-  ( B  +o  ( ( A  +o  B )  +o  A ) )  =  ( ( ( A  +o  B )  +o  A )  +o  B
)
1916, 18sseqtri 3541 . . . . . . 7  |-  B  C_  ( ( ( A  +o  B )  +o  A )  +o  B
)
20 nnaass 7283 . . . . . . . . 9  |-  ( ( ( A  +o  B
)  e.  om  /\  A  e.  om  /\  B  e.  om )  ->  (
( ( A  +o  B )  +o  A
)  +o  B )  =  ( ( A  +o  B )  +o  ( A  +o  B
) ) )
2113, 12, 11, 20mp3an 1324 . . . . . . . 8  |-  ( ( ( A  +o  B
)  +o  A )  +o  B )  =  ( ( A  +o  B )  +o  ( A  +o  B ) )
22 nnm2 7310 . . . . . . . . 9  |-  ( ( A  +o  B )  e.  om  ->  (
( A  +o  B
)  .o  2o )  =  ( ( A  +o  B )  +o  ( A  +o  B
) ) )
2313, 22ax-mp 5 . . . . . . . 8  |-  ( ( A  +o  B )  .o  2o )  =  ( ( A  +o  B )  +o  ( A  +o  B ) )
2421, 23eqtr4i 2499 . . . . . . 7  |-  ( ( ( A  +o  B
)  +o  A )  +o  B )  =  ( ( A  +o  B )  .o  2o )
2519, 24sseqtri 3541 . . . . . 6  |-  B  C_  ( ( A  +o  B )  .o  2o )
26 2onn 7301 . . . . . . . 8  |-  2o  e.  om
2713, 26nnmcli 7276 . . . . . . 7  |-  ( ( A  +o  B )  .o  2o )  e. 
om
2813, 13nnmcli 7276 . . . . . . 7  |-  ( ( A  +o  B )  .o  ( A  +o  B ) )  e. 
om
29 nnawordi 7282 . . . . . . 7  |-  ( ( B  e.  om  /\  ( ( A  +o  B )  .o  2o )  e.  om  /\  (
( A  +o  B
)  .o  ( A  +o  B ) )  e.  om )  -> 
( B  C_  (
( A  +o  B
)  .o  2o )  ->  ( B  +o  ( ( A  +o  B )  .o  ( A  +o  B ) ) )  C_  ( (
( A  +o  B
)  .o  2o )  +o  ( ( A  +o  B )  .o  ( A  +o  B
) ) ) ) )
3011, 27, 28, 29mp3an 1324 . . . . . 6  |-  ( B 
C_  ( ( A  +o  B )  .o  2o )  ->  ( B  +o  ( ( A  +o  B )  .o  ( A  +o  B
) ) )  C_  ( ( ( A  +o  B )  .o  2o )  +o  (
( A  +o  B
)  .o  ( A  +o  B ) ) ) )
3125, 30ax-mp 5 . . . . 5  |-  ( B  +o  ( ( A  +o  B )  .o  ( A  +o  B
) ) )  C_  ( ( ( A  +o  B )  .o  2o )  +o  (
( A  +o  B
)  .o  ( A  +o  B ) ) )
32 nnacom 7278 . . . . . 6  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  e.  om  /\  B  e.  om )  ->  (
( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  ( B  +o  (
( A  +o  B
)  .o  ( A  +o  B ) ) ) )
3328, 11, 32mp2an 672 . . . . 5  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  =  ( B  +o  (
( A  +o  B
)  .o  ( A  +o  B ) ) )
34 nnacom 7278 . . . . . 6  |-  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  e.  om  /\  (
( A  +o  B
)  .o  2o )  e.  om )  -> 
( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  (
( A  +o  B
)  .o  2o ) )  =  ( ( ( A  +o  B
)  .o  2o )  +o  ( ( A  +o  B )  .o  ( A  +o  B
) ) ) )
3528, 27, 34mp2an 672 . . . . 5  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  ( ( A  +o  B )  .o  2o ) )  =  ( ( ( A  +o  B )  .o  2o )  +o  (
( A  +o  B
)  .o  ( A  +o  B ) ) )
3631, 33, 353sstr4i 3548 . . . 4  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  C_  ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  (
( A  +o  B
)  .o  2o ) )
3713, 1omopthlem1 7316 . . . 4  |-  ( ( A  +o  B )  e.  C  ->  (
( ( A  +o  B )  .o  ( A  +o  B ) )  +o  ( ( A  +o  B )  .o  2o ) )  e.  ( C  .o  C
) )
3828, 11nnacli 7275 . . . . . 6  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  e. 
om
3938nnoni 6702 . . . . 5  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  e.  On
402nnoni 6702 . . . . 5  |-  ( C  .o  C )  e.  On
41 ontr2 4931 . . . . 5  |-  ( ( ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  B
)  e.  On  /\  ( C  .o  C
)  e.  On )  ->  ( ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  C_  ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  (
( A  +o  B
)  .o  2o ) )  /\  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  ( ( A  +o  B )  .o  2o ) )  e.  ( C  .o  C
) )  ->  (
( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  e.  ( C  .o  C
) ) )
4239, 40, 41mp2an 672 . . . 4  |-  ( ( ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  B
)  C_  ( (
( A  +o  B
)  .o  ( A  +o  B ) )  +o  ( ( A  +o  B )  .o  2o ) )  /\  ( ( ( A  +o  B )  .o  ( A  +o  B
) )  +o  (
( A  +o  B
)  .o  2o ) )  e.  ( C  .o  C ) )  ->  ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  e.  ( C  .o  C ) )
4336, 37, 42sylancr 663 . . 3  |-  ( ( A  +o  B )  e.  C  ->  (
( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  e.  ( C  .o  C
) )
4410, 43sseldi 3507 . 2  |-  ( ( A  +o  B )  e.  C  ->  (
( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  e.  ( ( C  .o  C )  +o  D
) )
458, 44nsyl3 119 1  |-  ( ( A  +o  B )  e.  C  ->  -.  ( ( C  .o  C )  +o  D
)  =  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3481   Oncon0 4884  (class class class)co 6295   omcom 6695   2oc2o 7136    +o coa 7139    .o comu 7140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147
This theorem is referenced by:  omopthi  7318
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