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Theorem omopthlem2 7107
Description: Lemma for omopthi 7108. (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 7066 . . . . . 6  |-  ( C  .o  C )  e. 
om
3 omopthlem2.4 . . . . . 6  |-  D  e. 
om
42, 3nnacli 7065 . . . . 5  |-  ( ( C  .o  C )  +o  D )  e. 
om
54nnoni 6495 . . . 4  |-  ( ( C  .o  C )  +o  D )  e.  On
65onirri 4837 . . 3  |-  -.  (
( C  .o  C
)  +o  D )  e.  ( ( C  .o  C )  +o  D )
7 eleq1 2503 . . 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 7080 . . . 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 7065 . . . . . . . . . 10  |-  ( A  +o  B )  e. 
om
1413, 12nnacli 7065 . . . . . . . . 9  |-  ( ( A  +o  B )  +o  A )  e. 
om
15 nnaword1 7080 . . . . . . . . 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 7068 . . . . . . . . 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 3400 . . . . . . 7  |-  B  C_  ( ( ( A  +o  B )  +o  A )  +o  B
)
20 nnaass 7073 . . . . . . . . 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 1314 . . . . . . . 8  |-  ( ( ( A  +o  B
)  +o  A )  +o  B )  =  ( ( A  +o  B )  +o  ( A  +o  B ) )
22 nnm2 7100 . . . . . . . . 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 2466 . . . . . . 7  |-  ( ( ( A  +o  B
)  +o  A )  +o  B )  =  ( ( A  +o  B )  .o  2o )
2519, 24sseqtri 3400 . . . . . 6  |-  B  C_  ( ( A  +o  B )  .o  2o )
26 2onn 7091 . . . . . . . 8  |-  2o  e.  om
2713, 26nnmcli 7066 . . . . . . 7  |-  ( ( A  +o  B )  .o  2o )  e. 
om
2813, 13nnmcli 7066 . . . . . . 7  |-  ( ( A  +o  B )  .o  ( A  +o  B ) )  e. 
om
29 nnawordi 7072 . . . . . . 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 1314 . . . . . 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 7068 . . . . . 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 7068 . . . . . 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 3407 . . . 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 7106 . . . 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 7065 . . . . . 6  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  e. 
om
3938nnoni 6495 . . . . 5  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  e.  On
402nnoni 6495 . . . . 5  |-  ( C  .o  C )  e.  On
41 ontr2 4778 . . . . 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 3366 . 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 1369    e. wcel 1756    C_ wss 3340   Oncon0 4731  (class class class)co 6103   omcom 6488   2oc2o 6926    +o coa 6929    .o comu 6930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-omul 6937
This theorem is referenced by:  omopthi  7108
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