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Theorem omopthlem2 6901
Description: Lemma for omopthi 6902. (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 6860 . . . . . 6  |-  ( C  .o  C )  e. 
om
3 omopthlem2.4 . . . . . 6  |-  D  e. 
om
42, 3nnacli 6859 . . . . 5  |-  ( ( C  .o  C )  +o  D )  e. 
om
54nnoni 4854 . . . 4  |-  ( ( C  .o  C )  +o  D )  e.  On
65onirri 4690 . . 3  |-  -.  (
( C  .o  C
)  +o  D )  e.  ( ( C  .o  C )  +o  D )
7 eleq1 2498 . . 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 295 . 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 6874 . . . 4  |-  ( ( ( C  .o  C
)  e.  om  /\  D  e.  om )  ->  ( C  .o  C
)  C_  ( ( C  .o  C )  +o  D ) )
102, 3, 9mp2an 655 . . 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 6859 . . . . . . . . . 10  |-  ( A  +o  B )  e. 
om
1413, 12nnacli 6859 . . . . . . . . 9  |-  ( ( A  +o  B )  +o  A )  e. 
om
15 nnaword1 6874 . . . . . . . . 9  |-  ( ( B  e.  om  /\  ( ( A  +o  B )  +o  A
)  e.  om )  ->  B  C_  ( B  +o  ( ( A  +o  B )  +o  A
) ) )
1611, 14, 15mp2an 655 . . . . . . . 8  |-  B  C_  ( B  +o  (
( A  +o  B
)  +o  A ) )
17 nnacom 6862 . . . . . . . . 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 655 . . . . . . . 8  |-  ( B  +o  ( ( A  +o  B )  +o  A ) )  =  ( ( ( A  +o  B )  +o  A )  +o  B
)
1916, 18sseqtri 3382 . . . . . . 7  |-  B  C_  ( ( ( A  +o  B )  +o  A )  +o  B
)
20 nnaass 6867 . . . . . . . . 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 1280 . . . . . . . 8  |-  ( ( ( A  +o  B
)  +o  A )  +o  B )  =  ( ( A  +o  B )  +o  ( A  +o  B ) )
22 nnm2 6894 . . . . . . . . 9  |-  ( ( A  +o  B )  e.  om  ->  (
( A  +o  B
)  .o  2o )  =  ( ( A  +o  B )  +o  ( A  +o  B
) ) )
2313, 22ax-mp 8 . . . . . . . 8  |-  ( ( A  +o  B )  .o  2o )  =  ( ( A  +o  B )  +o  ( A  +o  B ) )
2421, 23eqtr4i 2461 . . . . . . 7  |-  ( ( ( A  +o  B
)  +o  A )  +o  B )  =  ( ( A  +o  B )  .o  2o )
2519, 24sseqtri 3382 . . . . . 6  |-  B  C_  ( ( A  +o  B )  .o  2o )
26 2onn 6885 . . . . . . . 8  |-  2o  e.  om
2713, 26nnmcli 6860 . . . . . . 7  |-  ( ( A  +o  B )  .o  2o )  e. 
om
2813, 13nnmcli 6860 . . . . . . 7  |-  ( ( A  +o  B )  .o  ( A  +o  B ) )  e. 
om
29 nnawordi 6866 . . . . . . 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 1280 . . . . . 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 8 . . . . 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 6862 . . . . . 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 655 . . . . 5  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  =  ( B  +o  (
( A  +o  B
)  .o  ( A  +o  B ) ) )
34 nnacom 6862 . . . . . 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 655 . . . . 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 3389 . . . 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 6900 . . . 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 6859 . . . . . 6  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  e. 
om
3938nnoni 4854 . . . . 5  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  e.  On
402nnoni 4854 . . . . 5  |-  ( C  .o  C )  e.  On
41 ontr2 4630 . . . . 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 655 . . . 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 646 . . 3  |-  ( ( A  +o  B )  e.  C  ->  (
( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  e.  ( C  .o  C
) )
4410, 43sseldi 3348 . 2  |-  ( ( A  +o  B )  e.  C  ->  (
( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  e.  ( ( C  .o  C )  +o  D
) )
458, 44nsyl3 114 1  |-  ( ( A  +o  B )  e.  C  ->  -.  ( ( C  .o  C )  +o  D
)  =  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   Oncon0 4583   omcom 4847  (class class class)co 6083   2oc2o 6720    +o coa 6723    .o comu 6724
This theorem is referenced by:  omopthi  6902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-omul 6731
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