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Theorem omopthlem2 6670
Description: Lemma for omopthi 6671. (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 6629 . . . . . 6  |-  ( C  .o  C )  e. 
om
3 omopthlem2.4 . . . . . 6  |-  D  e. 
om
42, 3nnacli 6628 . . . . 5  |-  ( ( C  .o  C )  +o  D )  e. 
om
54nnoni 4679 . . . 4  |-  ( ( C  .o  C )  +o  D )  e.  On
65onirri 4515 . . 3  |-  -.  (
( C  .o  C
)  +o  D )  e.  ( ( C  .o  C )  +o  D )
7 eleq1 2356 . . 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 293 . 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 6643 . . . 4  |-  ( ( ( C  .o  C
)  e.  om  /\  D  e.  om )  ->  ( C  .o  C
)  C_  ( ( C  .o  C )  +o  D ) )
102, 3, 9mp2an 653 . . 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 6628 . . . . . . . . . 10  |-  ( A  +o  B )  e. 
om
1413, 12nnacli 6628 . . . . . . . . 9  |-  ( ( A  +o  B )  +o  A )  e. 
om
15 nnaword1 6643 . . . . . . . . 9  |-  ( ( B  e.  om  /\  ( ( A  +o  B )  +o  A
)  e.  om )  ->  B  C_  ( B  +o  ( ( A  +o  B )  +o  A
) ) )
1611, 14, 15mp2an 653 . . . . . . . 8  |-  B  C_  ( B  +o  (
( A  +o  B
)  +o  A ) )
17 nnacom 6631 . . . . . . . . 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 653 . . . . . . . 8  |-  ( B  +o  ( ( A  +o  B )  +o  A ) )  =  ( ( ( A  +o  B )  +o  A )  +o  B
)
1916, 18sseqtri 3223 . . . . . . 7  |-  B  C_  ( ( ( A  +o  B )  +o  A )  +o  B
)
20 nnaass 6636 . . . . . . . . 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 1277 . . . . . . . 8  |-  ( ( ( A  +o  B
)  +o  A )  +o  B )  =  ( ( A  +o  B )  +o  ( A  +o  B ) )
22 nnm2 6663 . . . . . . . . 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 2319 . . . . . . 7  |-  ( ( ( A  +o  B
)  +o  A )  +o  B )  =  ( ( A  +o  B )  .o  2o )
2519, 24sseqtri 3223 . . . . . 6  |-  B  C_  ( ( A  +o  B )  .o  2o )
26 2onn 6654 . . . . . . . 8  |-  2o  e.  om
2713, 26nnmcli 6629 . . . . . . 7  |-  ( ( A  +o  B )  .o  2o )  e. 
om
2813, 13nnmcli 6629 . . . . . . 7  |-  ( ( A  +o  B )  .o  ( A  +o  B ) )  e. 
om
29 nnawordi 6635 . . . . . . 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 1277 . . . . . 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 6631 . . . . . 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 653 . . . . 5  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  =  ( B  +o  (
( A  +o  B
)  .o  ( A  +o  B ) ) )
34 nnacom 6631 . . . . . 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 653 . . . . 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 3230 . . . 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 6669 . . . 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 6628 . . . . . 6  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  e. 
om
3938nnoni 4679 . . . . 5  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  e.  On
402nnoni 4679 . . . . 5  |-  ( C  .o  C )  e.  On
41 ontr2 4455 . . . . 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 653 . . . 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 644 . . 3  |-  ( ( A  +o  B )  e.  C  ->  (
( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  e.  ( C  .o  C
) )
4410, 43sseldi 3191 . 2  |-  ( ( A  +o  B )  e.  C  ->  (
( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  e.  ( ( C  .o  C )  +o  D
) )
458, 44nsyl3 111 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 358    = wceq 1632    e. wcel 1696    C_ wss 3165   Oncon0 4408   omcom 4672  (class class class)co 5874   2oc2o 6489    +o coa 6492    .o comu 6493
This theorem is referenced by:  omopthi  6671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500
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