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Theorem omopthlem2 7223
Description: Lemma for omopthi 7224. (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 7182 . . . . . 6  |-  ( C  .o  C )  e. 
om
3 omopthlem2.4 . . . . . 6  |-  D  e. 
om
42, 3nnacli 7181 . . . . 5  |-  ( ( C  .o  C )  +o  D )  e. 
om
54nnoni 6606 . . . 4  |-  ( ( C  .o  C )  +o  D )  e.  On
65onirri 4898 . . 3  |-  -.  (
( C  .o  C
)  +o  D )  e.  ( ( C  .o  C )  +o  D )
7 eleq1 2454 . . 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 300 . 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 7196 . . . 4  |-  ( ( ( C  .o  C
)  e.  om  /\  D  e.  om )  ->  ( C  .o  C
)  C_  ( ( C  .o  C )  +o  D ) )
102, 3, 9mp2an 670 . . 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 7181 . . . . . . . . . 10  |-  ( A  +o  B )  e. 
om
1413, 12nnacli 7181 . . . . . . . . 9  |-  ( ( A  +o  B )  +o  A )  e. 
om
15 nnaword1 7196 . . . . . . . . 9  |-  ( ( B  e.  om  /\  ( ( A  +o  B )  +o  A
)  e.  om )  ->  B  C_  ( B  +o  ( ( A  +o  B )  +o  A
) ) )
1611, 14, 15mp2an 670 . . . . . . . 8  |-  B  C_  ( B  +o  (
( A  +o  B
)  +o  A ) )
17 nnacom 7184 . . . . . . . . 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 670 . . . . . . . 8  |-  ( B  +o  ( ( A  +o  B )  +o  A ) )  =  ( ( ( A  +o  B )  +o  A )  +o  B
)
1916, 18sseqtri 3449 . . . . . . 7  |-  B  C_  ( ( ( A  +o  B )  +o  A )  +o  B
)
20 nnaass 7189 . . . . . . . . 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 1322 . . . . . . . 8  |-  ( ( ( A  +o  B
)  +o  A )  +o  B )  =  ( ( A  +o  B )  +o  ( A  +o  B ) )
22 nnm2 7216 . . . . . . . . 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 2414 . . . . . . 7  |-  ( ( ( A  +o  B
)  +o  A )  +o  B )  =  ( ( A  +o  B )  .o  2o )
2519, 24sseqtri 3449 . . . . . 6  |-  B  C_  ( ( A  +o  B )  .o  2o )
26 2onn 7207 . . . . . . . 8  |-  2o  e.  om
2713, 26nnmcli 7182 . . . . . . 7  |-  ( ( A  +o  B )  .o  2o )  e. 
om
2813, 13nnmcli 7182 . . . . . . 7  |-  ( ( A  +o  B )  .o  ( A  +o  B ) )  e. 
om
29 nnawordi 7188 . . . . . . 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 1322 . . . . . 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 7184 . . . . . 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 670 . . . . 5  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  =  ( B  +o  (
( A  +o  B
)  .o  ( A  +o  B ) ) )
34 nnacom 7184 . . . . . 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 670 . . . . 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 3456 . . . 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 7222 . . . 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 7181 . . . . . 6  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  e. 
om
3938nnoni 6606 . . . . 5  |-  ( ( ( A  +o  B
)  .o  ( A  +o  B ) )  +o  B )  e.  On
402nnoni 6606 . . . . 5  |-  ( C  .o  C )  e.  On
41 ontr2 4839 . . . . 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 670 . . . 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 661 . . 3  |-  ( ( A  +o  B )  e.  C  ->  (
( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  e.  ( C  .o  C
) )
4410, 43sseldi 3415 . 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 367    = wceq 1399    e. wcel 1826    C_ wss 3389   Oncon0 4792  (class class class)co 6196   omcom 6599   2oc2o 7042    +o coa 7045    .o comu 7046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-omul 7053
This theorem is referenced by:  omopthi  7224
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