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Theorem omeulem2 6597
Description: Lemma for omeu 6599: uniqueness part. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
omeulem2  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( B  e.  D  \/  ( B  =  D  /\  C  e.  E ) )  -> 
( ( A  .o  B )  +o  C
)  e.  ( ( A  .o  D )  +o  E ) ) )

Proof of Theorem omeulem2
StepHypRef Expression
1 simp3l 983 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  D  e.  On )
2 eloni 4418 . . . . . 6  |-  ( D  e.  On  ->  Ord  D )
3 ordsucss 4625 . . . . . 6  |-  ( Ord 
D  ->  ( B  e.  D  ->  suc  B  C_  D ) )
41, 2, 33syl 18 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( B  e.  D  ->  suc  B  C_  D
) )
5 simp2l 981 . . . . . . 7  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  B  e.  On )
6 suceloni 4620 . . . . . . 7  |-  ( B  e.  On  ->  suc  B  e.  On )
75, 6syl 15 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  suc  B  e.  On )
8 simp1l 979 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  A  e.  On )
9 simp1r 980 . . . . . . 7  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  A  =/=  (/) )
10 on0eln0 4463 . . . . . . . 8  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
118, 10syl 15 . . . . . . 7  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( (/)  e.  A  <->  A  =/=  (/) ) )
129, 11mpbird 223 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  (/) 
e.  A )
13 omword 6584 . . . . . 6  |-  ( ( ( suc  B  e.  On  /\  D  e.  On  /\  A  e.  On )  /\  (/)  e.  A
)  ->  ( suc  B 
C_  D  <->  ( A  .o  suc  B )  C_  ( A  .o  D
) ) )
147, 1, 8, 12, 13syl31anc 1185 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( suc  B  C_  D  <->  ( A  .o  suc  B
)  C_  ( A  .o  D ) ) )
154, 14sylibd 205 . . . 4  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( B  e.  D  ->  ( A  .o  suc  B )  C_  ( A  .o  D ) ) )
16 omcl 6551 . . . . . 6  |-  ( ( A  e.  On  /\  D  e.  On )  ->  ( A  .o  D
)  e.  On )
178, 1, 16syl2anc 642 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( A  .o  D
)  e.  On )
18 simp3r 984 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  E  e.  A )
19 onelon 4433 . . . . . 6  |-  ( ( A  e.  On  /\  E  e.  A )  ->  E  e.  On )
208, 18, 19syl2anc 642 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  E  e.  On )
21 oaword1 6566 . . . . . 6  |-  ( ( ( A  .o  D
)  e.  On  /\  E  e.  On )  ->  ( A  .o  D
)  C_  ( ( A  .o  D )  +o  E ) )
22 sstr 3200 . . . . . . 7  |-  ( ( ( A  .o  suc  B )  C_  ( A  .o  D )  /\  ( A  .o  D )  C_  ( ( A  .o  D )  +o  E
) )  ->  ( A  .o  suc  B ) 
C_  ( ( A  .o  D )  +o  E ) )
2322expcom 424 . . . . . 6  |-  ( ( A  .o  D ) 
C_  ( ( A  .o  D )  +o  E )  ->  (
( A  .o  suc  B )  C_  ( A  .o  D )  ->  ( A  .o  suc  B ) 
C_  ( ( A  .o  D )  +o  E ) ) )
2421, 23syl 15 . . . . 5  |-  ( ( ( A  .o  D
)  e.  On  /\  E  e.  On )  ->  ( ( A  .o  suc  B )  C_  ( A  .o  D )  -> 
( A  .o  suc  B )  C_  ( ( A  .o  D )  +o  E ) ) )
2517, 20, 24syl2anc 642 . . . 4  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( A  .o  suc  B )  C_  ( A  .o  D )  -> 
( A  .o  suc  B )  C_  ( ( A  .o  D )  +o  E ) ) )
2615, 25syld 40 . . 3  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( B  e.  D  ->  ( A  .o  suc  B )  C_  ( ( A  .o  D )  +o  E ) ) )
27 simp2r 982 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  C  e.  A )
28 onelon 4433 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  A )  ->  C  e.  On )
298, 27, 28syl2anc 642 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  C  e.  On )
30 omcl 6551 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
318, 5, 30syl2anc 642 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( A  .o  B
)  e.  On )
32 oaord 6561 . . . . . 6  |-  ( ( C  e.  On  /\  A  e.  On  /\  ( A  .o  B )  e.  On )  ->  ( C  e.  A  <->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  B
)  +o  A ) ) )
3332biimpa 470 . . . . 5  |-  ( ( ( C  e.  On  /\  A  e.  On  /\  ( A  .o  B
)  e.  On )  /\  C  e.  A
)  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  B
)  +o  A ) )
3429, 8, 31, 27, 33syl31anc 1185 . . . 4  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( A  .o  B )  +o  C
)  e.  ( ( A  .o  B )  +o  A ) )
35 omsuc 6541 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  +o  A ) )
368, 5, 35syl2anc 642 . . . 4  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( A  .o  suc  B )  =  ( ( A  .o  B )  +o  A ) )
3734, 36eleqtrrd 2373 . . 3  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( A  .o  B )  +o  C
)  e.  ( A  .o  suc  B ) )
38 ssel 3187 . . 3  |-  ( ( A  .o  suc  B
)  C_  ( ( A  .o  D )  +o  E )  ->  (
( ( A  .o  B )  +o  C
)  e.  ( A  .o  suc  B )  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) ) )
3926, 37, 38ee21 1365 . 2  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( B  e.  D  ->  ( ( A  .o  B )  +o  C
)  e.  ( ( A  .o  D )  +o  E ) ) )
40 simpr 447 . . . . 5  |-  ( ( B  =  D  /\  C  e.  E )  ->  C  e.  E )
41 oaord 6561 . . . . 5  |-  ( ( C  e.  On  /\  E  e.  On  /\  ( A  .o  B )  e.  On )  ->  ( C  e.  E  <->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  B
)  +o  E ) ) )
4240, 41syl5ib 210 . . . 4  |-  ( ( C  e.  On  /\  E  e.  On  /\  ( A  .o  B )  e.  On )  ->  (
( B  =  D  /\  C  e.  E
)  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  B
)  +o  E ) ) )
43 oveq2 5882 . . . . . . 7  |-  ( B  =  D  ->  ( A  .o  B )  =  ( A  .o  D
) )
4443oveq1d 5889 . . . . . 6  |-  ( B  =  D  ->  (
( A  .o  B
)  +o  E )  =  ( ( A  .o  D )  +o  E ) )
4544adantr 451 . . . . 5  |-  ( ( B  =  D  /\  C  e.  E )  ->  ( ( A  .o  B )  +o  E
)  =  ( ( A  .o  D )  +o  E ) )
4645eleq2d 2363 . . . 4  |-  ( ( B  =  D  /\  C  e.  E )  ->  ( ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  B
)  +o  E )  <-> 
( ( A  .o  B )  +o  C
)  e.  ( ( A  .o  D )  +o  E ) ) )
4742, 46mpbidi 207 . . 3  |-  ( ( C  e.  On  /\  E  e.  On  /\  ( A  .o  B )  e.  On )  ->  (
( B  =  D  /\  C  e.  E
)  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) ) )
4829, 20, 31, 47syl3anc 1182 . 2  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( B  =  D  /\  C  e.  E )  ->  (
( A  .o  B
)  +o  C )  e.  ( ( A  .o  D )  +o  E ) ) )
4939, 48jaod 369 1  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( B  e.  D  \/  ( B  =  D  /\  C  e.  E ) )  -> 
( ( A  .o  B )  +o  C
)  e.  ( ( A  .o  D )  +o  E ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   Ord word 4407   Oncon0 4408   suc csuc 4410  (class class class)co 5874    +o coa 6492    .o comu 6493
This theorem is referenced by:  omopth2  6598
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-rep 4147  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-recs 6404  df-rdg 6439  df-oadd 6499  df-omul 6500
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