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Theorem omeulem2 7122
Description: Lemma for omeu 7124: 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 1016 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  D  e.  On )
2 eloni 4827 . . . . . 6  |-  ( D  e.  On  ->  Ord  D )
3 ordsucss 6529 . . . . . 6  |-  ( Ord 
D  ->  ( B  e.  D  ->  suc  B  C_  D ) )
41, 2, 33syl 20 . . . . 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 1014 . . . . . . 7  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  B  e.  On )
6 suceloni 6524 . . . . . . 7  |-  ( B  e.  On  ->  suc  B  e.  On )
75, 6syl 16 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  suc  B  e.  On )
8 simp1l 1012 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  A  e.  On )
9 simp1r 1013 . . . . . . 7  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  A  =/=  (/) )
10 on0eln0 4872 . . . . . . . 8  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
118, 10syl 16 . . . . . . 7  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( (/)  e.  A  <->  A  =/=  (/) ) )
129, 11mpbird 232 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  (/) 
e.  A )
13 omword 7109 . . . . . 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 1222 . . . . 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 214 . . . 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 7076 . . . . . 6  |-  ( ( A  e.  On  /\  D  e.  On )  ->  ( A  .o  D
)  e.  On )
178, 1, 16syl2anc 661 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( A  .o  D
)  e.  On )
18 simp3r 1017 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  E  e.  A )
19 onelon 4842 . . . . . 6  |-  ( ( A  e.  On  /\  E  e.  A )  ->  E  e.  On )
208, 18, 19syl2anc 661 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  E  e.  On )
21 oaword1 7091 . . . . . 6  |-  ( ( ( A  .o  D
)  e.  On  /\  E  e.  On )  ->  ( A  .o  D
)  C_  ( ( A  .o  D )  +o  E ) )
22 sstr 3462 . . . . . . 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 435 . . . . . 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 16 . . . . 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 661 . . . 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 44 . . 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 1015 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  C  e.  A )
28 onelon 4842 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  A )  ->  C  e.  On )
298, 27, 28syl2anc 661 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  C  e.  On )
30 omcl 7076 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
318, 5, 30syl2anc 661 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( A  .o  B
)  e.  On )
32 oaord 7086 . . . . . 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 484 . . . . 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 1222 . . . 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 7066 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  +o  A ) )
368, 5, 35syl2anc 661 . . . 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 2542 . . 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 3448 . . 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, 38syl6ci 65 . 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 461 . . . . 5  |-  ( ( B  =  D  /\  C  e.  E )  ->  C  e.  E )
41 oaord 7086 . . . . 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 219 . . . 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 6198 . . . . . . 7  |-  ( B  =  D  ->  ( A  .o  B )  =  ( A  .o  D
) )
4443oveq1d 6205 . . . . . 6  |-  ( B  =  D  ->  (
( A  .o  B
)  +o  E )  =  ( ( A  .o  D )  +o  E ) )
4544adantr 465 . . . . 5  |-  ( ( B  =  D  /\  C  e.  E )  ->  ( ( A  .o  B )  +o  E
)  =  ( ( A  .o  D )  +o  E ) )
4645eleq2d 2521 . . . 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 216 . . 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 1219 . 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 380 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644    C_ wss 3426   (/)c0 3735   Ord word 4816   Oncon0 4817   suc csuc 4819  (class class class)co 6190    +o coa 7017    .o comu 7018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-recs 6932  df-rdg 6966  df-oadd 7024  df-omul 7025
This theorem is referenced by:  omopth2  7123
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