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Theorem omeulem2 6818
Description: Lemma for omeu 6820: 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 985 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  D  e.  On )
2 eloni 4583 . . . . . 6  |-  ( D  e.  On  ->  Ord  D )
3 ordsucss 4790 . . . . . 6  |-  ( Ord 
D  ->  ( B  e.  D  ->  suc  B  C_  D ) )
41, 2, 33syl 19 . . . . 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 983 . . . . . . 7  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  B  e.  On )
6 suceloni 4785 . . . . . . 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 981 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  A  e.  On )
9 simp1r 982 . . . . . . 7  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  A  =/=  (/) )
10 on0eln0 4628 . . . . . . . 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 224 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  (/) 
e.  A )
13 omword 6805 . . . . . 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 1187 . . . . 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 206 . . . 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 6772 . . . . . 6  |-  ( ( A  e.  On  /\  D  e.  On )  ->  ( A  .o  D
)  e.  On )
178, 1, 16syl2anc 643 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( A  .o  D
)  e.  On )
18 simp3r 986 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  E  e.  A )
19 onelon 4598 . . . . . 6  |-  ( ( A  e.  On  /\  E  e.  A )  ->  E  e.  On )
208, 18, 19syl2anc 643 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  E  e.  On )
21 oaword1 6787 . . . . . 6  |-  ( ( ( A  .o  D
)  e.  On  /\  E  e.  On )  ->  ( A  .o  D
)  C_  ( ( A  .o  D )  +o  E ) )
22 sstr 3348 . . . . . . 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 425 . . . . . 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 643 . . . 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 42 . . 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 984 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  C  e.  A )
28 onelon 4598 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  A )  ->  C  e.  On )
298, 27, 28syl2anc 643 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  C  e.  On )
30 omcl 6772 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
318, 5, 30syl2anc 643 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( A  .o  B
)  e.  On )
32 oaord 6782 . . . . . 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 471 . . . . 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 1187 . . . 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 6762 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  +o  A ) )
368, 5, 35syl2anc 643 . . . 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 2512 . . 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 3334 . . 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 1384 . 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 448 . . . . 5  |-  ( ( B  =  D  /\  C  e.  E )  ->  C  e.  E )
41 oaord 6782 . . . . 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 211 . . . 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 6081 . . . . . . 7  |-  ( B  =  D  ->  ( A  .o  B )  =  ( A  .o  D
) )
4443oveq1d 6088 . . . . . 6  |-  ( B  =  D  ->  (
( A  .o  B
)  +o  E )  =  ( ( A  .o  D )  +o  E ) )
4544adantr 452 . . . . 5  |-  ( ( B  =  D  /\  C  e.  E )  ->  ( ( A  .o  B )  +o  E
)  =  ( ( A  .o  D )  +o  E ) )
4645eleq2d 2502 . . . 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 208 . . 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 1184 . 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 370 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   (/)c0 3620   Ord word 4572   Oncon0 4573   suc csuc 4575  (class class class)co 6073    +o coa 6713    .o comu 6714
This theorem is referenced by:  omopth2  6819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-oadd 6720  df-omul 6721
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