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Theorem omeulem2 7239
Description: Lemma for omeu 7241: 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 1033 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  D  e.  On )
2 eloni 5395 . . . . . 6  |-  ( D  e.  On  ->  Ord  D )
3 ordsucss 6603 . . . . . 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 1031 . . . . . . 7  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  B  e.  On )
6 suceloni 6598 . . . . . . 7  |-  ( B  e.  On  ->  suc  B  e.  On )
75, 6syl 17 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  suc  B  e.  On )
8 simp1l 1029 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  A  e.  On )
9 simp1r 1030 . . . . . . 7  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  A  =/=  (/) )
10 on0eln0 5440 . . . . . . . 8  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
118, 10syl 17 . . . . . . 7  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( (/)  e.  A  <->  A  =/=  (/) ) )
129, 11mpbird 235 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  (/) 
e.  A )
13 omword 7226 . . . . . 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 1267 . . . . 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 217 . . . 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 7193 . . . . . 6  |-  ( ( A  e.  On  /\  D  e.  On )  ->  ( A  .o  D
)  e.  On )
178, 1, 16syl2anc 665 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( A  .o  D
)  e.  On )
18 simp3r 1034 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  E  e.  A )
19 onelon 5410 . . . . . 6  |-  ( ( A  e.  On  /\  E  e.  A )  ->  E  e.  On )
208, 18, 19syl2anc 665 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  E  e.  On )
21 oaword1 7208 . . . . . 6  |-  ( ( ( A  .o  D
)  e.  On  /\  E  e.  On )  ->  ( A  .o  D
)  C_  ( ( A  .o  D )  +o  E ) )
22 sstr 3415 . . . . . . 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 436 . . . . . 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 17 . . . . 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 665 . . . 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 45 . . 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 1032 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  C  e.  A )
28 onelon 5410 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  A )  ->  C  e.  On )
298, 27, 28syl2anc 665 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  C  e.  On )
30 omcl 7193 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
318, 5, 30syl2anc 665 . . . . 5  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( A  .o  B
)  e.  On )
32 oaord 7203 . . . . . 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 486 . . . . 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 1267 . . . 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 7183 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  +o  A ) )
368, 5, 35syl2anc 665 . . . 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 2509 . . 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 3401 . . 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 67 . 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 462 . . . . 5  |-  ( ( B  =  D  /\  C  e.  E )  ->  C  e.  E )
41 oaord 7203 . . . . 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 222 . . . 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 6257 . . . . . . 7  |-  ( B  =  D  ->  ( A  .o  B )  =  ( A  .o  D
) )
4443oveq1d 6264 . . . . . 6  |-  ( B  =  D  ->  (
( A  .o  B
)  +o  E )  =  ( ( A  .o  D )  +o  E ) )
4544adantr 466 . . . . 5  |-  ( ( B  =  D  /\  C  e.  E )  ->  ( ( A  .o  B )  +o  E
)  =  ( ( A  .o  D )  +o  E ) )
4645eleq2d 2491 . . . 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 219 . . 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 1264 . 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 381 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599    C_ wss 3379   (/)c0 3704   Ord word 5384   Oncon0 5385   suc csuc 5387  (class class class)co 6249    +o coa 7134    .o comu 7135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-oadd 7141  df-omul 7142
This theorem is referenced by:  omopth2  7240
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