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Theorem om00 6754
Description: The product of two ordinal numbers is zero iff at least one of them is zero. Proposition 8.22 of [TakeutiZaring] p. 64. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
om00  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  (/)  <->  ( A  =  (/)  \/  B  =  (/) ) ) )

Proof of Theorem om00
StepHypRef Expression
1 neanior 2635 . . . . 5  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  -.  ( A  =  (/)  \/  B  =  (/) ) )
2 eloni 4532 . . . . . . . . . 10  |-  ( A  e.  On  ->  Ord  A )
3 ordge1n0 6678 . . . . . . . . . 10  |-  ( Ord 
A  ->  ( 1o  C_  A  <->  A  =/=  (/) ) )
42, 3syl 16 . . . . . . . . 9  |-  ( A  e.  On  ->  ( 1o  C_  A  <->  A  =/=  (/) ) )
54biimprd 215 . . . . . . . 8  |-  ( A  e.  On  ->  ( A  =/=  (/)  ->  1o  C_  A
) )
65adantr 452 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =/=  (/)  ->  1o  C_  A ) )
7 on0eln0 4577 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  B  =/=  (/) ) )
87adantl 453 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  <->  B  =/=  (/) ) )
9 omword1 6752 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  A  C_  ( A  .o  B ) )
109ex 424 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  ->  A  C_  ( A  .o  B ) ) )
118, 10sylbird 227 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  =/=  (/)  ->  A  C_  ( A  .o  B
) ) )
126, 11anim12d 547 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  =/=  (/)  /\  B  =/=  (/) )  -> 
( 1o  C_  A  /\  A  C_  ( A  .o  B ) ) ) )
13 sstr 3299 . . . . . 6  |-  ( ( 1o  C_  A  /\  A  C_  ( A  .o  B ) )  ->  1o  C_  ( A  .o  B ) )
1412, 13syl6 31 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  1o  C_  ( A  .o  B ) ) )
151, 14syl5bir 210 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  ( A  =  (/)  \/  B  =  (/) )  ->  1o  C_  ( A  .o  B
) ) )
16 omcl 6716 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
17 eloni 4532 . . . . 5  |-  ( ( A  .o  B )  e.  On  ->  Ord  ( A  .o  B
) )
18 ordge1n0 6678 . . . . 5  |-  ( Ord  ( A  .o  B
)  ->  ( 1o  C_  ( A  .o  B
)  <->  ( A  .o  B )  =/=  (/) ) )
1916, 17, 183syl 19 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  C_  ( A  .o  B )  <->  ( A  .o  B )  =/=  (/) ) )
2015, 19sylibd 206 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  ( A  =  (/)  \/  B  =  (/) )  ->  ( A  .o  B )  =/=  (/) ) )
2120necon4bd 2612 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  (/)  ->  ( A  =  (/)  \/  B  =  (/) ) ) )
22 oveq1 6027 . . . . . 6  |-  ( A  =  (/)  ->  ( A  .o  B )  =  ( (/)  .o  B
) )
23 om0r 6719 . . . . . 6  |-  ( B  e.  On  ->  ( (/) 
.o  B )  =  (/) )
2422, 23sylan9eqr 2441 . . . . 5  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( A  .o  B
)  =  (/) )
2524ex 424 . . . 4  |-  ( B  e.  On  ->  ( A  =  (/)  ->  ( A  .o  B )  =  (/) ) )
2625adantl 453 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  (/)  ->  ( A  .o  B
)  =  (/) ) )
27 oveq2 6028 . . . . . 6  |-  ( B  =  (/)  ->  ( A  .o  B )  =  ( A  .o  (/) ) )
28 om0 6697 . . . . . 6  |-  ( A  e.  On  ->  ( A  .o  (/) )  =  (/) )
2927, 28sylan9eqr 2441 . . . . 5  |-  ( ( A  e.  On  /\  B  =  (/) )  -> 
( A  .o  B
)  =  (/) )
3029ex 424 . . . 4  |-  ( A  e.  On  ->  ( B  =  (/)  ->  ( A  .o  B )  =  (/) ) )
3130adantr 452 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  =  (/)  ->  ( A  .o  B
)  =  (/) ) )
3226, 31jaod 370 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  =  (/)  \/  B  =  (/) )  ->  ( A  .o  B )  =  (/) ) )
3321, 32impbid 184 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  (/)  <->  ( A  =  (/)  \/  B  =  (/) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550    C_ wss 3263   (/)c0 3571   Ord word 4521   Oncon0 4522  (class class class)co 6020   1oc1o 6653    .o comu 6658
This theorem is referenced by:  om00el  6755  omlimcl  6757  oeoe  6778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-omul 6665
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