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Theorem om00 6589
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 2544 . . . . 5  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  -.  ( A  =  (/)  \/  B  =  (/) ) )
2 eloni 4418 . . . . . . . . . 10  |-  ( A  e.  On  ->  Ord  A )
3 ordge1n0 6513 . . . . . . . . . 10  |-  ( Ord 
A  ->  ( 1o  C_  A  <->  A  =/=  (/) ) )
42, 3syl 15 . . . . . . . . 9  |-  ( A  e.  On  ->  ( 1o  C_  A  <->  A  =/=  (/) ) )
54biimprd 214 . . . . . . . 8  |-  ( A  e.  On  ->  ( A  =/=  (/)  ->  1o  C_  A
) )
65adantr 451 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =/=  (/)  ->  1o  C_  A ) )
7 on0eln0 4463 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  B  =/=  (/) ) )
87adantl 452 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  <->  B  =/=  (/) ) )
9 omword1 6587 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  A  C_  ( A  .o  B ) )
109ex 423 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  ->  A  C_  ( A  .o  B ) ) )
118, 10sylbird 226 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  =/=  (/)  ->  A  C_  ( A  .o  B
) ) )
126, 11anim12d 546 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  =/=  (/)  /\  B  =/=  (/) )  -> 
( 1o  C_  A  /\  A  C_  ( A  .o  B ) ) ) )
13 sstr 3200 . . . . . 6  |-  ( ( 1o  C_  A  /\  A  C_  ( A  .o  B ) )  ->  1o  C_  ( A  .o  B ) )
1412, 13syl6 29 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  1o  C_  ( A  .o  B ) ) )
151, 14syl5bir 209 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  ( A  =  (/)  \/  B  =  (/) )  ->  1o  C_  ( A  .o  B
) ) )
16 omcl 6551 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
17 eloni 4418 . . . . 5  |-  ( ( A  .o  B )  e.  On  ->  Ord  ( A  .o  B
) )
18 ordge1n0 6513 . . . . 5  |-  ( Ord  ( A  .o  B
)  ->  ( 1o  C_  ( A  .o  B
)  <->  ( A  .o  B )  =/=  (/) ) )
1916, 17, 183syl 18 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  C_  ( A  .o  B )  <->  ( A  .o  B )  =/=  (/) ) )
2015, 19sylibd 205 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  ( A  =  (/)  \/  B  =  (/) )  ->  ( A  .o  B )  =/=  (/) ) )
2120necon4bd 2521 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  (/)  ->  ( A  =  (/)  \/  B  =  (/) ) ) )
22 oveq1 5881 . . . . . 6  |-  ( A  =  (/)  ->  ( A  .o  B )  =  ( (/)  .o  B
) )
23 om0r 6554 . . . . . 6  |-  ( B  e.  On  ->  ( (/) 
.o  B )  =  (/) )
2422, 23sylan9eqr 2350 . . . . 5  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( A  .o  B
)  =  (/) )
2524ex 423 . . . 4  |-  ( B  e.  On  ->  ( A  =  (/)  ->  ( A  .o  B )  =  (/) ) )
2625adantl 452 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  (/)  ->  ( A  .o  B
)  =  (/) ) )
27 oveq2 5882 . . . . . 6  |-  ( B  =  (/)  ->  ( A  .o  B )  =  ( A  .o  (/) ) )
28 om0 6532 . . . . . 6  |-  ( A  e.  On  ->  ( A  .o  (/) )  =  (/) )
2927, 28sylan9eqr 2350 . . . . 5  |-  ( ( A  e.  On  /\  B  =  (/) )  -> 
( A  .o  B
)  =  (/) )
3029ex 423 . . . 4  |-  ( A  e.  On  ->  ( B  =  (/)  ->  ( A  .o  B )  =  (/) ) )
3130adantr 451 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  =  (/)  ->  ( A  .o  B
)  =  (/) ) )
3226, 31jaod 369 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  =  (/)  \/  B  =  (/) )  ->  ( A  .o  B )  =  (/) ) )
3321, 32impbid 183 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 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   Ord word 4407   Oncon0 4408  (class class class)co 5874   1oc1o 6488    .o comu 6493
This theorem is referenced by:  om00el  6590  omlimcl  6592  oeoe  6613
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-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500
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