Theorem om00 5254

Description: The product of two ordinal numbers is zero iff at least one of them is zero. Proposition 8.22 of [TakeutiZaring] p. 64.

Assertion

Ref	Expression
om00	|- ((A e. On /\ B e. On) -> ((A .o B) = (/) <-> (A = (/) \/ B = (/))))

Proof of Theorem om00

Step	Hyp	Ref	Expression
1		eloni 3667	. . . . . . . . . 10 |- (A e. On -> Ord A)
2		ordge1n0 5190	. . . . . . . . . 10 |- (Ord A -> (1o C_ A <-> A =/= (/)))
3	1, 2	syl 12	. . . . . . . . 9 |- (A e. On -> (1o C_ A <-> A =/= (/)))
4	3	biimprd 171	. . . . . . . 8 |- (A e. On -> (A =/= (/) -> 1o C_ A))
5	4	adantr 425	. . . . . . 7 |- ((A e. On /\ B e. On) -> (A =/= (/) -> 1o C_ A))
6		on0eln0 3718	. . . . . . . . 9 |- (B e. On -> ((/) e. B <-> B =/= (/)))
7	6	adantl 424	. . . . . . . 8 |- ((A e. On /\ B e. On) -> ((/) e. B <-> B =/= (/)))
8		omword1 5252	. . . . . . . . 9 |- (((A e. On /\ B e. On) /\ (/) e. B) -> A C_ (A .o B))
9	8	ex 402	. . . . . . . 8 |- ((A e. On /\ B e. On) -> ((/) e. B -> A C_ (A .o B)))
10	7, 9	sylbird 222	. . . . . . 7 |- ((A e. On /\ B e. On) -> (B =/= (/) -> A C_ (A .o B)))
11	5, 10	anim12d 617	. . . . . 6 |- ((A e. On /\ B e. On) -> ((A =/= (/) /\ B =/= (/)) -> (1o C_ A /\ A C_ (A .o B))))
12		sstr 2625	. . . . . 6 |- ((1o C_ A /\ A C_ (A .o B)) -> 1o C_ (A .o B))
13	11, 12	syl6 25	. . . . 5 |- ((A e. On /\ B e. On) -> ((A =/= (/) /\ B =/= (/)) -> 1o C_ (A .o B)))
14		neanior 2097	. . . . 5 |- ((A =/= (/) /\ B =/= (/)) <-> -. (A = (/) \/ B = (/)))
15	13, 14	syl5ibr 224	. . . 4 |- ((A e. On /\ B e. On) -> (-. (A = (/) \/ B = (/)) -> 1o C_ (A .o B)))
16		omcl 5216	. . . . 5 |- ((A e. On /\ B e. On) -> (A .o B) e. On)
17		eloni 3667	. . . . 5 |- ((A .o B) e. On -> Ord (A .o B))
18		ordge1n0 5190	. . . . 5 |- (Ord (A .o B) -> (1o C_ (A .o B) <-> (A .o B) =/= (/)))
19	16, 17, 18	3syl 24	. . . 4 |- ((A e. On /\ B e. On) -> (1o C_ (A .o B) <-> (A .o B) =/= (/)))
20	15, 19	sylibd 219	. . 3 |- ((A e. On /\ B e. On) -> (-. (A = (/) \/ B = (/)) -> (A .o B) =/= (/)))
21	20	necon4bd 2073	. 2 |- ((A e. On /\ B e. On) -> ((A .o B) = (/) -> (A = (/) \/ B = (/))))
22		opreq1 4889	. . . . . 6 |- (A = (/) -> (A .o B) = ((/) .o B))
23		om0r 5221	. . . . . 6 |- (B e. On -> ((/) .o B) = (/))
24	22, 23	sylan9eqr 1951	. . . . 5 |- ((B e. On /\ A = (/)) -> (A .o B) = (/))
25	24	ex 402	. . . 4 |- (B e. On -> (A = (/) -> (A .o B) = (/)))
26	25	adantl 424	. . 3 |- ((A e. On /\ B e. On) -> (A = (/) -> (A .o B) = (/)))
27		opreq2 4890	. . . . . 6 |- (B = (/) -> (A .o B) = (A .o (/)))
28		om0 5201	. . . . . 6 |- (A e. On -> (A .o (/)) = (/))
29	27, 28	sylan9eqr 1951	. . . . 5 |- ((A e. On /\ B = (/)) -> (A .o B) = (/))
30	29	ex 402	. . . 4 |- (A e. On -> (B = (/) -> (A .o B) = (/)))
31	30	adantr 425	. . 3 |- ((A e. On /\ B e. On) -> (B = (/) -> (A .o B) = (/)))
32	26, 31	jaod 469	. 2 |- ((A e. On /\ B e. On) -> ((A = (/) \/ B = (/)) -> (A .o B) = (/)))
33	21, 32	impbid 574	1 |- ((A e. On /\ B e. On) -> ((A .o B) = (/) <-> (A = (/) \/ B = (/))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   C_ wss 2593  (/)c0 2875  Ord word 3656  Oncon0 3657  (class class class)co 4884  1oc1o 5172   .o comu 5175

This theorem is referenced by:  om00el 5255  omlimcl 5257  oeoe 5274

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180

Copyright terms: Public domain