Theorem om00 4190

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 2948	. . . . . . . . . 10 |- (A e. On -> Ord A)
2		ordge1n0 4129	. . . . . . . . . 10 |- (Ord A -> (1o (_ A <-> A =/= (/)))
3	1, 2	syl 10	. . . . . . . . 9 |- (A e. On -> (1o (_ A <-> A =/= (/)))
4	3	biimprd 154	. . . . . . . 8 |- (A e. On -> (A =/= (/) -> 1o (_ A))
5	4	adantr 389	. . . . . . 7 |- ((A e. On /\ B e. On) -> (A =/= (/) -> 1o (_ A))
6		on0eln0 3014	. . . . . . . . 9 |- (B e. On -> ((/) e. B <-> B =/= (/)))
7	6	adantl 388	. . . . . . . 8 |- ((A e. On /\ B e. On) -> ((/) e. B <-> B =/= (/)))
8		omword1 4188	. . . . . . . . 9 |- (((A e. On /\ B e. On) /\ (/) e. B) -> A (_ (A .o B))
9	8	ex 373	. . . . . . . 8 |- ((A e. On /\ B e. On) -> ((/) e. B -> A (_ (A .o B)))
10	7, 9	sylbird 205	. . . . . . 7 |- ((A e. On /\ B e. On) -> (B =/= (/) -> A (_ (A .o B)))
11	5, 10	anim12d 556	. . . . . 6 |- ((A e. On /\ B e. On) -> ((A =/= (/) /\ B =/= (/)) -> (1o (_ A /\ A (_ (A .o B))))
12		sstr 2062	. . . . . 6 |- ((1o (_ A /\ A (_ (A .o B)) -> 1o (_ (A .o B))
13	11, 12	syl6 22	. . . . 5 |- ((A e. On /\ B e. On) -> ((A =/= (/) /\ B =/= (/)) -> 1o (_ (A .o B)))
14		neanior 1631	. . . . 5 |- ((A =/= (/) /\ B =/= (/)) <-> -. (A = (/) \/ B = (/)))
15	13, 14	syl5ibr 207	. . . 4 |- ((A e. On /\ B e. On) -> (-. (A = (/) \/ B = (/)) -> 1o (_ (A .o B)))
16		omcl 4155	. . . . 5 |- ((A e. On /\ B e. On) -> (A .o B) e. On)
17		eloni 2948	. . . . 5 |- ((A .o B) e. On -> Ord (A .o B))
18		ordge1n0 4129	. . . . 5 |- (Ord (A .o B) -> (1o (_ (A .o B) <-> (A .o B) =/= (/)))
19	16, 17, 18	3syl 20	. . . 4 |- ((A e. On /\ B e. On) -> (1o (_ (A .o B) <-> (A .o B) =/= (/)))
20	15, 19	sylibd 202	. . 3 |- ((A e. On /\ B e. On) -> (-. (A = (/) \/ B = (/)) -> (A .o B) =/= (/)))
21	20	necon4bd 1619	. 2 |- ((A e. On /\ B e. On) -> ((A .o B) = (/) -> (A = (/) \/ B = (/))))
22		opreq1 3953	. . . . . 6 |- (A = (/) -> (A .o B) = ((/) .o B))
23		om0r 4158	. . . . . 6 |- (B e. On -> ((/) .o B) = (/))
24	22, 23	sylan9eqr 1521	. . . . 5 |- ((B e. On /\ A = (/)) -> (A .o B) = (/))
25	24	ex 373	. . . 4 |- (B e. On -> (A = (/) -> (A .o B) = (/)))
26	25	adantl 388	. . 3 |- ((A e. On /\ B e. On) -> (A = (/) -> (A .o B) = (/)))
27		opreq2 3954	. . . . . 6 |- (B = (/) -> (A .o B) = (A .o (/)))
28		om0 4140	. . . . . 6 |- (A e. On -> (A .o (/)) = (/))
29	27, 28	sylan9eqr 1521	. . . . 5 |- ((A e. On /\ B = (/)) -> (A .o B) = (/))
30	29	ex 373	. . . 4 |- (A e. On -> (B = (/) -> (A .o B) = (/)))
31	30	adantr 389	. . 3 |- ((A e. On /\ B e. On) -> (B = (/) -> (A .o B) = (/)))
32	26, 31	jaod 424	. 2 |- ((A e. On /\ B e. On) -> ((A = (/) \/ B = (/)) -> (A .o B) = (/)))
33	21, 32	impbid 514	1 |- ((A e. On /\ B e. On) -> ((A .o B) = (/) <-> (A = (/) \/ B = (/))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577   (_ wss 2037  (/)c0 2270  Ord word 2937  Oncon0 2938  (class class class)co 3948  1oc1o 4112   .o comu 4115

This theorem is referenced by:  om00el 4191  omlimcl 4193

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1o 4117  df-oadd 4119  df-omul 4120

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