Theorem om00 4212

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 2964	. . . . . . . . . 10 |- (A e. On -> Ord A)
2		ordge1n0 4151	. . . . . . . . . 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 391	. . . . . . 7 |- ((A e. On /\ B e. On) -> (A =/= (/) -> 1o (_ A))
6		on0eln0 3030	. . . . . . . . 9 |- (B e. On -> ((/) e. B <-> B =/= (/)))
7	6	adantl 390	. . . . . . . 8 |- ((A e. On /\ B e. On) -> ((/) e. B <-> B =/= (/)))
8		omword1 4210	. . . . . . . . 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 560	. . . . . 6 |- ((A e. On /\ B e. On) -> ((A =/= (/) /\ B =/= (/)) -> (1o (_ A /\ A (_ (A .o B))))
12		sstr 2075	. . . . . 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 1642	. . . . 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 4177	. . . . 5 |- ((A e. On /\ B e. On) -> (A .o B) e. On)
17		eloni 2964	. . . . 5 |- ((A .o B) e. On -> Ord (A .o B))
18		ordge1n0 4151	. . . . 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 1630	. 2 |- ((A e. On /\ B e. On) -> ((A .o B) = (/) -> (A = (/) \/ B = (/))))
22		opreq1 3974	. . . . . 6 |- (A = (/) -> (A .o B) = ((/) .o B))
23		om0r 4180	. . . . . 6 |- (B e. On -> ((/) .o B) = (/))
24	22, 23	sylan9eqr 1532	. . . . 5 |- ((B e. On /\ A = (/)) -> (A .o B) = (/))
25	24	ex 373	. . . 4 |- (B e. On -> (A = (/) -> (A .o B) = (/)))
26	25	adantl 390	. . 3 |- ((A e. On /\ B e. On) -> (A = (/) -> (A .o B) = (/)))
27		opreq2 3975	. . . . . 6 |- (B = (/) -> (A .o B) = (A .o (/)))
28		om0 4162	. . . . . 6 |- (A e. On -> (A .o (/)) = (/))
29	27, 28	sylan9eqr 1532	. . . . 5 |- ((A e. On /\ B = (/)) -> (A .o B) = (/))
30	29	ex 373	. . . 4 |- (A e. On -> (B = (/) -> (A .o B) = (/)))
31	30	adantr 391	. . 3 |- ((A e. On /\ B e. On) -> (B = (/) -> (A .o B) = (/)))
32	26, 31	jaod 426	. 2 |- ((A e. On /\ B e. On) -> ((A = (/) \/ B = (/)) -> (A .o B) = (/)))
33	21, 32	impbid 518	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 958   e. wcel 960   =/= wne 1588   (_ wss 2050  (/)c0 2283  Ord word 2953  Oncon0 2954  (class class class)co 3969  1oc1o 4134   .o comu 4137

This theorem is referenced by:  om00el 4213  omlimcl 4215

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1o 4139  df-oadd 4141  df-omul 4142

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