Theorem om00 4206

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 2958	. . . . . . . . . 10 |- (A e. On -> Ord A)
2		ordge1n0 4145	. . . . . . . . . 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 3024	. . . . . . . . 9 |- (B e. On -> ((/) e. B <-> B =/= (/)))
7	6	adantl 388	. . . . . . . 8 |- ((A e. On /\ B e. On) -> ((/) e. B <-> B =/= (/)))
8		omword1 4204	. . . . . . . . 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 558	. . . . . 6 |- ((A e. On /\ B e. On) -> ((A =/= (/) /\ B =/= (/)) -> (1o (_ A /\ A (_ (A .o B))))
12		sstr 2072	. . . . . 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 1639	. . . . 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 4171	. . . . 5 |- ((A e. On /\ B e. On) -> (A .o B) e. On)
17		eloni 2958	. . . . 5 |- ((A .o B) e. On -> Ord (A .o B))
18		ordge1n0 4145	. . . . 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 1627	. 2 |- ((A e. On /\ B e. On) -> ((A .o B) = (/) -> (A = (/) \/ B = (/))))
22		opreq1 3968	. . . . . 6 |- (A = (/) -> (A .o B) = ((/) .o B))
23		om0r 4174	. . . . . 6 |- (B e. On -> ((/) .o B) = (/))
24	22, 23	sylan9eqr 1529	. . . . 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 3969	. . . . . 6 |- (B = (/) -> (A .o B) = (A .o (/)))
28		om0 4156	. . . . . 6 |- (A e. On -> (A .o (/)) = (/))
29	27, 28	sylan9eqr 1529	. . . . 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 516	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 956   e. wcel 958   =/= wne 1585   (_ wss 2047  (/)c0 2280  Ord word 2947  Oncon0 2948  (class class class)co 3963  1oc1o 4128   .o comu 4131

This theorem is referenced by:  om00el 4207  omlimcl 4209

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1o 4133  df-oadd 4135  df-omul 4136

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