nnm00 - Intuitionistic Logic Explorer

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Theorem nnm00 6433

Description: The product of two natural numbers is zero iff at least one of them is zero. (Contributed by Jim Kingdon, 11-Nov-2004.)

**Assertion**
Ref	Expression
nnm00	$/\$ $\/$

**Proof of Theorem nnm00**
Step	Hyp	Ref	Expression
1		simpl 108	. . . . . . 7 $/\$
2		simpl 108	. . . . . . 7 $/\$
3	1, 2	jaoi 706	. . . . . 6 $/\$ $\/$ $/\$
4	3	orcd 723	. . . . 5 $/\$ $\/$ $/\$ $\/$
5	4	a1i 9	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
6		simpr 109	. . . . . . 7 $/\$
7	6	olcd 724	. . . . . 6 $/\$ $\/$
8	7	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $\/$
9		simplr 520	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
10		nnmordi 6420	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	expimpd 361	. . . . . . . . . . . 12 $/\$ $/\$
12	11	ancoms 266	. . . . . . . . . . 11 $/\$ $/\$
13		nnm0 6379	. . . . . . . . . . . . 13
14	13	adantr 274	. . . . . . . . . . . 12 $/\$
15	14	eleq1d 2209	. . . . . . . . . . 11 $/\$
16	12, 15	sylibd 148	. . . . . . . . . 10 $/\$ $/\$
17	16	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$
18	17	imp 123	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
19		n0i 3373	. . . . . . . 8
20	18, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
21	9, 20	pm2.21dd 610	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$
22	21	ex 114	. . . . 5 $/\$ $/\$ $/\$ $\/$
23	8, 22	jaod 707	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
24		0elnn 4540	. . . . . . 7 $\/$
25		0elnn 4540	. . . . . . 7 $\/$
26	24, 25	anim12i 336	. . . . . 6 $/\$ $\/$ $/\$ $\/$
27		anddi 811	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28	26, 27	sylib 121	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
29	28	adantr 274	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
30	5, 23, 29	mpjaod 708	. . 3 $/\$ $/\$ $\/$
31	30	ex 114	. 2 $/\$ $\/$
32		oveq1 5789	. . . . . 6
33		nnm0r 6383	. . . . . 6
34	32, 33	sylan9eqr 2195	. . . . 5 $/\$
35	34	ex 114	. . . 4
36	35	adantl 275	. . 3 $/\$
37		oveq2 5790	. . . . . 6
38	37, 13	sylan9eqr 2195	. . . . 5 $/\$
39	38	ex 114	. . . 4
40	39	adantr 274	. . 3 $/\$
41	36, 40	jaod 707	. 2 $/\$ $\/$
42	31, 41	impbid 128	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698

wceq 1332

wcel 1481

c0 3368

com 4512 (class class class)co 5782

comu 6319

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-oadd 6325 df-omul 6326

This theorem is referenced by: enq0tr 7266 nqnq0pi 7270

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