nnm00 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > nnm00		Unicode version

Theorem nnm00 6102

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 102	. . . . . . 7 $/\$
2		simpl 102	. . . . . . 7 $/\$
3	1, 2	jaoi 636	. . . . . 6 $/\$ $\/$ $/\$
4	3	orcd 652	. . . . 5 $/\$ $\/$ $/\$ $\/$
5	4	a1i 9	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
6		simpr 103	. . . . . . 7 $/\$
7	6	olcd 653	. . . . . 6 $/\$ $\/$
8	7	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $\/$
9		simplr 482	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
10		nnmordi 6089	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	expimpd 345	. . . . . . . . . . . 12 $/\$ $/\$
12	11	ancoms 255	. . . . . . . . . . 11 $/\$ $/\$
13		nnm0 6054	. . . . . . . . . . . . 13
14	13	adantr 261	. . . . . . . . . . . 12 $/\$
15	14	eleq1d 2106	. . . . . . . . . . 11 $/\$
16	12, 15	sylibd 138	. . . . . . . . . 10 $/\$ $/\$
17	16	adantr 261	. . . . . . . . 9 $/\$ $/\$ $/\$
18	17	imp 115	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
19		n0i 3229	. . . . . . . 8
20	18, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
21	9, 20	pm2.21dd 550	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$
22	21	ex 108	. . . . 5 $/\$ $/\$ $/\$ $\/$
23	8, 22	jaod 637	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
24		0elnn 4340	. . . . . . 7 $\/$
25		0elnn 4340	. . . . . . 7 $\/$
26	24, 25	anim12i 321	. . . . . 6 $/\$ $\/$ $/\$ $\/$
27		anddi 734	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28	26, 27	sylib 127	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
29	28	adantr 261	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
30	5, 23, 29	mpjaod 638	. . 3 $/\$ $/\$ $\/$
31	30	ex 108	. 2 $/\$ $\/$
32		oveq1 5519	. . . . . 6
33		nnm0r 6058	. . . . . 6
34	32, 33	sylan9eqr 2094	. . . . 5 $/\$
35	34	ex 108	. . . 4
36	35	adantl 262	. . 3 $/\$
37		oveq2 5520	. . . . . 6
38	37, 13	sylan9eqr 2094	. . . . 5 $/\$
39	38	ex 108	. . . 4
40	39	adantr 261	. . 3 $/\$
41	36, 40	jaod 637	. 2 $/\$ $\/$
42	31, 41	impbid 120	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 97

wb 98 $\/$ wo 629

wceq 1243

wcel 1393

c0 3224

com 4313 (class class class)co 5512

comu 5999

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006

This theorem is referenced by: enq0tr 6532 nqnq0pi 6536

W3C validator