nnm00 - Intuitionistic Logic Explorer

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

Theorem nnm00 6741

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 109	. . . . . . 7 $/\$
2		simpl 109	. . . . . . 7 $/\$
3	1, 2	jaoi 724	. . . . . 6 $/\$ $\/$ $/\$
4	3	orcd 741	. . . . 5 $/\$ $\/$ $/\$ $\/$
5	4	a1i 9	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
6		simpr 110	. . . . . . 7 $/\$
7	6	olcd 742	. . . . . 6 $/\$ $\/$
8	7	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $\/$
9		simplr 529	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
10		nnmordi 6727	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	expimpd 363	. . . . . . . . . . . 12 $/\$ $/\$
12	11	ancoms 268	. . . . . . . . . . 11 $/\$ $/\$
13		nnm0 6686	. . . . . . . . . . . . 13
14	13	adantr 276	. . . . . . . . . . . 12 $/\$
15	14	eleq1d 2300	. . . . . . . . . . 11 $/\$
16	12, 15	sylibd 149	. . . . . . . . . 10 $/\$ $/\$
17	16	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
18	17	imp 124	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
19		n0i 3502	. . . . . . . 8
20	18, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
21	9, 20	pm2.21dd 625	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$
22	21	ex 115	. . . . 5 $/\$ $/\$ $/\$ $\/$
23	8, 22	jaod 725	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
24		0elnn 4723	. . . . . . 7 $\/$
25		0elnn 4723	. . . . . . 7 $\/$
26	24, 25	anim12i 338	. . . . . 6 $/\$ $\/$ $/\$ $\/$
27		anddi 829	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28	26, 27	sylib 122	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
29	28	adantr 276	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
30	5, 23, 29	mpjaod 726	. . 3 $/\$ $/\$ $\/$
31	30	ex 115	. 2 $/\$ $\/$
32		oveq1 6035	. . . . . 6
33		nnm0r 6690	. . . . . 6
34	32, 33	sylan9eqr 2286	. . . . 5 $/\$
35	34	ex 115	. . . 4
36	35	adantl 277	. . 3 $/\$
37		oveq2 6036	. . . . . 6
38	37, 13	sylan9eqr 2286	. . . . 5 $/\$
39	38	ex 115	. . . 4
40	39	adantr 276	. . 3 $/\$
41	36, 40	jaod 725	. 2 $/\$ $\/$
42	31, 41	impbid 129	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716

wceq 1398

wcel 2202

c0 3496

com 4694 (class class class)co 6028

comu 6623

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-iord 4469 df-on 4471 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-oadd 6629 df-omul 6630

This theorem is referenced by: enq0tr 7714 nqnq0pi 7718

W3C validator