nnm00 - Intuitionistic Logic Explorer

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Theorem nnm00 6697

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 723	. . . . . 6 $/\$ $\/$ $/\$
4	3	orcd 740	. . . . 5 $/\$ $\/$ $/\$ $\/$
5	4	a1i 9	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
6		simpr 110	. . . . . . 7 $/\$
7	6	olcd 741	. . . . . 6 $/\$ $\/$
8	7	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $\/$
9		simplr 529	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
10		nnmordi 6683	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	expimpd 363	. . . . . . . . . . . 12 $/\$ $/\$
12	11	ancoms 268	. . . . . . . . . . 11 $/\$ $/\$
13		nnm0 6642	. . . . . . . . . . . . 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 3500	. . . . . . . 8
20	18, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
21	9, 20	pm2.21dd 625	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$
22	21	ex 115	. . . . 5 $/\$ $/\$ $/\$ $\/$
23	8, 22	jaod 724	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
24		0elnn 4717	. . . . . . 7 $\/$
25		0elnn 4717	. . . . . . 7 $\/$
26	24, 25	anim12i 338	. . . . . 6 $/\$ $\/$ $/\$ $\/$
27		anddi 828	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28	26, 27	sylib 122	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
29	28	adantr 276	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
30	5, 23, 29	mpjaod 725	. . 3 $/\$ $/\$ $\/$
31	30	ex 115	. 2 $/\$ $\/$
32		oveq1 6024	. . . . . 6
33		nnm0r 6646	. . . . . 6
34	32, 33	sylan9eqr 2286	. . . . 5 $/\$
35	34	ex 115	. . . 4
36	35	adantl 277	. . 3 $/\$
37		oveq2 6025	. . . . . 6
38	37, 13	sylan9eqr 2286	. . . . 5 $/\$
39	38	ex 115	. . . 4
40	39	adantr 276	. . 3 $/\$
41	36, 40	jaod 724	. 2 $/\$ $\/$
42	31, 41	impbid 129	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 715

wceq 1397

wcel 2202

c0 3494

com 4688 (class class class)co 6017

comu 6579

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-iord 4463 df-on 4465 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-irdg 6535 df-oadd 6585 df-omul 6586

This theorem is referenced by: enq0tr 7653 nqnq0pi 7657

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