nnm00 - Intuitionistic Logic Explorer

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Theorem nnm00 6497

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 6484	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	expimpd 361	. . . . . . . . . . . 12 $/\$ $/\$
12	11	ancoms 266	. . . . . . . . . . 11 $/\$ $/\$
13		nnm0 6443	. . . . . . . . . . . . 13
14	13	adantr 274	. . . . . . . . . . . 12 $/\$
15	14	eleq1d 2235	. . . . . . . . . . 11 $/\$
16	12, 15	sylibd 148	. . . . . . . . . 10 $/\$ $/\$
17	16	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$
18	17	imp 123	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
19		n0i 3414	. . . . . . . 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 4596	. . . . . . 7 $\/$
25		0elnn 4596	. . . . . . 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 5849	. . . . . 6
33		nnm0r 6447	. . . . . 6
34	32, 33	sylan9eqr 2221	. . . . 5 $/\$
35	34	ex 114	. . . 4
36	35	adantl 275	. . 3 $/\$
37		oveq2 5850	. . . . . 6
38	37, 13	sylan9eqr 2221	. . . . 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 1343

wcel 2136

c0 3409

com 4567 (class class class)co 5842

comu 6382

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-iord 4344 df-on 4346 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-oadd 6388 df-omul 6389

This theorem is referenced by: enq0tr 7375 nqnq0pi 7379

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