nnm00 - Intuitionistic Logic Explorer

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Theorem nnm00 6583

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 717	. . . . . 6 $/\$ $\/$ $/\$
4	3	orcd 734	. . . . 5 $/\$ $\/$ $/\$ $\/$
5	4	a1i 9	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
6		simpr 110	. . . . . . 7 $/\$
7	6	olcd 735	. . . . . 6 $/\$ $\/$
8	7	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $\/$
9		simplr 528	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
10		nnmordi 6569	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	expimpd 363	. . . . . . . . . . . 12 $/\$ $/\$
12	11	ancoms 268	. . . . . . . . . . 11 $/\$ $/\$
13		nnm0 6528	. . . . . . . . . . . . 13
14	13	adantr 276	. . . . . . . . . . . 12 $/\$
15	14	eleq1d 2262	. . . . . . . . . . 11 $/\$
16	12, 15	sylibd 149	. . . . . . . . . 10 $/\$ $/\$
17	16	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
18	17	imp 124	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
19		n0i 3452	. . . . . . . 8
20	18, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
21	9, 20	pm2.21dd 621	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$
22	21	ex 115	. . . . 5 $/\$ $/\$ $/\$ $\/$
23	8, 22	jaod 718	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
24		0elnn 4651	. . . . . . 7 $\/$
25		0elnn 4651	. . . . . . 7 $\/$
26	24, 25	anim12i 338	. . . . . 6 $/\$ $\/$ $/\$ $\/$
27		anddi 822	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28	26, 27	sylib 122	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
29	28	adantr 276	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
30	5, 23, 29	mpjaod 719	. . 3 $/\$ $/\$ $\/$
31	30	ex 115	. 2 $/\$ $\/$
32		oveq1 5925	. . . . . 6
33		nnm0r 6532	. . . . . 6
34	32, 33	sylan9eqr 2248	. . . . 5 $/\$
35	34	ex 115	. . . 4
36	35	adantl 277	. . 3 $/\$
37		oveq2 5926	. . . . . 6
38	37, 13	sylan9eqr 2248	. . . . 5 $/\$
39	38	ex 115	. . . 4
40	39	adantr 276	. . 3 $/\$
41	36, 40	jaod 718	. 2 $/\$ $\/$
42	31, 41	impbid 129	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709

wceq 1364

wcel 2164

c0 3446

com 4622 (class class class)co 5918

comu 6467

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-iord 4397 df-on 4399 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-oadd 6473 df-omul 6474

This theorem is referenced by: enq0tr 7494 nqnq0pi 7498

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