nnm00 - Intuitionistic Logic Explorer

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Theorem nnm00 6793

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 728	. . . . . 6 $/\$ $\/$ $/\$
4	3	orcd 745	. . . . 5 $/\$ $\/$ $/\$ $\/$
5	4	a1i 9	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
6		simpr 110	. . . . . . 7 $/\$
7	6	olcd 746	. . . . . 6 $/\$ $\/$
8	7	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $\/$
9		simplr 533	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
10		nnmordi 6779	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	expimpd 363	. . . . . . . . . . . 12 $/\$ $/\$
12	11	ancoms 268	. . . . . . . . . . 11 $/\$ $/\$
13		nnm0 6738	. . . . . . . . . . . . 13
14	13	adantr 276	. . . . . . . . . . . 12 $/\$
15	14	eleq1d 2307	. . . . . . . . . . 11 $/\$
16	12, 15	sylibd 149	. . . . . . . . . 10 $/\$ $/\$
17	16	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
18	17	imp 124	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
19		n0i 3527	. . . . . . . 8
20	18, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
21	9, 20	pm2.21dd 629	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$
22	21	ex 115	. . . . 5 $/\$ $/\$ $/\$ $\/$
23	8, 22	jaod 729	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
24		0elnn 4761	. . . . . . 7 $\/$
25		0elnn 4761	. . . . . . 7 $\/$
26	24, 25	anim12i 338	. . . . . 6 $/\$ $\/$ $/\$ $\/$
27		anddi 833	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28	26, 27	sylib 122	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
29	28	adantr 276	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
30	5, 23, 29	mpjaod 730	. . 3 $/\$ $/\$ $\/$
31	30	ex 115	. 2 $/\$ $\/$
32		oveq1 6082	. . . . . 6
33		nnm0r 6742	. . . . . 6
34	32, 33	sylan9eqr 2293	. . . . 5 $/\$
35	34	ex 115	. . . 4
36	35	adantl 277	. . 3 $/\$
37		oveq2 6083	. . . . . 6
38	37, 13	sylan9eqr 2293	. . . . 5 $/\$
39	38	ex 115	. . . 4
40	39	adantr 276	. . 3 $/\$
41	36, 40	jaod 729	. 2 $/\$ $\/$
42	31, 41	impbid 129	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720

wceq 1402

wcel 2209

c0 3520

com 4732 (class class class)co 6075

comu 6675

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-iord 4506 df-on 4508 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-oadd 6681 df-omul 6682

This theorem is referenced by: enq0tr 7791 nqnq0pi 7795

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