nnm00 - Intuitionistic Logic Explorer

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Theorem nnm00 6125

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 107	. . . . . . 7 $/\$
2		simpl 107	. . . . . . 7 $/\$
3	1, 2	jaoi 668	. . . . . 6 $/\$ $\/$ $/\$
4	3	orcd 684	. . . . 5 $/\$ $\/$ $/\$ $\/$
5	4	a1i 9	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
6		simpr 108	. . . . . . 7 $/\$
7	6	olcd 685	. . . . . 6 $/\$ $\/$
8	7	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $\/$
9		simplr 496	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
10		nnmordi 6112	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	expimpd 355	. . . . . . . . . . . 12 $/\$ $/\$
12	11	ancoms 264	. . . . . . . . . . 11 $/\$ $/\$
13		nnm0 6077	. . . . . . . . . . . . 13
14	13	adantr 270	. . . . . . . . . . . 12 $/\$
15	14	eleq1d 2147	. . . . . . . . . . 11 $/\$
16	12, 15	sylibd 147	. . . . . . . . . 10 $/\$ $/\$
17	16	adantr 270	. . . . . . . . 9 $/\$ $/\$ $/\$
18	17	imp 122	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
19		n0i 3256	. . . . . . . 8
20	18, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
21	9, 20	pm2.21dd 582	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$
22	21	ex 113	. . . . 5 $/\$ $/\$ $/\$ $\/$
23	8, 22	jaod 669	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
24		0elnn 4358	. . . . . . 7 $\/$
25		0elnn 4358	. . . . . . 7 $\/$
26	24, 25	anim12i 331	. . . . . 6 $/\$ $\/$ $/\$ $\/$
27		anddi 767	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28	26, 27	sylib 120	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
29	28	adantr 270	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
30	5, 23, 29	mpjaod 670	. . 3 $/\$ $/\$ $\/$
31	30	ex 113	. 2 $/\$ $\/$
32		oveq1 5539	. . . . . 6
33		nnm0r 6081	. . . . . 6
34	32, 33	sylan9eqr 2135	. . . . 5 $/\$
35	34	ex 113	. . . 4
36	35	adantl 271	. . 3 $/\$
37		oveq2 5540	. . . . . 6
38	37, 13	sylan9eqr 2135	. . . . 5 $/\$
39	38	ex 113	. . . 4
40	39	adantr 270	. . 3 $/\$
41	36, 40	jaod 669	. 2 $/\$ $\/$
42	31, 41	impbid 127	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 102

wb 103 $\/$ wo 661

wceq 1284

wcel 1433

c0 3251

com 4331 (class class class)co 5532

comu 6022

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029

This theorem is referenced by: enq0tr 6624 nqnq0pi 6628

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