nnm00 - Intuitionistic Logic Explorer

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Theorem nnm00 6530

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 716	. . . . . 6 $/\$ $\/$ $/\$
4	3	orcd 733	. . . . 5 $/\$ $\/$ $/\$ $\/$
5	4	a1i 9	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
6		simpr 110	. . . . . . 7 $/\$
7	6	olcd 734	. . . . . 6 $/\$ $\/$
8	7	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $\/$
9		simplr 528	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
10		nnmordi 6516	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	expimpd 363	. . . . . . . . . . . 12 $/\$ $/\$
12	11	ancoms 268	. . . . . . . . . . 11 $/\$ $/\$
13		nnm0 6475	. . . . . . . . . . . . 13
14	13	adantr 276	. . . . . . . . . . . 12 $/\$
15	14	eleq1d 2246	. . . . . . . . . . 11 $/\$
16	12, 15	sylibd 149	. . . . . . . . . 10 $/\$ $/\$
17	16	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
18	17	imp 124	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
19		n0i 3428	. . . . . . . 8
20	18, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
21	9, 20	pm2.21dd 620	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$
22	21	ex 115	. . . . 5 $/\$ $/\$ $/\$ $\/$
23	8, 22	jaod 717	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
24		0elnn 4618	. . . . . . 7 $\/$
25		0elnn 4618	. . . . . . 7 $\/$
26	24, 25	anim12i 338	. . . . . 6 $/\$ $\/$ $/\$ $\/$
27		anddi 821	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28	26, 27	sylib 122	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
29	28	adantr 276	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
30	5, 23, 29	mpjaod 718	. . 3 $/\$ $/\$ $\/$
31	30	ex 115	. 2 $/\$ $\/$
32		oveq1 5881	. . . . . 6
33		nnm0r 6479	. . . . . 6
34	32, 33	sylan9eqr 2232	. . . . 5 $/\$
35	34	ex 115	. . . 4
36	35	adantl 277	. . 3 $/\$
37		oveq2 5882	. . . . . 6
38	37, 13	sylan9eqr 2232	. . . . 5 $/\$
39	38	ex 115	. . . 4
40	39	adantr 276	. . 3 $/\$
41	36, 40	jaod 717	. 2 $/\$ $\/$
42	31, 41	impbid 129	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 708

wceq 1353

wcel 2148

c0 3422

com 4589 (class class class)co 5874

comu 6414

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-oadd 6420 df-omul 6421

This theorem is referenced by: enq0tr 7432 nqnq0pi 7436

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