nnm00 - Intuitionistic Logic Explorer

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Theorem nnm00 6509

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 711	. . . . . 6 $/\$ $\/$ $/\$
4	3	orcd 728	. . . . 5 $/\$ $\/$ $/\$ $\/$
5	4	a1i 9	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
6		simpr 109	. . . . . . 7 $/\$
7	6	olcd 729	. . . . . 6 $/\$ $\/$
8	7	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $\/$
9		simplr 525	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
10		nnmordi 6495	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	expimpd 361	. . . . . . . . . . . 12 $/\$ $/\$
12	11	ancoms 266	. . . . . . . . . . 11 $/\$ $/\$
13		nnm0 6454	. . . . . . . . . . . . 13
14	13	adantr 274	. . . . . . . . . . . 12 $/\$
15	14	eleq1d 2239	. . . . . . . . . . 11 $/\$
16	12, 15	sylibd 148	. . . . . . . . . 10 $/\$ $/\$
17	16	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$
18	17	imp 123	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
19		n0i 3420	. . . . . . . 8
20	18, 19	syl 14	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
21	9, 20	pm2.21dd 615	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$
22	21	ex 114	. . . . 5 $/\$ $/\$ $/\$ $\/$
23	8, 22	jaod 712	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
24		0elnn 4603	. . . . . . 7 $\/$
25		0elnn 4603	. . . . . . 7 $\/$
26	24, 25	anim12i 336	. . . . . 6 $/\$ $\/$ $/\$ $\/$
27		anddi 816	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28	26, 27	sylib 121	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
29	28	adantr 274	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
30	5, 23, 29	mpjaod 713	. . 3 $/\$ $/\$ $\/$
31	30	ex 114	. 2 $/\$ $\/$
32		oveq1 5860	. . . . . 6
33		nnm0r 6458	. . . . . 6
34	32, 33	sylan9eqr 2225	. . . . 5 $/\$
35	34	ex 114	. . . 4
36	35	adantl 275	. . 3 $/\$
37		oveq2 5861	. . . . . 6
38	37, 13	sylan9eqr 2225	. . . . 5 $/\$
39	38	ex 114	. . . 4
40	39	adantr 274	. . 3 $/\$
41	36, 40	jaod 712	. 2 $/\$ $\/$
42	31, 41	impbid 128	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703

wceq 1348

wcel 2141

c0 3414

com 4574 (class class class)co 5853

comu 6393

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-oadd 6399 df-omul 6400

This theorem is referenced by: enq0tr 7396 nqnq0pi 7400

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