zmulcl - Intuitionistic Logic Explorer

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Theorem zmulcl 8297

Description: Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)

**Assertion**
Ref	Expression
zmulcl	$/\$

**Proof of Theorem zmulcl**
Step	Hyp	Ref	Expression
1		elznn0 8260	. 2 $/\$ $\/$
2		elznn0 8260	. 2 $/\$ $\/$
3		nn0mulcl 8218	. . . . . . . . 9 $/\$
4	3	orcd 652	. . . . . . . 8 $/\$ $\/$
5	4	a1i 9	. . . . . . 7 $/\$ $/\$ $\/$
6		remulcl 7009	. . . . . . 7 $/\$
7	5, 6	jctild 299	. . . . . 6 $/\$ $/\$ $/\$ $\/$
8		nn0mulcl 8218	. . . . . . . . 9 $/\$
9		recn 7014	. . . . . . . . . . 11
10		recn 7014	. . . . . . . . . . 11
11		mulneg1 7392	. . . . . . . . . . 11 $/\$
12	9, 10, 11	syl2an 273	. . . . . . . . . 10 $/\$
13	12	eleq1d 2106	. . . . . . . . 9 $/\$
14	8, 13	syl5ib 143	. . . . . . . 8 $/\$ $/\$
15		olc 632	. . . . . . . 8 $\/$
16	14, 15	syl6 29	. . . . . . 7 $/\$ $/\$ $\/$
17	16, 6	jctild 299	. . . . . 6 $/\$ $/\$ $/\$ $\/$
18		nn0mulcl 8218	. . . . . . . . 9 $/\$
19		mulneg2 7393	. . . . . . . . . . 11 $/\$
20	9, 10, 19	syl2an 273	. . . . . . . . . 10 $/\$
21	20	eleq1d 2106	. . . . . . . . 9 $/\$
22	18, 21	syl5ib 143	. . . . . . . 8 $/\$ $/\$
23	22, 15	syl6 29	. . . . . . 7 $/\$ $/\$ $\/$
24	23, 6	jctild 299	. . . . . 6 $/\$ $/\$ $/\$ $\/$
25		nn0mulcl 8218	. . . . . . . . 9 $/\$
26		mul2neg 7395	. . . . . . . . . . 11 $/\$
27	9, 10, 26	syl2an 273	. . . . . . . . . 10 $/\$
28	27	eleq1d 2106	. . . . . . . . 9 $/\$
29	25, 28	syl5ib 143	. . . . . . . 8 $/\$ $/\$
30		orc 633	. . . . . . . 8 $\/$
31	29, 30	syl6 29	. . . . . . 7 $/\$ $/\$ $\/$
32	31, 6	jctild 299	. . . . . 6 $/\$ $/\$ $/\$ $\/$
33	7, 17, 24, 32	ccased 872	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$
34		elznn0 8260	. . . . 5 $/\$ $\/$
35	33, 34	syl6ibr 151	. . . 4 $/\$ $\/$ $/\$ $\/$
36	35	imp 115	. . 3 $/\$ $/\$ $\/$ $/\$ $\/$
37	36	an4s 522	. 2 $/\$ $\/$ $/\$ $/\$ $\/$
38	1, 2, 37	syl2anb 275	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 97 $\/$ wo 629

wceq 1243

wcel 1393 (class class class)co 5512

cc 6887

cr 6888

cmul 6894

cneg 7183

cn0 8181

cz 8245

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995

This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-neg 7185 df-inn 7915 df-n0 8182 df-z 8246

This theorem is referenced by: zdivmul 8330 msqznn 8338 zmulcld 8366 uz2mulcl 8545 qaddcl 8570 qmulcl 8572 qreccl 8576 fzctr 8991 flqmulnn0 9141 zexpcl 9270 zesq 9367

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