Theorem nnmulcl 7716

Description: Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)

Assertion

Ref	Expression
nnmulcl	|- ((A e. NN /\ B e. NN) -> (A x. B) e. NN)

Proof of Theorem nnmulcl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5463	. . . . 5 |- (x = 1 -> (A x. x) = (A x. 1))
2	1	eleq1d 2103	. . . 4 |- (x = 1 -> ((A x. x) e. NN <-> (A x. 1) e. NN))
3	2	imbi2d 219	. . 3 |- (x = 1 -> ((A e. NN -> (A x. x) e. NN) <-> (A e. NN -> (A x. 1) e. NN)))
4		oveq2 5463	. . . . 5 |- (x = y -> (A x. x) = (A x. y))
5	4	eleq1d 2103	. . . 4 |- (x = y -> ((A x. x) e. NN <-> (A x. y) e. NN))
6	5	imbi2d 219	. . 3 |- (x = y -> ((A e. NN -> (A x. x) e. NN) <-> (A e. NN -> (A x. y) e. NN)))
7		oveq2 5463	. . . . 5 |- (x = (y + 1) -> (A x. x) = (A x. (y + 1)))
8	7	eleq1d 2103	. . . 4 |- (x = (y + 1) -> ((A x. x) e. NN <-> (A x. (y + 1)) e. NN))
9	8	imbi2d 219	. . 3 |- (x = (y + 1) -> ((A e. NN -> (A x. x) e. NN) <-> (A e. NN -> (A x. (y + 1)) e. NN)))
10		oveq2 5463	. . . . 5 |- (x = B -> (A x. x) = (A x. B))
11	10	eleq1d 2103	. . . 4 |- (x = B -> ((A x. x) e. NN <-> (A x. B) e. NN))
12	11	imbi2d 219	. . 3 |- (x = B -> ((A e. NN -> (A x. x) e. NN) <-> (A e. NN -> (A x. B) e. NN)))
13		nncn 7703	. . . 4 |- (A e. NN -> A e. CC)
14		mulid1 6822	. . . . . 6 |- (A e. CC -> (A x. 1) = A)
15	14	eleq1d 2103	. . . . 5 |- (A e. CC -> ((A x. 1) e. NN <-> A e. NN))
16	15	biimprd 147	. . . 4 |- (A e. CC -> (A e. NN -> (A x. 1) e. NN))
17	13, 16	mpcom 32	. . 3 |- (A e. NN -> (A x. 1) e. NN)
18		nnaddcl 7715	. . . . . . . 8 |- (((A x. y) e. NN /\ A e. NN) -> ((A x. y) + A) e. NN)
19	18	ancoms 255	. . . . . . 7 |- ((A e. NN /\ (A x. y) e. NN) -> ((A x. y) + A) e. NN)
20		nncn 7703	. . . . . . . . 9 |- (y e. NN -> y e. CC)
21		ax-1cn 6776	. . . . . . . . . . 11 |- 1 e. CC
22		adddi 6811	. . . . . . . . . . 11 |- ((A e. CC /\ y e. CC /\ 1 e. CC) -> (A x. (y + 1)) = ((A x. y) + (A x. 1)))
23	21, 22	mp3an3 1220	. . . . . . . . . 10 |- ((A e. CC /\ y e. CC) -> (A x. (y + 1)) = ((A x. y) + (A x. 1)))
24	14	oveq2d 5471	. . . . . . . . . . 11 |- (A e. CC -> ((A x. y) + (A x. 1)) = ((A x. y) + A))
25	24	adantr 261	. . . . . . . . . 10 |- ((A e. CC /\ y e. CC) -> ((A x. y) + (A x. 1)) = ((A x. y) + A))
26	23, 25	eqtrd 2069	. . . . . . . . 9 |- ((A e. CC /\ y e. CC) -> (A x. (y + 1)) = ((A x. y) + A))
27	13, 20, 26	syl2an 273	. . . . . . . 8 |- ((A e. NN /\ y e. NN) -> (A x. (y + 1)) = ((A x. y) + A))
28	27	eleq1d 2103	. . . . . . 7 |- ((A e. NN /\ y e. NN) -> ((A x. (y + 1)) e. NN <-> ((A x. y) + A) e. NN))
29	19, 28	syl5ibr 145	. . . . . 6 |- ((A e. NN /\ y e. NN) -> ((A e. NN /\ (A x. y) e. NN) -> (A x. (y + 1)) e. NN))
30	29	exp4b 349	. . . . 5 |- (A e. NN -> (y e. NN -> (A e. NN -> ((A x. y) e. NN -> (A x. (y + 1)) e. NN))))
31	30	pm2.43b 46	. . . 4 |- (y e. NN -> (A e. NN -> ((A x. y) e. NN -> (A x. (y + 1)) e. NN)))
32	31	a2d 23	. . 3 |- (y e. NN -> ((A e. NN -> (A x. y) e. NN) -> (A e. NN -> (A x. (y + 1)) e. NN)))
33	3, 6, 9, 12, 17, 32	nnind 7711	. 2 |- (B e. NN -> (A e. NN -> (A x. B) e. NN))
34	33	impcom 116	1 |- ((A e. NN /\ B e. NN) -> (A x. B) e. NN)

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   e. wcel 1390  (class class class)co 5455   CCcc 6709   1c1 6712   + caddc 6714   x. cmul 6716   NNcn 7695

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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-mulcom 6784  ax-addass 6785  ax-mulass 6786  ax-distr 6787  ax-1rid 6790  ax-cnre 6794

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458  df-inn 7696

This theorem is referenced by:  nnmulcli  7717  nndivtr  7736  nnmulcld  7742  nn0mulcl  7994  qaddcl  8346  qmulcl  8348  nnexpcl  8922  nnsqcl  8976