Theorem un0mulcl 11327

Description: If S is closed under multiplication, then so is S u. { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)

Hypotheses

Ref	Expression
un0addcl.1	|- ( ph -> S C_ CC )
un0addcl.2	|- T = ( S u. { 0 } )
un0mulcl.3	|- ( ( ph /\ ( M e. S /\ N e. S ) ) -> ( M x. N ) e. S )

Assertion

Ref	Expression
un0mulcl	|- ( ( ph /\ ( M e. T /\ N e. T ) ) -> ( M x. N ) e. T )

Proof of Theorem un0mulcl

Step	Hyp	Ref	Expression
1		un0addcl.2	. . . . 5 |- T = ( S u. { 0 } )
2	1	eleq2i 2693	. . . 4 |- ( N e. T <-> N e. ( S u. { 0 } ) )
3		elun 3753	. . . 4 |- ( N e. ( S u. { 0 } ) <-> ( N e. S \/ N e. { 0 } ) )
4	2, 3	bitri 264	. . 3 |- ( N e. T <-> ( N e. S \/ N e. { 0 } ) )
5	1	eleq2i 2693	. . . . . 6 |- ( M e. T <-> M e. ( S u. { 0 } ) )
6		elun 3753	. . . . . 6 |- ( M e. ( S u. { 0 } ) <-> ( M e. S \/ M e. { 0 } ) )
7	5, 6	bitri 264	. . . . 5 |- ( M e. T <-> ( M e. S \/ M e. { 0 } ) )
8		ssun1 3776	. . . . . . . . 9 |- S C_ ( S u. { 0 } )
9	8, 1	sseqtr4i 3638	. . . . . . . 8 |- S C_ T
10		un0mulcl.3	. . . . . . . 8 |- ( ( ph /\ ( M e. S /\ N e. S ) ) -> ( M x. N ) e. S )
11	9, 10	sseldi 3601	. . . . . . 7 |- ( ( ph /\ ( M e. S /\ N e. S ) ) -> ( M x. N ) e. T )
12	11	expr 643	. . . . . 6 |- ( ( ph /\ M e. S ) -> ( N e. S -> ( M x. N ) e. T ) )
13		un0addcl.1	. . . . . . . . . . 11 |- ( ph -> S C_ CC )
14	13	sselda 3603	. . . . . . . . . 10 |- ( ( ph /\ N e. S ) -> N e. CC )
15	14	mul02d 10234	. . . . . . . . 9 |- ( ( ph /\ N e. S ) -> ( 0 x. N ) = 0 )
16		ssun2 3777	. . . . . . . . . . 11 |- { 0 } C_ ( S u. { 0 } )
17	16, 1	sseqtr4i 3638	. . . . . . . . . 10 |- { 0 } C_ T
18		c0ex 10034	. . . . . . . . . . 11 |- 0 e. _V
19	18	snss 4316	. . . . . . . . . 10 |- ( 0 e. T <-> { 0 } C_ T )
20	17, 19	mpbir 221	. . . . . . . . 9 |- 0 e. T
21	15, 20	syl6eqel 2709	. . . . . . . 8 |- ( ( ph /\ N e. S ) -> ( 0 x. N ) e. T )
22		elsni 4194	. . . . . . . . . 10 |- ( M e. { 0 } -> M = 0 )
23	22	oveq1d 6665	. . . . . . . . 9 |- ( M e. { 0 } -> ( M x. N ) = ( 0 x. N ) )
24	23	eleq1d 2686	. . . . . . . 8 |- ( M e. { 0 } -> ( ( M x. N ) e. T <-> ( 0 x. N ) e. T ) )
25	21, 24	syl5ibrcom 237	. . . . . . 7 |- ( ( ph /\ N e. S ) -> ( M e. { 0 } -> ( M x. N ) e. T ) )
26	25	impancom 456	. . . . . 6 |- ( ( ph /\ M e. { 0 } ) -> ( N e. S -> ( M x. N ) e. T ) )
27	12, 26	jaodan 826	. . . . 5 |- ( ( ph /\ ( M e. S \/ M e. { 0 } ) ) -> ( N e. S -> ( M x. N ) e. T ) )
28	7, 27	sylan2b 492	. . . 4 |- ( ( ph /\ M e. T ) -> ( N e. S -> ( M x. N ) e. T ) )
29		0cnd 10033	. . . . . . . . . . 11 |- ( ph -> 0 e. CC )
30	29	snssd 4340	. . . . . . . . . 10 |- ( ph -> { 0 } C_ CC )
31	13, 30	unssd 3789	. . . . . . . . 9 |- ( ph -> ( S u. { 0 } ) C_ CC )
32	1, 31	syl5eqss 3649	. . . . . . . 8 |- ( ph -> T C_ CC )
33	32	sselda 3603	. . . . . . 7 |- ( ( ph /\ M e. T ) -> M e. CC )
34	33	mul01d 10235	. . . . . 6 |- ( ( ph /\ M e. T ) -> ( M x. 0 ) = 0 )
35	34, 20	syl6eqel 2709	. . . . 5 |- ( ( ph /\ M e. T ) -> ( M x. 0 ) e. T )
36		elsni 4194	. . . . . . 7 |- ( N e. { 0 } -> N = 0 )
37	36	oveq2d 6666	. . . . . 6 |- ( N e. { 0 } -> ( M x. N ) = ( M x. 0 ) )
38	37	eleq1d 2686	. . . . 5 |- ( N e. { 0 } -> ( ( M x. N ) e. T <-> ( M x. 0 ) e. T ) )
39	35, 38	syl5ibrcom 237	. . . 4 |- ( ( ph /\ M e. T ) -> ( N e. { 0 } -> ( M x. N ) e. T ) )
40	28, 39	jaod 395	. . 3 |- ( ( ph /\ M e. T ) -> ( ( N e. S \/ N e. { 0 } ) -> ( M x. N ) e. T ) )
41	4, 40	syl5bi 232	. 2 |- ( ( ph /\ M e. T ) -> ( N e. T -> ( M x. N ) e. T ) )
42	41	impr 649	1 |- ( ( ph /\ ( M e. T /\ N e. T ) ) -> ( M x. N ) e. T )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   {csn 4177  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079

This theorem is referenced by:  nn0mulcl  11329  plymullem  23972