Theorem unitmulcl 14120

Description: The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
unitmulcl.1	|- U = (Unit ` R )
unitmulcl.2	|- .x. = ( .r ` R )

Assertion

Ref	Expression
unitmulcl	|- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) e. U )

Proof of Theorem unitmulcl

Step	Hyp	Ref	Expression
1		simp1 1021	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> R e. Ring )
2		eqidd 2230	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( Base ` R ) = ( Base ` R ) )
3		unitmulcl.1	. . . . . . 7 |- U = (Unit ` R )
4	3	a1i 9	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> U = (Unit ` R ) )
5		ringsrg 14053	. . . . . . 7 |- ( R e. Ring -> R e. SRing )
6	1, 5	syl 14	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> R e. SRing )
7		simp3 1023	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> Y e. U )
8	2, 4, 6, 7	unitcld 14115	. . . . 5 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> Y e. ( Base ` R ) )
9		simp2 1022	. . . . . . 7 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X e. U )
10		eqidd 2230	. . . . . . . 8 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( 1r ` R ) = ( 1r ` R ) )
11		eqidd 2230	. . . . . . . 8 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ||r ` R ) = ( ||r ` R ) )
12		eqidd 2230	. . . . . . . 8 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> (oppr ` R ) = (oppr ` R ) )
13		eqidd 2230	. . . . . . . 8 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
14	4, 10, 11, 12, 13, 6	isunitd 14113	. . . . . . 7 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X e. U <-> ( X ( ||r ` R ) ( 1r ` R ) /\ X ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) ) )
15	9, 14	mpbid 147	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X ( ||r ` R ) ( 1r ` R ) /\ X ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
16	15	simpld 112	. . . . 5 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X ( ||r ` R ) ( 1r ` R ) )
17		eqid 2229	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
18		eqid 2229	. . . . . 6 |- ( ||r ` R ) = ( ||r ` R )
19		unitmulcl.2	. . . . . 6 |- .x. = ( .r ` R )
20	17, 18, 19	dvdsrmul1 14109	. . . . 5 |- ( ( R e. Ring /\ Y e. ( Base ` R ) /\ X ( ||r ` R ) ( 1r ` R ) ) -> ( X .x. Y ) ( ||r ` R ) ( ( 1r ` R ) .x. Y ) )
21	1, 8, 16, 20	syl3anc 1271	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` R ) ( ( 1r ` R ) .x. Y ) )
22		eqid 2229	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
23	17, 19, 22	ringlidm 14029	. . . . 5 |- ( ( R e. Ring /\ Y e. ( Base ` R ) ) -> ( ( 1r ` R ) .x. Y ) = Y )
24	1, 8, 23	syl2anc 411	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ( 1r ` R ) .x. Y ) = Y )
25	21, 24	breqtrd 4112	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` R ) Y )
26	4, 10, 11, 12, 13, 6	isunitd 14113	. . . . 5 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( Y e. U <-> ( Y ( ||r ` R ) ( 1r ` R ) /\ Y ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) ) )
27	7, 26	mpbid 147	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( Y ( ||r ` R ) ( 1r ` R ) /\ Y ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
28	27	simpld 112	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> Y ( ||r ` R ) ( 1r ` R ) )
29	17, 18	dvdsrtr 14108	. . 3 |- ( ( R e. Ring /\ ( X .x. Y ) ( ||r ` R ) Y /\ Y ( ||r ` R ) ( 1r ` R ) ) -> ( X .x. Y ) ( ||r ` R ) ( 1r ` R ) )
30	1, 25, 28, 29	syl3anc 1271	. 2 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` R ) ( 1r ` R ) )
31		eqid 2229	. . . . 5 |- (oppr ` R ) = (oppr ` R )
32	31	opprring 14085	. . . 4 |- ( R e. Ring -> (oppr ` R ) e. Ring )
33	1, 32	syl 14	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> (oppr ` R ) e. Ring )
34	2, 4, 6, 9	unitcld 14115	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X e. ( Base ` R ) )
35	31, 17	opprbasg 14081	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
36	1, 35	syl 14	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
37	34, 36	eleqtrd 2308	. . . . 5 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X e. ( Base ` (oppr ` R ) ) )
38	27	simprd 114	. . . . 5 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> Y ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
39		eqid 2229	. . . . . 6 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
40		eqid 2229	. . . . . 6 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
41		eqid 2229	. . . . . 6 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
42	39, 40, 41	dvdsrmul1 14109	. . . . 5 |- ( ( (oppr ` R ) e. Ring /\ X e. ( Base ` (oppr ` R ) ) /\ Y ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) -> ( Y ( .r ` (oppr ` R ) ) X ) ( ||r ` (oppr ` R ) ) ( ( 1r ` R ) ( .r ` (oppr ` R ) ) X ) )
43	33, 37, 38, 42	syl3anc 1271	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( Y ( .r ` (oppr ` R ) ) X ) ( ||r ` (oppr ` R ) ) ( ( 1r ` R ) ( .r ` (oppr ` R ) ) X ) )
44	17, 19, 31, 41	opprmulg 14077	. . . . 5 |- ( ( R e. Ring /\ Y e. U /\ X e. U ) -> ( Y ( .r ` (oppr ` R ) ) X ) = ( X .x. Y ) )
45	44	3com23 1233	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( Y ( .r ` (oppr ` R ) ) X ) = ( X .x. Y ) )
46	17, 22	srgidcl 13982	. . . . . . 7 |- ( R e. SRing -> ( 1r ` R ) e. ( Base ` R ) )
47	6, 46	syl 14	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( 1r ` R ) e. ( Base ` R ) )
48	17, 19, 31, 41	opprmulg 14077	. . . . . 6 |- ( ( R e. Ring /\ ( 1r ` R ) e. ( Base ` R ) /\ X e. U ) -> ( ( 1r ` R ) ( .r ` (oppr ` R ) ) X ) = ( X .x. ( 1r ` R ) ) )
49	1, 47, 9, 48	syl3anc 1271	. . . . 5 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ( 1r ` R ) ( .r ` (oppr ` R ) ) X ) = ( X .x. ( 1r ` R ) ) )
50	17, 19, 22	ringridm 14030	. . . . . 6 |- ( ( R e. Ring /\ X e. ( Base ` R ) ) -> ( X .x. ( 1r ` R ) ) = X )
51	1, 34, 50	syl2anc 411	. . . . 5 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. ( 1r ` R ) ) = X )
52	49, 51	eqtrd 2262	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ( 1r ` R ) ( .r ` (oppr ` R ) ) X ) = X )
53	43, 45, 52	3brtr3d 4117	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` (oppr ` R ) ) X )
54	15	simprd 114	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
55	39, 40	dvdsrtr 14108	. . 3 |- ( ( (oppr ` R ) e. Ring /\ ( X .x. Y ) ( ||r ` (oppr ` R ) ) X /\ X ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) -> ( X .x. Y ) ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
56	33, 53, 54, 55	syl3anc 1271	. 2 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
57	4, 10, 11, 12, 13, 6	isunitd 14113	. 2 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ( X .x. Y ) e. U <-> ( ( X .x. Y ) ( ||r ` R ) ( 1r ` R ) /\ ( X .x. Y ) ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) ) )
58	30, 56, 57	mpbir2and 950	1 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) e. U )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   Basecbs 13075   .rcmulr 13154   1rcur 13965  SRingcsrg 13969   Ringcrg 14002  opp_rcoppr 14073   ||_rcdsr 14092  Unitcui 14093

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-pre-ltirr 8137  ax-pre-lttrn 8139  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-tpos 6406  df-pnf 8209  df-mnf 8210  df-ltxr 8212  df-inn 9137  df-2 9195  df-3 9196  df-ndx 13078  df-slot 13079  df-base 13081  df-sets 13082  df-plusg 13166  df-mulr 13167  df-0g 13334  df-mgm 13432  df-sgrp 13478  df-mnd 13493  df-grp 13579  df-minusg 13580  df-cmn 13866  df-abl 13867  df-mgp 13927  df-ur 13966  df-srg 13970  df-ring 14004  df-oppr 14074  df-dvdsr 14095  df-unit 14096

This theorem is referenced by:  unitmulclb  14121  unitgrp  14123  unitdvcl  14143  rdivmuldivd  14151  lringuplu  14203  subrgugrp  14247