Theorem unitmulcl 14085

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 14018	. . . . . . 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 14080	. . . . 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 14078	. . . . . . 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 14074	. . . . 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 13994	. . . . 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 4109	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` R ) Y )
26	4, 10, 11, 12, 13, 6	isunitd 14078	. . . . 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 14073	. . 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 14050	. . . 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 14080	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X e. ( Base ` R ) )
35	31, 17	opprbasg 14046	. . . . . . 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 14074	. . . . 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 14042	. . . . 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 13947	. . . . . . 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 14042	. . . . . 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 13995	. . . . . 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 4114	. . 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 14073	. . 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 14078	. 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 4083   ` cfv 5318  (class class class)co 6007   Basecbs 13040   .rcmulr 13119   1rcur 13930  SRingcsrg 13934   Ringcrg 13967  opp_rcoppr 14038   ||_rcdsr 14057  Unitcui 14058

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-pre-ltirr 8119  ax-pre-lttrn 8121  ax-pre-ltadd 8123

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-tpos 6397  df-pnf 8191  df-mnf 8192  df-ltxr 8194  df-inn 9119  df-2 9177  df-3 9178  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-plusg 13131  df-mulr 13132  df-0g 13299  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-grp 13544  df-minusg 13545  df-cmn 13831  df-abl 13832  df-mgp 13892  df-ur 13931  df-srg 13935  df-ring 13969  df-oppr 14039  df-dvdsr 14060  df-unit 14061

This theorem is referenced by:  unitmulclb  14086  unitgrp  14088  unitdvcl  14108  rdivmuldivd  14116  lringuplu  14168  subrgugrp  14212