Theorem unitmulcl 14189

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 1024	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> R e. Ring )
2		eqidd 2232	. . . . . 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 14122	. . . . . . 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 1026	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> Y e. U )
8	2, 4, 6, 7	unitcld 14184	. . . . 5 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> Y e. ( Base ` R ) )
9		simp2 1025	. . . . . . 7 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X e. U )
10		eqidd 2232	. . . . . . . 8 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( 1r ` R ) = ( 1r ` R ) )
11		eqidd 2232	. . . . . . . 8 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ||r ` R ) = ( ||r ` R ) )
12		eqidd 2232	. . . . . . . 8 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> (oppr ` R ) = (oppr ` R ) )
13		eqidd 2232	. . . . . . . 8 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
14	4, 10, 11, 12, 13, 6	isunitd 14182	. . . . . . 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 2231	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
18		eqid 2231	. . . . . 6 |- ( ||r ` R ) = ( ||r ` R )
19		unitmulcl.2	. . . . . 6 |- .x. = ( .r ` R )
20	17, 18, 19	dvdsrmul1 14178	. . . . 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 1274	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` R ) ( ( 1r ` R ) .x. Y ) )
22		eqid 2231	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
23	17, 19, 22	ringlidm 14098	. . . . 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 4119	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` R ) Y )
26	4, 10, 11, 12, 13, 6	isunitd 14182	. . . . 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 14177	. . 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 1274	. 2 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` R ) ( 1r ` R ) )
31		eqid 2231	. . . . 5 |- (oppr ` R ) = (oppr ` R )
32	31	opprring 14154	. . . 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 14184	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X e. ( Base ` R ) )
35	31, 17	opprbasg 14150	. . . . . . 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 2310	. . . . 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 2231	. . . . . 6 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
40		eqid 2231	. . . . . 6 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
41		eqid 2231	. . . . . 6 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
42	39, 40, 41	dvdsrmul1 14178	. . . . 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 1274	. . . 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 14146	. . . . 5 |- ( ( R e. Ring /\ Y e. U /\ X e. U ) -> ( Y ( .r ` (oppr ` R ) ) X ) = ( X .x. Y ) )
45	44	3com23 1236	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( Y ( .r ` (oppr ` R ) ) X ) = ( X .x. Y ) )
46	17, 22	srgidcl 14051	. . . . . . 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 14146	. . . . . 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 1274	. . . . 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 14099	. . . . . 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 2264	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ( 1r ` R ) ( .r ` (oppr ` R ) ) X ) = X )
53	43, 45, 52	3brtr3d 4124	. . 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 14177	. . 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 1274	. 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 14182	. 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 953	1 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) e. U )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   Basecbs 13143   .rcmulr 13222   1rcur 14034  SRingcsrg 14038   Ringcrg 14071  opp_rcoppr 14142   ||_rcdsr 14161  Unitcui 14162

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-tpos 6454  df-pnf 8259  df-mnf 8260  df-ltxr 8262  df-inn 9187  df-2 9245  df-3 9246  df-ndx 13146  df-slot 13147  df-base 13149  df-sets 13150  df-plusg 13234  df-mulr 13235  df-0g 13402  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-grp 13647  df-minusg 13648  df-cmn 13934  df-abl 13935  df-mgp 13996  df-ur 14035  df-srg 14039  df-ring 14073  df-oppr 14143  df-dvdsr 14164  df-unit 14165

This theorem is referenced by:  unitmulclb  14190  unitgrp  14192  unitdvcl  14212  rdivmuldivd  14220  lringuplu  14272  subrgugrp  14316