Theorem unitmulcl 13669

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 999	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> R e. Ring )
2		eqidd 2197	. . . . . 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 13603	. . . . . . 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 1001	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> Y e. U )
8	2, 4, 6, 7	unitcld 13664	. . . . 5 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> Y e. ( Base ` R ) )
9		simp2 1000	. . . . . . 7 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X e. U )
10		eqidd 2197	. . . . . . . 8 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( 1r ` R ) = ( 1r ` R ) )
11		eqidd 2197	. . . . . . . 8 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ||r ` R ) = ( ||r ` R ) )
12		eqidd 2197	. . . . . . . 8 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> (oppr ` R ) = (oppr ` R ) )
13		eqidd 2197	. . . . . . . 8 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
14	4, 10, 11, 12, 13, 6	isunitd 13662	. . . . . . 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 2196	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
18		eqid 2196	. . . . . 6 |- ( ||r ` R ) = ( ||r ` R )
19		unitmulcl.2	. . . . . 6 |- .x. = ( .r ` R )
20	17, 18, 19	dvdsrmul1 13658	. . . . 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 1249	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` R ) ( ( 1r ` R ) .x. Y ) )
22		eqid 2196	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
23	17, 19, 22	ringlidm 13579	. . . . 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 4059	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` R ) Y )
26	4, 10, 11, 12, 13, 6	isunitd 13662	. . . . 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 13657	. . 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 1249	. 2 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` R ) ( 1r ` R ) )
31		eqid 2196	. . . . 5 |- (oppr ` R ) = (oppr ` R )
32	31	opprring 13635	. . . 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 13664	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X e. ( Base ` R ) )
35	31, 17	opprbasg 13631	. . . . . . 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 2275	. . . . 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 2196	. . . . . 6 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
40		eqid 2196	. . . . . 6 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
41		eqid 2196	. . . . . 6 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
42	39, 40, 41	dvdsrmul1 13658	. . . . 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 1249	. . . 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 13627	. . . . 5 |- ( ( R e. Ring /\ Y e. U /\ X e. U ) -> ( Y ( .r ` (oppr ` R ) ) X ) = ( X .x. Y ) )
45	44	3com23 1211	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( Y ( .r ` (oppr ` R ) ) X ) = ( X .x. Y ) )
46	17, 22	srgidcl 13532	. . . . . . 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 13627	. . . . . 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 1249	. . . . 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 13580	. . . . . 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 2229	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ( 1r ` R ) ( .r ` (oppr ` R ) ) X ) = X )
53	43, 45, 52	3brtr3d 4064	. . 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 13657	. . 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 1249	. 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 13662	. 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 946	1 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) e. U )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   Basecbs 12678   .rcmulr 12756   1rcur 13515  SRingcsrg 13519   Ringcrg 13552  opp_rcoppr 13623   ||_rcdsr 13642  Unitcui 13643

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-cmn 13416  df-abl 13417  df-mgp 13477  df-ur 13516  df-srg 13520  df-ring 13554  df-oppr 13624  df-dvdsr 13645  df-unit 13646

This theorem is referenced by:  unitmulclb  13670  unitgrp  13672  unitdvcl  13692  rdivmuldivd  13700  lringuplu  13752  subrgugrp  13796