Theorem unitmulcl 18664

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 1061	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> R e. Ring )
2		simp3 1063	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> Y e. U )
3		eqid 2622	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
4		unitmulcl.1	. . . . . . 7 |- U = (Unit ` R )
5	3, 4	unitcl 18659	. . . . . 6 |- ( Y e. U -> Y e. ( Base ` R ) )
6	2, 5	syl 17	. . . . 5 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> Y e. ( Base ` R ) )
7		simp2 1062	. . . . . . 7 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X e. U )
8		eqid 2622	. . . . . . . 8 |- ( 1r ` R ) = ( 1r ` R )
9		eqid 2622	. . . . . . . 8 |- ( ||r ` R ) = ( ||r ` R )
10		eqid 2622	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
11		eqid 2622	. . . . . . . 8 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
12	4, 8, 9, 10, 11	isunit 18657	. . . . . . 7 |- ( X e. U <-> ( X ( ||r ` R ) ( 1r ` R ) /\ X ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
13	7, 12	sylib 208	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X ( ||r ` R ) ( 1r ` R ) /\ X ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
14	13	simpld 475	. . . . 5 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X ( ||r ` R ) ( 1r ` R ) )
15		unitmulcl.2	. . . . . 6 |- .x. = ( .r ` R )
16	3, 9, 15	dvdsrmul1 18653	. . . . 5 |- ( ( R e. Ring /\ Y e. ( Base ` R ) /\ X ( ||r ` R ) ( 1r ` R ) ) -> ( X .x. Y ) ( ||r ` R ) ( ( 1r ` R ) .x. Y ) )
17	1, 6, 14, 16	syl3anc 1326	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` R ) ( ( 1r ` R ) .x. Y ) )
18	3, 15, 8	ringlidm 18571	. . . . 5 |- ( ( R e. Ring /\ Y e. ( Base ` R ) ) -> ( ( 1r ` R ) .x. Y ) = Y )
19	1, 6, 18	syl2anc 693	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ( 1r ` R ) .x. Y ) = Y )
20	17, 19	breqtrd 4679	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` R ) Y )
21	4, 8, 9, 10, 11	isunit 18657	. . . . 5 |- ( Y e. U <-> ( Y ( ||r ` R ) ( 1r ` R ) /\ Y ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
22	2, 21	sylib 208	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( Y ( ||r ` R ) ( 1r ` R ) /\ Y ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
23	22	simpld 475	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> Y ( ||r ` R ) ( 1r ` R ) )
24	3, 9	dvdsrtr 18652	. . 3 |- ( ( R e. Ring /\ ( X .x. Y ) ( ||r ` R ) Y /\ Y ( ||r ` R ) ( 1r ` R ) ) -> ( X .x. Y ) ( ||r ` R ) ( 1r ` R ) )
25	1, 20, 23, 24	syl3anc 1326	. 2 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` R ) ( 1r ` R ) )
26	10	opprring 18631	. . . 4 |- ( R e. Ring -> (oppr ` R ) e. Ring )
27	1, 26	syl 17	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> (oppr ` R ) e. Ring )
28		eqid 2622	. . . . 5 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
29	3, 15, 10, 28	opprmul 18626	. . . 4 |- ( Y ( .r ` (oppr ` R ) ) X ) = ( X .x. Y )
30	3, 4	unitcl 18659	. . . . . . 7 |- ( X e. U -> X e. ( Base ` R ) )
31	7, 30	syl 17	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X e. ( Base ` R ) )
32	22	simprd 479	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> Y ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
33	10, 3	opprbas 18629	. . . . . . 7 |- ( Base ` R ) = ( Base ` (oppr ` R ) )
34	33, 11, 28	dvdsrmul1 18653	. . . . . 6 |- ( ( (oppr ` R ) e. Ring /\ X e. ( Base ` R ) /\ Y ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) -> ( Y ( .r ` (oppr ` R ) ) X ) ( ||r ` (oppr ` R ) ) ( ( 1r ` R ) ( .r ` (oppr ` R ) ) X ) )
35	27, 31, 32, 34	syl3anc 1326	. . . . 5 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( Y ( .r ` (oppr ` R ) ) X ) ( ||r ` (oppr ` R ) ) ( ( 1r ` R ) ( .r ` (oppr ` R ) ) X ) )
36	3, 15, 10, 28	opprmul 18626	. . . . . 6 |- ( ( 1r ` R ) ( .r ` (oppr ` R ) ) X ) = ( X .x. ( 1r ` R ) )
37	3, 15, 8	ringridm 18572	. . . . . . 7 |- ( ( R e. Ring /\ X e. ( Base ` R ) ) -> ( X .x. ( 1r ` R ) ) = X )
38	1, 31, 37	syl2anc 693	. . . . . 6 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. ( 1r ` R ) ) = X )
39	36, 38	syl5eq 2668	. . . . 5 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( ( 1r ` R ) ( .r ` (oppr ` R ) ) X ) = X )
40	35, 39	breqtrd 4679	. . . 4 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( Y ( .r ` (oppr ` R ) ) X ) ( ||r ` (oppr ` R ) ) X )
41	29, 40	syl5eqbrr 4689	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` (oppr ` R ) ) X )
42	13	simprd 479	. . 3 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> X ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
43	33, 11	dvdsrtr 18652	. . 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 ) )
44	27, 41, 42, 43	syl3anc 1326	. 2 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
45	4, 8, 9, 10, 11	isunit 18657	. 2 |- ( ( X .x. Y ) e. U <-> ( ( X .x. Y ) ( ||r ` R ) ( 1r ` R ) /\ ( X .x. Y ) ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
46	25, 44, 45	sylanbrc 698	1 |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) e. U )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   1rcur 18501   Ringcrg 18547  opp_rcoppr 18622   ||_rcdsr 18638  Unitcui 18639

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  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  ax-pre-mulgt0 10013

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-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642

This theorem is referenced by:  unitmulclb  18665  unitgrp  18667  unitdvcl  18687  irredrmul  18707  subrgugrp  18799  dchrelbasd  24964  dchrptlem2  24990  rdivmuldivd  29791  dvrcan5  29793  qqhghm  30032  qqhrhm  30033