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Theorem unitmulcl 17090
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
StepHypRef Expression
1 simp1 991 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  R  e.  Ring )
2 simp3 993 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y  e.  U )
3 eqid 2460 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
4 unitmulcl.1 . . . . . . 7  |-  U  =  (Unit `  R )
53, 4unitcl 17085 . . . . . 6  |-  ( Y  e.  U  ->  Y  e.  ( Base `  R
) )
62, 5syl 16 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y  e.  ( Base `  R
) )
7 simp2 992 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X  e.  U )
8 eqid 2460 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
9 eqid 2460 . . . . . . . 8  |-  ( ||r `  R
)  =  ( ||r `  R
)
10 eqid 2460 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
11 eqid 2460 . . . . . . . 8  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
124, 8, 9, 10, 11isunit 17083 . . . . . . 7  |-  ( X  e.  U  <->  ( X
( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
137, 12sylib 196 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X ( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
1413simpld 459 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X
( ||r `
 R ) ( 1r `  R ) )
15 unitmulcl.2 . . . . . 6  |-  .x.  =  ( .r `  R )
163, 9, 15dvdsrmul1 17079 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  ( Base `  R
)  /\  X ( ||r `  R ) ( 1r
`  R ) )  ->  ( X  .x.  Y ) ( ||r `  R
) ( ( 1r
`  R )  .x.  Y ) )
171, 6, 14, 16syl3anc 1223 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) ( ( 1r `  R ) 
.x.  Y ) )
183, 15, 8rnglidm 17002 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  ( Base `  R
) )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
191, 6, 18syl2anc 661 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
2017, 19breqtrd 4464 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) Y )
214, 8, 9, 10, 11isunit 17083 . . . . 5  |-  ( Y  e.  U  <->  ( Y
( ||r `
 R ) ( 1r `  R )  /\  Y ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
222, 21sylib 196 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Y ( ||r `
 R ) ( 1r `  R )  /\  Y ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
2322simpld 459 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y
( ||r `
 R ) ( 1r `  R ) )
243, 9dvdsrtr 17078 . . 3  |-  ( ( R  e.  Ring  /\  ( X  .x.  Y ) (
||r `  R ) Y  /\  Y ( ||r `
 R ) ( 1r `  R ) )  ->  ( X  .x.  Y ) ( ||r `  R
) ( 1r `  R ) )
251, 20, 23, 24syl3anc 1223 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) ( 1r
`  R ) )
2610opprrng 17057 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
271, 26syl 16 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (oppr `  R
)  e.  Ring )
28 eqid 2460 . . . . 5  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
293, 15, 10, 28opprmul 17052 . . . 4  |-  ( Y ( .r `  (oppr `  R
) ) X )  =  ( X  .x.  Y )
303, 4unitcl 17085 . . . . . . 7  |-  ( X  e.  U  ->  X  e.  ( Base `  R
) )
317, 30syl 16 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X  e.  ( Base `  R
) )
3222simprd 463 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
3310, 3opprbas 17055 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
3433, 11, 28dvdsrmul1 17079 . . . . . 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 ) )
3527, 31, 32, 34syl3anc 1223 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Y ( .r `  (oppr `  R ) ) X ) ( ||r `
 (oppr
`  R ) ) ( ( 1r `  R ) ( .r
`  (oppr
`  R ) ) X ) )
363, 15, 10, 28opprmul 17052 . . . . . 6  |-  ( ( 1r `  R ) ( .r `  (oppr `  R
) ) X )  =  ( X  .x.  ( 1r `  R ) )
373, 15, 8rngridm 17003 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  ( Base `  R
) )  ->  ( X  .x.  ( 1r `  R ) )  =  X )
381, 31, 37syl2anc 661 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  ( 1r `  R ) )  =  X )
3936, 38syl5eq 2513 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (
( 1r `  R
) ( .r `  (oppr `  R ) ) X )  =  X )
4035, 39breqtrd 4464 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Y ( .r `  (oppr `  R ) ) X ) ( ||r `
 (oppr
`  R ) ) X )
4129, 40syl5eqbrr 4474 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  (oppr
`  R ) ) X )
4213simprd 463 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
4333, 11dvdsrtr 17078 . . 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
) )
4427, 41, 42, 43syl3anc 1223 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  (oppr
`  R ) ) ( 1r `  R
) )
454, 8, 9, 10, 11isunit 17083 . 2  |-  ( ( X  .x.  Y )  e.  U  <->  ( ( X  .x.  Y ) (
||r `  R ) ( 1r
`  R )  /\  ( X  .x.  Y ) ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
4625, 44, 45sylanbrc 664 1  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y )  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   .rcmulr 14545   1rcur 16936   Ringcrg 16979  opprcoppr 17048   ||rcdsr 17064  Unitcui 17065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-tpos 6945  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-mulr 14558  df-0g 14686  df-mnd 15721  df-grp 15851  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-dvdsr 17067  df-unit 17068
This theorem is referenced by:  unitmulclb  17091  unitgrp  17093  unitdvcl  17113  irredrmul  17133  subrgugrp  17224  dchrelbasd  23235  dchrptlem2  23261  rdivmuldivd  27430  dvrcan5  27432  qqhghm  27591  qqhrhm  27592
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