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Theorem unitmulcl 15769
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 957 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  R  e.  Ring )
2 simp3 959 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y  e.  U )
3 eqid 2436 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
4 unitmulcl.1 . . . . . . 7  |-  U  =  (Unit `  R )
53, 4unitcl 15764 . . . . . 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 958 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X  e.  U )
8 eqid 2436 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
9 eqid 2436 . . . . . . . 8  |-  ( ||r `  R
)  =  ( ||r `  R
)
10 eqid 2436 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
11 eqid 2436 . . . . . . . 8  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
124, 8, 9, 10, 11isunit 15762 . . . . . . 7  |-  ( X  e.  U  <->  ( X
( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
137, 12sylib 189 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X ( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
1413simpld 446 . . . . 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 15758 . . . . 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 1184 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) ( ( 1r `  R ) 
.x.  Y ) )
183, 15, 8rnglidm 15687 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  ( Base `  R
) )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
191, 6, 18syl2anc 643 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
2017, 19breqtrd 4236 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) Y )
214, 8, 9, 10, 11isunit 15762 . . . . 5  |-  ( Y  e.  U  <->  ( Y
( ||r `
 R ) ( 1r `  R )  /\  Y ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
222, 21sylib 189 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Y ( ||r `
 R ) ( 1r `  R )  /\  Y ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
2322simpld 446 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y
( ||r `
 R ) ( 1r `  R ) )
243, 9dvdsrtr 15757 . . 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 1184 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) ( 1r
`  R ) )
2610opprrng 15736 . . . 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 2436 . . . . 5  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
293, 15, 10, 28opprmul 15731 . . . 4  |-  ( Y ( .r `  (oppr `  R
) ) X )  =  ( X  .x.  Y )
303, 4unitcl 15764 . . . . . . 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 450 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
3310, 3opprbas 15734 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
3433, 11, 28dvdsrmul1 15758 . . . . . 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 1184 . . . . 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 15731 . . . . . 6  |-  ( ( 1r `  R ) ( .r `  (oppr `  R
) ) X )  =  ( X  .x.  ( 1r `  R ) )
373, 15, 8rngridm 15688 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  ( Base `  R
) )  ->  ( X  .x.  ( 1r `  R ) )  =  X )
381, 31, 37syl2anc 643 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  ( 1r `  R ) )  =  X )
3936, 38syl5eq 2480 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (
( 1r `  R
) ( .r `  (oppr `  R ) ) X )  =  X )
4035, 39breqtrd 4236 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Y ( .r `  (oppr `  R ) ) X ) ( ||r `
 (oppr
`  R ) ) X )
4129, 40syl5eqbrr 4246 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  (oppr
`  R ) ) X )
4213simprd 450 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
4333, 11dvdsrtr 15757 . . 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 1184 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  (oppr
`  R ) ) ( 1r `  R
) )
454, 8, 9, 10, 11isunit 15762 . 2  |-  ( ( X  .x.  Y )  e.  U  <->  ( ( X  .x.  Y ) (
||r `  R ) ( 1r
`  R )  /\  ( X  .x.  Y ) ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
4625, 44, 45sylanbrc 646 1  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y )  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   .rcmulr 13530   Ringcrg 15660   1rcur 15662  opprcoppr 15727   ||rcdsr 15743  Unitcui 15744
This theorem is referenced by:  unitmulclb  15770  unitgrp  15772  unitdvcl  15792  irredrmul  15812  subrgugrp  15887  dchrelbasd  21023  dchrptlem2  21049  rdivmuldivd  24227  dvrcan5  24229  qqhghm  24372  qqhrhm  24373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-grp 14812  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747
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