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Theorem unitmulcl 16775
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 988 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  R  e.  Ring )
2 simp3 990 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y  e.  U )
3 eqid 2443 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
4 unitmulcl.1 . . . . . . 7  |-  U  =  (Unit `  R )
53, 4unitcl 16770 . . . . . 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 989 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X  e.  U )
8 eqid 2443 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
9 eqid 2443 . . . . . . . 8  |-  ( ||r `  R
)  =  ( ||r `  R
)
10 eqid 2443 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
11 eqid 2443 . . . . . . . 8  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
124, 8, 9, 10, 11isunit 16768 . . . . . . 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 16764 . . . . 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 1218 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) ( ( 1r `  R ) 
.x.  Y ) )
183, 15, 8rnglidm 16687 . . . . 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 4335 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) Y )
214, 8, 9, 10, 11isunit 16768 . . . . 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 16763 . . 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 1218 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) ( 1r
`  R ) )
2610opprrng 16742 . . . 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 2443 . . . . 5  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
293, 15, 10, 28opprmul 16737 . . . 4  |-  ( Y ( .r `  (oppr `  R
) ) X )  =  ( X  .x.  Y )
303, 4unitcl 16770 . . . . . . 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 16740 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
3433, 11, 28dvdsrmul1 16764 . . . . . 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 1218 . . . . 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 16737 . . . . . 6  |-  ( ( 1r `  R ) ( .r `  (oppr `  R
) ) X )  =  ( X  .x.  ( 1r `  R ) )
373, 15, 8rngridm 16688 . . . . . . 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 2487 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (
( 1r `  R
) ( .r `  (oppr `  R ) ) X )  =  X )
4035, 39breqtrd 4335 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Y ( .r `  (oppr `  R ) ) X ) ( ||r `
 (oppr
`  R ) ) X )
4129, 40syl5eqbrr 4345 . . 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 16763 . . 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 1218 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  (oppr
`  R ) ) ( 1r `  R
) )
454, 8, 9, 10, 11isunit 16768 . 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 965    = wceq 1369    e. wcel 1756   class class class wbr 4311   ` cfv 5437  (class class class)co 6110   Basecbs 14193   .rcmulr 14258   1rcur 16622   Ringcrg 16664  opprcoppr 16733   ||rcdsr 16749  Unitcui 16750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-tpos 6764  df-recs 6851  df-rdg 6885  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-2 10399  df-3 10400  df-ndx 14196  df-slot 14197  df-base 14198  df-sets 14199  df-plusg 14270  df-mulr 14271  df-0g 14399  df-mnd 15434  df-grp 15564  df-mgp 16611  df-ur 16623  df-rng 16666  df-oppr 16734  df-dvdsr 16752  df-unit 16753
This theorem is referenced by:  unitmulclb  16776  unitgrp  16778  unitdvcl  16798  irredrmul  16818  subrgugrp  16903  dchrelbasd  22597  dchrptlem2  22623  rdivmuldivd  26278  dvrcan5  26280  qqhghm  26436  qqhrhm  26437
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