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Theorem unitmulcl 13287
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 997 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  R  e.  Ring )
2 eqidd 2178 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Base `  R )  =  ( Base `  R
) )
3 unitmulcl.1 . . . . . . 7  |-  U  =  (Unit `  R )
43a1i 9 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  U  =  (Unit `  R )
)
5 ringsrg 13229 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. SRing
)
61, 5syl 14 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  R  e. SRing )
7 simp3 999 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y  e.  U )
82, 4, 6, 7unitcld 13282 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y  e.  ( Base `  R
) )
9 simp2 998 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X  e.  U )
10 eqidd 2178 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( 1r `  R )  =  ( 1r `  R
) )
11 eqidd 2178 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( ||r `  R )  =  (
||r `  R ) )
12 eqidd 2178 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (oppr `  R
)  =  (oppr `  R
) )
13 eqidd 2178 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( ||r `  (oppr
`  R ) )  =  ( ||r `
 (oppr
`  R ) ) )
144, 10, 11, 12, 13, 6isunitd 13280 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  e.  U  <->  ( X
( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) ) )
159, 14mpbid 147 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X ( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
1615simpld 112 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X
( ||r `
 R ) ( 1r `  R ) )
17 eqid 2177 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
18 eqid 2177 . . . . . 6  |-  ( ||r `  R
)  =  ( ||r `  R
)
19 unitmulcl.2 . . . . . 6  |-  .x.  =  ( .r `  R )
2017, 18, 19dvdsrmul1 13276 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  ( Base `  R
)  /\  X ( ||r `  R ) ( 1r
`  R ) )  ->  ( X  .x.  Y ) ( ||r `  R
) ( ( 1r
`  R )  .x.  Y ) )
211, 8, 16, 20syl3anc 1238 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) ( ( 1r `  R ) 
.x.  Y ) )
22 eqid 2177 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
2317, 19, 22ringlidm 13211 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  ( Base `  R
) )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
241, 8, 23syl2anc 411 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
2521, 24breqtrd 4031 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) Y )
264, 10, 11, 12, 13, 6isunitd 13280 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Y  e.  U  <->  ( Y
( ||r `
 R ) ( 1r `  R )  /\  Y ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) ) )
277, 26mpbid 147 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Y ( ||r `
 R ) ( 1r `  R )  /\  Y ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
2827simpld 112 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y
( ||r `
 R ) ( 1r `  R ) )
2917, 18dvdsrtr 13275 . . 3  |-  ( ( R  e.  Ring  /\  ( X  .x.  Y ) (
||r `  R ) Y  /\  Y ( ||r `
 R ) ( 1r `  R ) )  ->  ( X  .x.  Y ) ( ||r `  R
) ( 1r `  R ) )
301, 25, 28, 29syl3anc 1238 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) ( 1r
`  R ) )
31 eqid 2177 . . . . 5  |-  (oppr `  R
)  =  (oppr `  R
)
3231opprring 13254 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
331, 32syl 14 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (oppr `  R
)  e.  Ring )
342, 4, 6, 9unitcld 13282 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X  e.  ( Base `  R
) )
3531, 17opprbasg 13252 . . . . . . 7  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  (oppr
`  R ) ) )
361, 35syl 14 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Base `  R )  =  ( Base `  (oppr `  R
) ) )
3734, 36eleqtrd 2256 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X  e.  ( Base `  (oppr `  R
) ) )
3827simprd 114 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
39 eqid 2177 . . . . . 6  |-  ( Base `  (oppr
`  R ) )  =  ( Base `  (oppr `  R
) )
40 eqid 2177 . . . . . 6  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
41 eqid 2177 . . . . . 6  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
4239, 40, 41dvdsrmul1 13276 . . . . 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 ) )
4333, 37, 38, 42syl3anc 1238 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Y ( .r `  (oppr `  R ) ) X ) ( ||r `
 (oppr
`  R ) ) ( ( 1r `  R ) ( .r
`  (oppr
`  R ) ) X ) )
4417, 19, 31, 41opprmulg 13248 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  U  /\  X  e.  U )  ->  ( Y ( .r `  (oppr `  R ) ) X )  =  ( X 
.x.  Y ) )
45443com23 1209 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Y ( .r `  (oppr `  R ) ) X )  =  ( X 
.x.  Y ) )
4617, 22srgidcl 13164 . . . . . . 7  |-  ( R  e. SRing  ->  ( 1r `  R )  e.  (
Base `  R )
)
476, 46syl 14 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( 1r `  R )  e.  ( Base `  R
) )
4817, 19, 31, 41opprmulg 13248 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( 1r `  R )  e.  ( Base `  R
)  /\  X  e.  U )  ->  (
( 1r `  R
) ( .r `  (oppr `  R ) ) X )  =  ( X 
.x.  ( 1r `  R ) ) )
491, 47, 9, 48syl3anc 1238 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (
( 1r `  R
) ( .r `  (oppr `  R ) ) X )  =  ( X 
.x.  ( 1r `  R ) ) )
5017, 19, 22ringridm 13212 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  ( Base `  R
) )  ->  ( X  .x.  ( 1r `  R ) )  =  X )
511, 34, 50syl2anc 411 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  ( 1r `  R ) )  =  X )
5249, 51eqtrd 2210 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (
( 1r `  R
) ( .r `  (oppr `  R ) ) X )  =  X )
5343, 45, 523brtr3d 4036 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  (oppr
`  R ) ) X )
5415simprd 114 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
5539, 40dvdsrtr 13275 . . 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
) )
5633, 53, 54, 55syl3anc 1238 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  (oppr
`  R ) ) ( 1r `  R
) )
574, 10, 11, 12, 13, 6isunitd 13280 . 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
) ) ) )
5830, 56, 57mpbir2and 944 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 104    /\ w3a 978    = wceq 1353    e. wcel 2148   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   Basecbs 12464   .rcmulr 12539   1rcur 13147  SRingcsrg 13151   Ringcrg 13184  opprcoppr 13244   ||rcdsr 13260  Unitcui 13261
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-tpos 6248  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-cmn 13095  df-abl 13096  df-mgp 13136  df-ur 13148  df-srg 13152  df-ring 13186  df-oppr 13245  df-dvdsr 13263  df-unit 13264
This theorem is referenced by:  unitmulclb  13288  unitgrp  13290  unitdvcl  13310  rdivmuldivd  13318  lringuplu  13342  subrgugrp  13366
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