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Theorem unitpropd 15761
Description: The set of units depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
Hypotheses
Ref Expression
rngidpropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
rngidpropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
rngidpropd.3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
Assertion
Ref Expression
unitpropd  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
Distinct variable groups:    x, y, B    x, K, y    x, L, y    ph, x, y

Proof of Theorem unitpropd
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 rngidpropd.1 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  K ) )
2 rngidpropd.2 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  L ) )
3 rngidpropd.3 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
41, 2, 3rngidpropd 15759 . . . . . 6  |-  ( ph  ->  ( 1r `  K
)  =  ( 1r
`  L ) )
54breq2d 4188 . . . . 5  |-  ( ph  ->  ( z ( ||r `  K
) ( 1r `  K )  <->  z ( ||r `  K ) ( 1r
`  L ) ) )
64breq2d 4188 . . . . 5  |-  ( ph  ->  ( z ( ||r `  (oppr `  K
) ) ( 1r
`  K )  <->  z ( ||r `  (oppr
`  K ) ) ( 1r `  L
) ) )
75, 6anbi12d 692 . . . 4  |-  ( ph  ->  ( ( z (
||r `  K ) ( 1r
`  K )  /\  z ( ||r `
 (oppr
`  K ) ) ( 1r `  K
) )  <->  ( z
( ||r `
 K ) ( 1r `  L )  /\  z ( ||r `  (oppr `  K
) ) ( 1r
`  L ) ) ) )
81, 2, 3dvdsrpropd 15760 . . . . . 6  |-  ( ph  ->  ( ||r `
 K )  =  ( ||r `
 L ) )
98breqd 4187 . . . . 5  |-  ( ph  ->  ( z ( ||r `  K
) ( 1r `  L )  <->  z ( ||r `  L ) ( 1r
`  L ) ) )
10 eqid 2408 . . . . . . . . 9  |-  (oppr `  K
)  =  (oppr `  K
)
11 eqid 2408 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
1210, 11opprbas 15693 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  (oppr
`  K ) )
131, 12syl6eq 2456 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (oppr
`  K ) ) )
14 eqid 2408 . . . . . . . . 9  |-  (oppr `  L
)  =  (oppr `  L
)
15 eqid 2408 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
1614, 15opprbas 15693 . . . . . . . 8  |-  ( Base `  L )  =  (
Base `  (oppr
`  L ) )
172, 16syl6eq 2456 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (oppr
`  L ) ) )
183ancom2s 778 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  x  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
19 eqid 2408 . . . . . . . . 9  |-  ( .r
`  K )  =  ( .r `  K
)
20 eqid 2408 . . . . . . . . 9  |-  ( .r
`  (oppr
`  K ) )  =  ( .r `  (oppr `  K ) )
2111, 19, 10, 20opprmul 15690 . . . . . . . 8  |-  ( y ( .r `  (oppr `  K
) ) x )  =  ( x ( .r `  K ) y )
22 eqid 2408 . . . . . . . . 9  |-  ( .r
`  L )  =  ( .r `  L
)
23 eqid 2408 . . . . . . . . 9  |-  ( .r
`  (oppr
`  L ) )  =  ( .r `  (oppr `  L ) )
2415, 22, 14, 23opprmul 15690 . . . . . . . 8  |-  ( y ( .r `  (oppr `  L
) ) x )  =  ( x ( .r `  L ) y )
2518, 21, 243eqtr4g 2465 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  x  e.  B ) )  -> 
( y ( .r
`  (oppr
`  K ) ) x )  =  ( y ( .r `  (oppr `  L ) ) x ) )
2613, 17, 25dvdsrpropd 15760 . . . . . 6  |-  ( ph  ->  ( ||r `
 (oppr
`  K ) )  =  ( ||r `
 (oppr
`  L ) ) )
2726breqd 4187 . . . . 5  |-  ( ph  ->  ( z ( ||r `  (oppr `  K
) ) ( 1r
`  L )  <->  z ( ||r `  (oppr
`  L ) ) ( 1r `  L
) ) )
289, 27anbi12d 692 . . . 4  |-  ( ph  ->  ( ( z (
||r `  K ) ( 1r
`  L )  /\  z ( ||r `
 (oppr
`  K ) ) ( 1r `  L
) )  <->  ( z
( ||r `
 L ) ( 1r `  L )  /\  z ( ||r `  (oppr `  L
) ) ( 1r
`  L ) ) ) )
297, 28bitrd 245 . . 3  |-  ( ph  ->  ( ( z (
||r `  K ) ( 1r
`  K )  /\  z ( ||r `
 (oppr
`  K ) ) ( 1r `  K
) )  <->  ( z
( ||r `
 L ) ( 1r `  L )  /\  z ( ||r `  (oppr `  L
) ) ( 1r
`  L ) ) ) )
30 eqid 2408 . . . 4  |-  (Unit `  K )  =  (Unit `  K )
31 eqid 2408 . . . 4  |-  ( 1r
`  K )  =  ( 1r `  K
)
32 eqid 2408 . . . 4  |-  ( ||r `  K
)  =  ( ||r `  K
)
33 eqid 2408 . . . 4  |-  ( ||r `  (oppr `  K
) )  =  (
||r `  (oppr
`  K ) )
3430, 31, 32, 10, 33isunit 15721 . . 3  |-  ( z  e.  (Unit `  K
)  <->  ( z (
||r `  K ) ( 1r
`  K )  /\  z ( ||r `
 (oppr
`  K ) ) ( 1r `  K
) ) )
35 eqid 2408 . . . 4  |-  (Unit `  L )  =  (Unit `  L )
36 eqid 2408 . . . 4  |-  ( 1r
`  L )  =  ( 1r `  L
)
37 eqid 2408 . . . 4  |-  ( ||r `  L
)  =  ( ||r `  L
)
38 eqid 2408 . . . 4  |-  ( ||r `  (oppr `  L
) )  =  (
||r `  (oppr
`  L ) )
3935, 36, 37, 14, 38isunit 15721 . . 3  |-  ( z  e.  (Unit `  L
)  <->  ( z (
||r `  L ) ( 1r
`  L )  /\  z ( ||r `
 (oppr
`  L ) ) ( 1r `  L
) ) )
4029, 34, 393bitr4g 280 . 2  |-  ( ph  ->  ( z  e.  (Unit `  K )  <->  z  e.  (Unit `  L ) ) )
4140eqrdv 2406 1  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   Basecbs 13428   .rcmulr 13489   1rcur 15621  opprcoppr 15686   ||rcdsr 15702  Unitcui 15703
This theorem is referenced by:  invrpropd  15762  drngprop  15805  drngpropd  15821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-tpos 6442  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-plusg 13501  df-mulr 13502  df-0g 13686  df-mgp 15608  df-ur 15624  df-oppr 15687  df-dvdsr 15705  df-unit 15706
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