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Theorem unitpropd 15479
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 15477 . . . . . 6  |-  ( ph  ->  ( 1r `  K
)  =  ( 1r
`  L ) )
54breq2d 4035 . . . . 5  |-  ( ph  ->  ( z ( ||r `  K
) ( 1r `  K )  <->  z ( ||r `  K ) ( 1r
`  L ) ) )
64breq2d 4035 . . . . 5  |-  ( ph  ->  ( z ( ||r `  (oppr `  K
) ) ( 1r
`  K )  <->  z ( ||r `  (oppr
`  K ) ) ( 1r `  L
) ) )
75, 6anbi12d 691 . . . 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 15478 . . . . . 6  |-  ( ph  ->  ( ||r `
 K )  =  ( ||r `
 L ) )
98breqd 4034 . . . . 5  |-  ( ph  ->  ( z ( ||r `  K
) ( 1r `  L )  <->  z ( ||r `  L ) ( 1r
`  L ) ) )
10 eqid 2283 . . . . . . . . 9  |-  (oppr `  K
)  =  (oppr `  K
)
11 eqid 2283 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
1210, 11opprbas 15411 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  (oppr
`  K ) )
131, 12syl6eq 2331 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (oppr
`  K ) ) )
14 eqid 2283 . . . . . . . . 9  |-  (oppr `  L
)  =  (oppr `  L
)
15 eqid 2283 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
1614, 15opprbas 15411 . . . . . . . 8  |-  ( Base `  L )  =  (
Base `  (oppr
`  L ) )
172, 16syl6eq 2331 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (oppr
`  L ) ) )
183ancom2s 777 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  x  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
19 eqid 2283 . . . . . . . . 9  |-  ( .r
`  K )  =  ( .r `  K
)
20 eqid 2283 . . . . . . . . 9  |-  ( .r
`  (oppr
`  K ) )  =  ( .r `  (oppr `  K ) )
2111, 19, 10, 20opprmul 15408 . . . . . . . 8  |-  ( y ( .r `  (oppr `  K
) ) x )  =  ( x ( .r `  K ) y )
22 eqid 2283 . . . . . . . . 9  |-  ( .r
`  L )  =  ( .r `  L
)
23 eqid 2283 . . . . . . . . 9  |-  ( .r
`  (oppr
`  L ) )  =  ( .r `  (oppr `  L ) )
2415, 22, 14, 23opprmul 15408 . . . . . . . 8  |-  ( y ( .r `  (oppr `  L
) ) x )  =  ( x ( .r `  L ) y )
2518, 21, 243eqtr4g 2340 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  x  e.  B ) )  -> 
( y ( .r
`  (oppr
`  K ) ) x )  =  ( y ( .r `  (oppr `  L ) ) x ) )
2613, 17, 25dvdsrpropd 15478 . . . . . 6  |-  ( ph  ->  ( ||r `
 (oppr
`  K ) )  =  ( ||r `
 (oppr
`  L ) ) )
2726breqd 4034 . . . . 5  |-  ( ph  ->  ( z ( ||r `  (oppr `  K
) ) ( 1r
`  L )  <->  z ( ||r `  (oppr
`  L ) ) ( 1r `  L
) ) )
289, 27anbi12d 691 . . . 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 244 . . 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 2283 . . . 4  |-  (Unit `  K )  =  (Unit `  K )
31 eqid 2283 . . . 4  |-  ( 1r
`  K )  =  ( 1r `  K
)
32 eqid 2283 . . . 4  |-  ( ||r `  K
)  =  ( ||r `  K
)
33 eqid 2283 . . . 4  |-  ( ||r `  (oppr `  K
) )  =  (
||r `  (oppr
`  K ) )
3430, 31, 32, 10, 33isunit 15439 . . 3  |-  ( z  e.  (Unit `  K
)  <->  ( z (
||r `  K ) ( 1r
`  K )  /\  z ( ||r `
 (oppr
`  K ) ) ( 1r `  K
) ) )
35 eqid 2283 . . . 4  |-  (Unit `  L )  =  (Unit `  L )
36 eqid 2283 . . . 4  |-  ( 1r
`  L )  =  ( 1r `  L
)
37 eqid 2283 . . . 4  |-  ( ||r `  L
)  =  ( ||r `  L
)
38 eqid 2283 . . . 4  |-  ( ||r `  (oppr `  L
) )  =  (
||r `  (oppr
`  L ) )
3935, 36, 37, 14, 38isunit 15439 . . 3  |-  ( z  e.  (Unit `  L
)  <->  ( z (
||r `  L ) ( 1r
`  L )  /\  z ( ||r `
 (oppr
`  L ) ) ( 1r `  L
) ) )
4029, 34, 393bitr4g 279 . 2  |-  ( ph  ->  ( z  e.  (Unit `  K )  <->  z  e.  (Unit `  L ) ) )
4140eqrdv 2281 1  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   1rcur 15339  opprcoppr 15404   ||rcdsr 15420  Unitcui 15421
This theorem is referenced by:  invrpropd  15480  drngprop  15523  drngpropd  15539
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mgp 15326  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424
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