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Theorem unitpropd 17159
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 17157 . . . . . 6  |-  ( ph  ->  ( 1r `  K
)  =  ( 1r
`  L ) )
54breq2d 4459 . . . . 5  |-  ( ph  ->  ( z ( ||r `  K
) ( 1r `  K )  <->  z ( ||r `  K ) ( 1r
`  L ) ) )
64breq2d 4459 . . . . 5  |-  ( ph  ->  ( z ( ||r `  (oppr `  K
) ) ( 1r
`  K )  <->  z ( ||r `  (oppr
`  K ) ) ( 1r `  L
) ) )
75, 6anbi12d 710 . . . 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 17158 . . . . . 6  |-  ( ph  ->  ( ||r `
 K )  =  ( ||r `
 L ) )
98breqd 4458 . . . . 5  |-  ( ph  ->  ( z ( ||r `  K
) ( 1r `  L )  <->  z ( ||r `  L ) ( 1r
`  L ) ) )
10 eqid 2467 . . . . . . . . 9  |-  (oppr `  K
)  =  (oppr `  K
)
11 eqid 2467 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
1210, 11opprbas 17091 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  (oppr
`  K ) )
131, 12syl6eq 2524 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (oppr
`  K ) ) )
14 eqid 2467 . . . . . . . . 9  |-  (oppr `  L
)  =  (oppr `  L
)
15 eqid 2467 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
1614, 15opprbas 17091 . . . . . . . 8  |-  ( Base `  L )  =  (
Base `  (oppr
`  L ) )
172, 16syl6eq 2524 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (oppr
`  L ) ) )
183ancom2s 800 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  x  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
19 eqid 2467 . . . . . . . . 9  |-  ( .r
`  K )  =  ( .r `  K
)
20 eqid 2467 . . . . . . . . 9  |-  ( .r
`  (oppr
`  K ) )  =  ( .r `  (oppr `  K ) )
2111, 19, 10, 20opprmul 17088 . . . . . . . 8  |-  ( y ( .r `  (oppr `  K
) ) x )  =  ( x ( .r `  K ) y )
22 eqid 2467 . . . . . . . . 9  |-  ( .r
`  L )  =  ( .r `  L
)
23 eqid 2467 . . . . . . . . 9  |-  ( .r
`  (oppr
`  L ) )  =  ( .r `  (oppr `  L ) )
2415, 22, 14, 23opprmul 17088 . . . . . . . 8  |-  ( y ( .r `  (oppr `  L
) ) x )  =  ( x ( .r `  L ) y )
2518, 21, 243eqtr4g 2533 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  x  e.  B ) )  -> 
( y ( .r
`  (oppr
`  K ) ) x )  =  ( y ( .r `  (oppr `  L ) ) x ) )
2613, 17, 25dvdsrpropd 17158 . . . . . 6  |-  ( ph  ->  ( ||r `
 (oppr
`  K ) )  =  ( ||r `
 (oppr
`  L ) ) )
2726breqd 4458 . . . . 5  |-  ( ph  ->  ( z ( ||r `  (oppr `  K
) ) ( 1r
`  L )  <->  z ( ||r `  (oppr
`  L ) ) ( 1r `  L
) ) )
289, 27anbi12d 710 . . . 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 253 . . 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 2467 . . . 4  |-  (Unit `  K )  =  (Unit `  K )
31 eqid 2467 . . . 4  |-  ( 1r
`  K )  =  ( 1r `  K
)
32 eqid 2467 . . . 4  |-  ( ||r `  K
)  =  ( ||r `  K
)
33 eqid 2467 . . . 4  |-  ( ||r `  (oppr `  K
) )  =  (
||r `  (oppr
`  K ) )
3430, 31, 32, 10, 33isunit 17119 . . 3  |-  ( z  e.  (Unit `  K
)  <->  ( z (
||r `  K ) ( 1r
`  K )  /\  z ( ||r `
 (oppr
`  K ) ) ( 1r `  K
) ) )
35 eqid 2467 . . . 4  |-  (Unit `  L )  =  (Unit `  L )
36 eqid 2467 . . . 4  |-  ( 1r
`  L )  =  ( 1r `  L
)
37 eqid 2467 . . . 4  |-  ( ||r `  L
)  =  ( ||r `  L
)
38 eqid 2467 . . . 4  |-  ( ||r `  (oppr `  L
) )  =  (
||r `  (oppr
`  L ) )
3935, 36, 37, 14, 38isunit 17119 . . 3  |-  ( z  e.  (Unit `  L
)  <->  ( z (
||r `  L ) ( 1r
`  L )  /\  z ( ||r `
 (oppr
`  L ) ) ( 1r `  L
) ) )
4029, 34, 393bitr4g 288 . 2  |-  ( ph  ->  ( z  e.  (Unit `  K )  <->  z  e.  (Unit `  L ) ) )
4140eqrdv 2464 1  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   .rcmulr 14559   1rcur 16967  opprcoppr 17084   ||rcdsr 17100  Unitcui 17101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-tpos 6956  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-plusg 14571  df-mulr 14572  df-0g 14700  df-mgp 16956  df-ur 16968  df-oppr 17085  df-dvdsr 17103  df-unit 17104
This theorem is referenced by:  invrpropd  17160  drngprop  17219  drngpropd  17235
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