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Theorem unitpropd 15578
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 15576 . . . . . 6  |-  ( ph  ->  ( 1r `  K
)  =  ( 1r
`  L ) )
54breq2d 4116 . . . . 5  |-  ( ph  ->  ( z ( ||r `  K
) ( 1r `  K )  <->  z ( ||r `  K ) ( 1r
`  L ) ) )
64breq2d 4116 . . . . 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 15577 . . . . . 6  |-  ( ph  ->  ( ||r `
 K )  =  ( ||r `
 L ) )
98breqd 4115 . . . . 5  |-  ( ph  ->  ( z ( ||r `  K
) ( 1r `  L )  <->  z ( ||r `  L ) ( 1r
`  L ) ) )
10 eqid 2358 . . . . . . . . 9  |-  (oppr `  K
)  =  (oppr `  K
)
11 eqid 2358 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
1210, 11opprbas 15510 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  (oppr
`  K ) )
131, 12syl6eq 2406 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (oppr
`  K ) ) )
14 eqid 2358 . . . . . . . . 9  |-  (oppr `  L
)  =  (oppr `  L
)
15 eqid 2358 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
1614, 15opprbas 15510 . . . . . . . 8  |-  ( Base `  L )  =  (
Base `  (oppr
`  L ) )
172, 16syl6eq 2406 . . . . . . 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 2358 . . . . . . . . 9  |-  ( .r
`  K )  =  ( .r `  K
)
20 eqid 2358 . . . . . . . . 9  |-  ( .r
`  (oppr
`  K ) )  =  ( .r `  (oppr `  K ) )
2111, 19, 10, 20opprmul 15507 . . . . . . . 8  |-  ( y ( .r `  (oppr `  K
) ) x )  =  ( x ( .r `  K ) y )
22 eqid 2358 . . . . . . . . 9  |-  ( .r
`  L )  =  ( .r `  L
)
23 eqid 2358 . . . . . . . . 9  |-  ( .r
`  (oppr
`  L ) )  =  ( .r `  (oppr `  L ) )
2415, 22, 14, 23opprmul 15507 . . . . . . . 8  |-  ( y ( .r `  (oppr `  L
) ) x )  =  ( x ( .r `  L ) y )
2518, 21, 243eqtr4g 2415 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  x  e.  B ) )  -> 
( y ( .r
`  (oppr
`  K ) ) x )  =  ( y ( .r `  (oppr `  L ) ) x ) )
2613, 17, 25dvdsrpropd 15577 . . . . . 6  |-  ( ph  ->  ( ||r `
 (oppr
`  K ) )  =  ( ||r `
 (oppr
`  L ) ) )
2726breqd 4115 . . . . 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 2358 . . . 4  |-  (Unit `  K )  =  (Unit `  K )
31 eqid 2358 . . . 4  |-  ( 1r
`  K )  =  ( 1r `  K
)
32 eqid 2358 . . . 4  |-  ( ||r `  K
)  =  ( ||r `  K
)
33 eqid 2358 . . . 4  |-  ( ||r `  (oppr `  K
) )  =  (
||r `  (oppr
`  K ) )
3430, 31, 32, 10, 33isunit 15538 . . 3  |-  ( z  e.  (Unit `  K
)  <->  ( z (
||r `  K ) ( 1r
`  K )  /\  z ( ||r `
 (oppr
`  K ) ) ( 1r `  K
) ) )
35 eqid 2358 . . . 4  |-  (Unit `  L )  =  (Unit `  L )
36 eqid 2358 . . . 4  |-  ( 1r
`  L )  =  ( 1r `  L
)
37 eqid 2358 . . . 4  |-  ( ||r `  L
)  =  ( ||r `  L
)
38 eqid 2358 . . . 4  |-  ( ||r `  (oppr `  L
) )  =  (
||r `  (oppr
`  L ) )
3935, 36, 37, 14, 38isunit 15538 . . 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 2356 1  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   Basecbs 13245   .rcmulr 13306   1rcur 15438  opprcoppr 15503   ||rcdsr 15519  Unitcui 15520
This theorem is referenced by:  invrpropd  15579  drngprop  15622  drngpropd  15638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-tpos 6321  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-plusg 13318  df-mulr 13319  df-0g 13503  df-mgp 15425  df-ur 15441  df-oppr 15504  df-dvdsr 15522  df-unit 15523
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