MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unitpropd Structured version   Unicode version

Theorem unitpropd 17541
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 17539 . . . . . 6  |-  ( ph  ->  ( 1r `  K
)  =  ( 1r
`  L ) )
54breq2d 4451 . . . . 5  |-  ( ph  ->  ( z ( ||r `  K
) ( 1r `  K )  <->  z ( ||r `  K ) ( 1r
`  L ) ) )
64breq2d 4451 . . . . 5  |-  ( ph  ->  ( z ( ||r `  (oppr `  K
) ) ( 1r
`  K )  <->  z ( ||r `  (oppr
`  K ) ) ( 1r `  L
) ) )
75, 6anbi12d 708 . . . 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 17540 . . . . . 6  |-  ( ph  ->  ( ||r `
 K )  =  ( ||r `
 L ) )
98breqd 4450 . . . . 5  |-  ( ph  ->  ( z ( ||r `  K
) ( 1r `  L )  <->  z ( ||r `  L ) ( 1r
`  L ) ) )
10 eqid 2454 . . . . . . . . 9  |-  (oppr `  K
)  =  (oppr `  K
)
11 eqid 2454 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
1210, 11opprbas 17473 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  (oppr
`  K ) )
131, 12syl6eq 2511 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (oppr
`  K ) ) )
14 eqid 2454 . . . . . . . . 9  |-  (oppr `  L
)  =  (oppr `  L
)
15 eqid 2454 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
1614, 15opprbas 17473 . . . . . . . 8  |-  ( Base `  L )  =  (
Base `  (oppr
`  L ) )
172, 16syl6eq 2511 . . . . . . 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 2454 . . . . . . . . 9  |-  ( .r
`  K )  =  ( .r `  K
)
20 eqid 2454 . . . . . . . . 9  |-  ( .r
`  (oppr
`  K ) )  =  ( .r `  (oppr `  K ) )
2111, 19, 10, 20opprmul 17470 . . . . . . . 8  |-  ( y ( .r `  (oppr `  K
) ) x )  =  ( x ( .r `  K ) y )
22 eqid 2454 . . . . . . . . 9  |-  ( .r
`  L )  =  ( .r `  L
)
23 eqid 2454 . . . . . . . . 9  |-  ( .r
`  (oppr
`  L ) )  =  ( .r `  (oppr `  L ) )
2415, 22, 14, 23opprmul 17470 . . . . . . . 8  |-  ( y ( .r `  (oppr `  L
) ) x )  =  ( x ( .r `  L ) y )
2518, 21, 243eqtr4g 2520 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  x  e.  B ) )  -> 
( y ( .r
`  (oppr
`  K ) ) x )  =  ( y ( .r `  (oppr `  L ) ) x ) )
2613, 17, 25dvdsrpropd 17540 . . . . . 6  |-  ( ph  ->  ( ||r `
 (oppr
`  K ) )  =  ( ||r `
 (oppr
`  L ) ) )
2726breqd 4450 . . . . 5  |-  ( ph  ->  ( z ( ||r `  (oppr `  K
) ) ( 1r
`  L )  <->  z ( ||r `  (oppr
`  L ) ) ( 1r `  L
) ) )
289, 27anbi12d 708 . . . 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 2454 . . . 4  |-  (Unit `  K )  =  (Unit `  K )
31 eqid 2454 . . . 4  |-  ( 1r
`  K )  =  ( 1r `  K
)
32 eqid 2454 . . . 4  |-  ( ||r `  K
)  =  ( ||r `  K
)
33 eqid 2454 . . . 4  |-  ( ||r `  (oppr `  K
) )  =  (
||r `  (oppr
`  K ) )
3430, 31, 32, 10, 33isunit 17501 . . 3  |-  ( z  e.  (Unit `  K
)  <->  ( z (
||r `  K ) ( 1r
`  K )  /\  z ( ||r `
 (oppr
`  K ) ) ( 1r `  K
) ) )
35 eqid 2454 . . . 4  |-  (Unit `  L )  =  (Unit `  L )
36 eqid 2454 . . . 4  |-  ( 1r
`  L )  =  ( 1r `  L
)
37 eqid 2454 . . . 4  |-  ( ||r `  L
)  =  ( ||r `  L
)
38 eqid 2454 . . . 4  |-  ( ||r `  (oppr `  L
) )  =  (
||r `  (oppr
`  L ) )
3935, 36, 37, 14, 38isunit 17501 . . 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 2451 1  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   .rcmulr 14785   1rcur 17348  opprcoppr 17466   ||rcdsr 17482  Unitcui 17483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-plusg 14797  df-mulr 14798  df-0g 14931  df-mgp 17337  df-ur 17349  df-oppr 17467  df-dvdsr 17485  df-unit 17486
This theorem is referenced by:  invrpropd  17542  drngprop  17602  drngpropd  17618
  Copyright terms: Public domain W3C validator