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Theorem invrpropd 15732
Description: The ring inverse function depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
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
invrpropd  |-  ( ph  ->  ( invr `  K
)  =  ( invr `  L ) )
Distinct variable groups:    x, y, B    x, K, y    x, L, y    ph, x, y

Proof of Theorem invrpropd
StepHypRef Expression
1 eqid 2389 . . . . 5  |-  (Unit `  K )  =  (Unit `  K )
2 eqid 2389 . . . . 5  |-  ( (mulGrp `  K )s  (Unit `  K )
)  =  ( (mulGrp `  K )s  (Unit `  K )
)
31, 2unitgrpbas 15700 . . . 4  |-  (Unit `  K )  =  (
Base `  ( (mulGrp `  K )s  (Unit `  K )
) )
43a1i 11 . . 3  |-  ( ph  ->  (Unit `  K )  =  ( Base `  (
(mulGrp `  K )s  (Unit `  K ) ) ) )
5 rngidpropd.1 . . . . 5  |-  ( ph  ->  B  =  ( Base `  K ) )
6 rngidpropd.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  L ) )
7 rngidpropd.3 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
85, 6, 7unitpropd 15731 . . . 4  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
9 eqid 2389 . . . . 5  |-  (Unit `  L )  =  (Unit `  L )
10 eqid 2389 . . . . 5  |-  ( (mulGrp `  L )s  (Unit `  L )
)  =  ( (mulGrp `  L )s  (Unit `  L )
)
119, 10unitgrpbas 15700 . . . 4  |-  (Unit `  L )  =  (
Base `  ( (mulGrp `  L )s  (Unit `  L )
) )
128, 11syl6eq 2437 . . 3  |-  ( ph  ->  (Unit `  K )  =  ( Base `  (
(mulGrp `  L )s  (Unit `  L ) ) ) )
13 eqid 2389 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
1413, 1unitss 15694 . . . . . . . 8  |-  (Unit `  K )  C_  ( Base `  K )
1514, 5syl5sseqr 3342 . . . . . . 7  |-  ( ph  ->  (Unit `  K )  C_  B )
1615sselda 3293 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  K ) )  ->  x  e.  B
)
1715sselda 3293 . . . . . 6  |-  ( (
ph  /\  y  e.  (Unit `  K ) )  ->  y  e.  B
)
1816, 17anim12dan 811 . . . . 5  |-  ( (
ph  /\  ( x  e.  (Unit `  K )  /\  y  e.  (Unit `  K ) ) )  ->  ( x  e.  B  /\  y  e.  B ) )
1918, 7syldan 457 . . . 4  |-  ( (
ph  /\  ( x  e.  (Unit `  K )  /\  y  e.  (Unit `  K ) ) )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )
20 fvex 5684 . . . . . 6  |-  (Unit `  K )  e.  _V
21 eqid 2389 . . . . . . . 8  |-  (mulGrp `  K )  =  (mulGrp `  K )
22 eqid 2389 . . . . . . . 8  |-  ( .r
`  K )  =  ( .r `  K
)
2321, 22mgpplusg 15581 . . . . . . 7  |-  ( .r
`  K )  =  ( +g  `  (mulGrp `  K ) )
242, 23ressplusg 13500 . . . . . 6  |-  ( (Unit `  K )  e.  _V  ->  ( .r `  K
)  =  ( +g  `  ( (mulGrp `  K
)s  (Unit `  K )
) ) )
2520, 24ax-mp 8 . . . . 5  |-  ( .r
`  K )  =  ( +g  `  (
(mulGrp `  K )s  (Unit `  K ) ) )
2625oveqi 6035 . . . 4  |-  ( x ( .r `  K
) y )  =  ( x ( +g  `  ( (mulGrp `  K
)s  (Unit `  K )
) ) y )
27 fvex 5684 . . . . . 6  |-  (Unit `  L )  e.  _V
28 eqid 2389 . . . . . . . 8  |-  (mulGrp `  L )  =  (mulGrp `  L )
29 eqid 2389 . . . . . . . 8  |-  ( .r
`  L )  =  ( .r `  L
)
3028, 29mgpplusg 15581 . . . . . . 7  |-  ( .r
`  L )  =  ( +g  `  (mulGrp `  L ) )
3110, 30ressplusg 13500 . . . . . 6  |-  ( (Unit `  L )  e.  _V  ->  ( .r `  L
)  =  ( +g  `  ( (mulGrp `  L
)s  (Unit `  L )
) ) )
3227, 31ax-mp 8 . . . . 5  |-  ( .r
`  L )  =  ( +g  `  (
(mulGrp `  L )s  (Unit `  L ) ) )
3332oveqi 6035 . . . 4  |-  ( x ( .r `  L
) y )  =  ( x ( +g  `  ( (mulGrp `  L
)s  (Unit `  L )
) ) y )
3419, 26, 333eqtr3g 2444 . . 3  |-  ( (
ph  /\  ( x  e.  (Unit `  K )  /\  y  e.  (Unit `  K ) ) )  ->  ( x ( +g  `  ( (mulGrp `  K )s  (Unit `  K )
) ) y )  =  ( x ( +g  `  ( (mulGrp `  L )s  (Unit `  L )
) ) y ) )
354, 12, 34grpinvpropd 14795 . 2  |-  ( ph  ->  ( inv g `  ( (mulGrp `  K )s  (Unit `  K ) ) )  =  ( inv g `  ( (mulGrp `  L
)s  (Unit `  L )
) ) )
36 eqid 2389 . . 3  |-  ( invr `  K )  =  (
invr `  K )
371, 2, 36invrfval 15707 . 2  |-  ( invr `  K )  =  ( inv g `  (
(mulGrp `  K )s  (Unit `  K ) ) )
38 eqid 2389 . . 3  |-  ( invr `  L )  =  (
invr `  L )
399, 10, 38invrfval 15707 . 2  |-  ( invr `  L )  =  ( inv g `  (
(mulGrp `  L )s  (Unit `  L ) ) )
4035, 37, 393eqtr4g 2446 1  |-  ( ph  ->  ( invr `  K
)  =  ( invr `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2901   ` cfv 5396  (class class class)co 6022   Basecbs 13398   ↾s cress 13399   +g cplusg 13458   .rcmulr 13459   inv gcminusg 14615  mulGrpcmgp 15577  Unitcui 15673   invrcinvr 15705
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-tpos 6417  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-0g 13656  df-minusg 14742  df-mgp 15578  df-ur 15594  df-oppr 15657  df-dvdsr 15675  df-unit 15676  df-invr 15706
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