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Theorem invrpropd 15480
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 2283 . . . . 5  |-  (Unit `  K )  =  (Unit `  K )
2 eqid 2283 . . . . 5  |-  ( (mulGrp `  K )s  (Unit `  K )
)  =  ( (mulGrp `  K )s  (Unit `  K )
)
31, 2unitgrpbas 15448 . . . 4  |-  (Unit `  K )  =  (
Base `  ( (mulGrp `  K )s  (Unit `  K )
) )
43a1i 10 . . 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 15479 . . . 4  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
9 eqid 2283 . . . . 5  |-  (Unit `  L )  =  (Unit `  L )
10 eqid 2283 . . . . 5  |-  ( (mulGrp `  L )s  (Unit `  L )
)  =  ( (mulGrp `  L )s  (Unit `  L )
)
119, 10unitgrpbas 15448 . . . 4  |-  (Unit `  L )  =  (
Base `  ( (mulGrp `  L )s  (Unit `  L )
) )
128, 11syl6eq 2331 . . 3  |-  ( ph  ->  (Unit `  K )  =  ( Base `  (
(mulGrp `  L )s  (Unit `  L ) ) ) )
13 eqid 2283 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
1413, 1unitss 15442 . . . . . . . 8  |-  (Unit `  K )  C_  ( Base `  K )
1514, 5syl5sseqr 3227 . . . . . . 7  |-  ( ph  ->  (Unit `  K )  C_  B )
1615sselda 3180 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  K ) )  ->  x  e.  B
)
1715sselda 3180 . . . . . 6  |-  ( (
ph  /\  y  e.  (Unit `  K ) )  ->  y  e.  B
)
1816, 17anim12dan 810 . . . . 5  |-  ( (
ph  /\  ( x  e.  (Unit `  K )  /\  y  e.  (Unit `  K ) ) )  ->  ( x  e.  B  /\  y  e.  B ) )
1918, 7syldan 456 . . . 4  |-  ( (
ph  /\  ( x  e.  (Unit `  K )  /\  y  e.  (Unit `  K ) ) )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )
20 fvex 5539 . . . . . 6  |-  (Unit `  K )  e.  _V
21 eqid 2283 . . . . . . . 8  |-  (mulGrp `  K )  =  (mulGrp `  K )
22 eqid 2283 . . . . . . . 8  |-  ( .r
`  K )  =  ( .r `  K
)
2321, 22mgpplusg 15329 . . . . . . 7  |-  ( .r
`  K )  =  ( +g  `  (mulGrp `  K ) )
242, 23ressplusg 13250 . . . . . 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 5871 . . . 4  |-  ( x ( .r `  K
) y )  =  ( x ( +g  `  ( (mulGrp `  K
)s  (Unit `  K )
) ) y )
27 fvex 5539 . . . . . 6  |-  (Unit `  L )  e.  _V
28 eqid 2283 . . . . . . . 8  |-  (mulGrp `  L )  =  (mulGrp `  L )
29 eqid 2283 . . . . . . . 8  |-  ( .r
`  L )  =  ( .r `  L
)
3028, 29mgpplusg 15329 . . . . . . 7  |-  ( .r
`  L )  =  ( +g  `  (mulGrp `  L ) )
3110, 30ressplusg 13250 . . . . . 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 5871 . . . 4  |-  ( x ( .r `  L
) y )  =  ( x ( +g  `  ( (mulGrp `  L
)s  (Unit `  L )
) ) y )
3419, 26, 333eqtr3g 2338 . . 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 14543 . 2  |-  ( ph  ->  ( inv g `  ( (mulGrp `  K )s  (Unit `  K ) ) )  =  ( inv g `  ( (mulGrp `  L
)s  (Unit `  L )
) ) )
36 eqid 2283 . . 3  |-  ( invr `  K )  =  (
invr `  K )
371, 2, 36invrfval 15455 . 2  |-  ( invr `  K )  =  ( inv g `  (
(mulGrp `  K )s  (Unit `  K ) ) )
38 eqid 2283 . . 3  |-  ( invr `  L )  =  (
invr `  L )
399, 10, 38invrfval 15455 . 2  |-  ( invr `  L )  =  ( inv g `  (
(mulGrp `  L )s  (Unit `  L ) ) )
4035, 37, 393eqtr4g 2340 1  |-  ( ph  ->  ( invr `  K
)  =  ( invr `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   .rcmulr 13209   inv gcminusg 14363  mulGrpcmgp 15325  Unitcui 15421   invrcinvr 15453
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-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-minusg 14490  df-mgp 15326  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454
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