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Theorem invrpropd 16914
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 2454 . . . . 5  |-  (Unit `  K )  =  (Unit `  K )
2 eqid 2454 . . . . 5  |-  ( (mulGrp `  K )s  (Unit `  K )
)  =  ( (mulGrp `  K )s  (Unit `  K )
)
31, 2unitgrpbas 16882 . . . 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 16913 . . . 4  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
9 eqid 2454 . . . . 5  |-  (Unit `  L )  =  (Unit `  L )
10 eqid 2454 . . . . 5  |-  ( (mulGrp `  L )s  (Unit `  L )
)  =  ( (mulGrp `  L )s  (Unit `  L )
)
119, 10unitgrpbas 16882 . . . 4  |-  (Unit `  L )  =  (
Base `  ( (mulGrp `  L )s  (Unit `  L )
) )
128, 11syl6eq 2511 . . 3  |-  ( ph  ->  (Unit `  K )  =  ( Base `  (
(mulGrp `  L )s  (Unit `  L ) ) ) )
13 eqid 2454 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
1413, 1unitss 16876 . . . . . . . 8  |-  (Unit `  K )  C_  ( Base `  K )
1514, 5syl5sseqr 3514 . . . . . . 7  |-  ( ph  ->  (Unit `  K )  C_  B )
1615sselda 3465 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  K ) )  ->  x  e.  B
)
1715sselda 3465 . . . . . 6  |-  ( (
ph  /\  y  e.  (Unit `  K ) )  ->  y  e.  B
)
1816, 17anim12dan 833 . . . . 5  |-  ( (
ph  /\  ( x  e.  (Unit `  K )  /\  y  e.  (Unit `  K ) ) )  ->  ( x  e.  B  /\  y  e.  B ) )
1918, 7syldan 470 . . . 4  |-  ( (
ph  /\  ( x  e.  (Unit `  K )  /\  y  e.  (Unit `  K ) ) )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )
20 fvex 5810 . . . . . 6  |-  (Unit `  K )  e.  _V
21 eqid 2454 . . . . . . . 8  |-  (mulGrp `  K )  =  (mulGrp `  K )
22 eqid 2454 . . . . . . . 8  |-  ( .r
`  K )  =  ( .r `  K
)
2321, 22mgpplusg 16718 . . . . . . 7  |-  ( .r
`  K )  =  ( +g  `  (mulGrp `  K ) )
242, 23ressplusg 14400 . . . . . 6  |-  ( (Unit `  K )  e.  _V  ->  ( .r `  K
)  =  ( +g  `  ( (mulGrp `  K
)s  (Unit `  K )
) ) )
2520, 24ax-mp 5 . . . . 5  |-  ( .r
`  K )  =  ( +g  `  (
(mulGrp `  K )s  (Unit `  K ) ) )
2625oveqi 6214 . . . 4  |-  ( x ( .r `  K
) y )  =  ( x ( +g  `  ( (mulGrp `  K
)s  (Unit `  K )
) ) y )
27 fvex 5810 . . . . . 6  |-  (Unit `  L )  e.  _V
28 eqid 2454 . . . . . . . 8  |-  (mulGrp `  L )  =  (mulGrp `  L )
29 eqid 2454 . . . . . . . 8  |-  ( .r
`  L )  =  ( .r `  L
)
3028, 29mgpplusg 16718 . . . . . . 7  |-  ( .r
`  L )  =  ( +g  `  (mulGrp `  L ) )
3110, 30ressplusg 14400 . . . . . 6  |-  ( (Unit `  L )  e.  _V  ->  ( .r `  L
)  =  ( +g  `  ( (mulGrp `  L
)s  (Unit `  L )
) ) )
3227, 31ax-mp 5 . . . . 5  |-  ( .r
`  L )  =  ( +g  `  (
(mulGrp `  L )s  (Unit `  L ) ) )
3332oveqi 6214 . . . 4  |-  ( x ( .r `  L
) y )  =  ( x ( +g  `  ( (mulGrp `  L
)s  (Unit `  L )
) ) y )
3419, 26, 333eqtr3g 2518 . . 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 15721 . 2  |-  ( ph  ->  ( invg `  ( (mulGrp `  K )s  (Unit `  K ) ) )  =  ( invg `  ( (mulGrp `  L
)s  (Unit `  L )
) ) )
36 eqid 2454 . . 3  |-  ( invr `  K )  =  (
invr `  K )
371, 2, 36invrfval 16889 . 2  |-  ( invr `  K )  =  ( invg `  (
(mulGrp `  K )s  (Unit `  K ) ) )
38 eqid 2454 . . 3  |-  ( invr `  L )  =  (
invr `  L )
399, 10, 38invrfval 16889 . 2  |-  ( invr `  L )  =  ( invg `  (
(mulGrp `  L )s  (Unit `  L ) ) )
4035, 37, 393eqtr4g 2520 1  |-  ( ph  ->  ( invr `  K
)  =  ( invr `  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   ` cfv 5527  (class class class)co 6201   Basecbs 14293   ↾s cress 14294   +g cplusg 14358   .rcmulr 14359   invgcminusg 15531  mulGrpcmgp 16714  Unitcui 16855   invrcinvr 16887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-tpos 6856  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-0g 14500  df-minusg 15666  df-mgp 16715  df-ur 16727  df-oppr 16839  df-dvdsr 16857  df-unit 16858  df-invr 16888
This theorem is referenced by: (None)
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