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Theorem invrpropd 17460
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 2382 . . . . 5  |-  (Unit `  K )  =  (Unit `  K )
2 eqid 2382 . . . . 5  |-  ( (mulGrp `  K )s  (Unit `  K )
)  =  ( (mulGrp `  K )s  (Unit `  K )
)
31, 2unitgrpbas 17428 . . . 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 17459 . . . 4  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
9 eqid 2382 . . . . 5  |-  (Unit `  L )  =  (Unit `  L )
10 eqid 2382 . . . . 5  |-  ( (mulGrp `  L )s  (Unit `  L )
)  =  ( (mulGrp `  L )s  (Unit `  L )
)
119, 10unitgrpbas 17428 . . . 4  |-  (Unit `  L )  =  (
Base `  ( (mulGrp `  L )s  (Unit `  L )
) )
128, 11syl6eq 2439 . . 3  |-  ( ph  ->  (Unit `  K )  =  ( Base `  (
(mulGrp `  L )s  (Unit `  L ) ) ) )
13 eqid 2382 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
1413, 1unitss 17422 . . . . . . . 8  |-  (Unit `  K )  C_  ( Base `  K )
1514, 5syl5sseqr 3466 . . . . . . 7  |-  ( ph  ->  (Unit `  K )  C_  B )
1615sselda 3417 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  K ) )  ->  x  e.  B
)
1715sselda 3417 . . . . . 6  |-  ( (
ph  /\  y  e.  (Unit `  K ) )  ->  y  e.  B
)
1816, 17anim12dan 835 . . . . 5  |-  ( (
ph  /\  ( x  e.  (Unit `  K )  /\  y  e.  (Unit `  K ) ) )  ->  ( x  e.  B  /\  y  e.  B ) )
1918, 7syldan 468 . . . 4  |-  ( (
ph  /\  ( x  e.  (Unit `  K )  /\  y  e.  (Unit `  K ) ) )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )
20 fvex 5784 . . . . . 6  |-  (Unit `  K )  e.  _V
21 eqid 2382 . . . . . . . 8  |-  (mulGrp `  K )  =  (mulGrp `  K )
22 eqid 2382 . . . . . . . 8  |-  ( .r
`  K )  =  ( .r `  K
)
2321, 22mgpplusg 17258 . . . . . . 7  |-  ( .r
`  K )  =  ( +g  `  (mulGrp `  K ) )
242, 23ressplusg 14748 . . . . . 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 6209 . . . 4  |-  ( x ( .r `  K
) y )  =  ( x ( +g  `  ( (mulGrp `  K
)s  (Unit `  K )
) ) y )
27 fvex 5784 . . . . . 6  |-  (Unit `  L )  e.  _V
28 eqid 2382 . . . . . . . 8  |-  (mulGrp `  L )  =  (mulGrp `  L )
29 eqid 2382 . . . . . . . 8  |-  ( .r
`  L )  =  ( .r `  L
)
3028, 29mgpplusg 17258 . . . . . . 7  |-  ( .r
`  L )  =  ( +g  `  (mulGrp `  L ) )
3110, 30ressplusg 14748 . . . . . 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 6209 . . . 4  |-  ( x ( .r `  L
) y )  =  ( x ( +g  `  ( (mulGrp `  L
)s  (Unit `  L )
) ) y )
3419, 26, 333eqtr3g 2446 . . 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 16230 . 2  |-  ( ph  ->  ( invg `  ( (mulGrp `  K )s  (Unit `  K ) ) )  =  ( invg `  ( (mulGrp `  L
)s  (Unit `  L )
) ) )
36 eqid 2382 . . 3  |-  ( invr `  K )  =  (
invr `  K )
371, 2, 36invrfval 17435 . 2  |-  ( invr `  K )  =  ( invg `  (
(mulGrp `  K )s  (Unit `  K ) ) )
38 eqid 2382 . . 3  |-  ( invr `  L )  =  (
invr `  L )
399, 10, 38invrfval 17435 . 2  |-  ( invr `  L )  =  ( invg `  (
(mulGrp `  L )s  (Unit `  L ) ) )
4035, 37, 393eqtr4g 2448 1  |-  ( ph  ->  ( invr `  K
)  =  ( invr `  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   _Vcvv 3034   ` cfv 5496  (class class class)co 6196   Basecbs 14634   ↾s cress 14635   +g cplusg 14702   .rcmulr 14703   invgcminusg 16171  mulGrpcmgp 17254  Unitcui 17401   invrcinvr 17433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-tpos 6873  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-0g 14849  df-minusg 16175  df-mgp 17255  df-ur 17267  df-oppr 17385  df-dvdsr 17403  df-unit 17404  df-invr 17434
This theorem is referenced by: (None)
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