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Theorem invrpropd 17215
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 2441 . . . . 5  |-  (Unit `  K )  =  (Unit `  K )
2 eqid 2441 . . . . 5  |-  ( (mulGrp `  K )s  (Unit `  K )
)  =  ( (mulGrp `  K )s  (Unit `  K )
)
31, 2unitgrpbas 17183 . . . 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 17214 . . . 4  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
9 eqid 2441 . . . . 5  |-  (Unit `  L )  =  (Unit `  L )
10 eqid 2441 . . . . 5  |-  ( (mulGrp `  L )s  (Unit `  L )
)  =  ( (mulGrp `  L )s  (Unit `  L )
)
119, 10unitgrpbas 17183 . . . 4  |-  (Unit `  L )  =  (
Base `  ( (mulGrp `  L )s  (Unit `  L )
) )
128, 11syl6eq 2498 . . 3  |-  ( ph  ->  (Unit `  K )  =  ( Base `  (
(mulGrp `  L )s  (Unit `  L ) ) ) )
13 eqid 2441 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
1413, 1unitss 17177 . . . . . . . 8  |-  (Unit `  K )  C_  ( Base `  K )
1514, 5syl5sseqr 3535 . . . . . . 7  |-  ( ph  ->  (Unit `  K )  C_  B )
1615sselda 3486 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  K ) )  ->  x  e.  B
)
1715sselda 3486 . . . . . 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 470 . . . 4  |-  ( (
ph  /\  ( x  e.  (Unit `  K )  /\  y  e.  (Unit `  K ) ) )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )
20 fvex 5862 . . . . . 6  |-  (Unit `  K )  e.  _V
21 eqid 2441 . . . . . . . 8  |-  (mulGrp `  K )  =  (mulGrp `  K )
22 eqid 2441 . . . . . . . 8  |-  ( .r
`  K )  =  ( .r `  K
)
2321, 22mgpplusg 17013 . . . . . . 7  |-  ( .r
`  K )  =  ( +g  `  (mulGrp `  K ) )
242, 23ressplusg 14611 . . . . . 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 6290 . . . 4  |-  ( x ( .r `  K
) y )  =  ( x ( +g  `  ( (mulGrp `  K
)s  (Unit `  K )
) ) y )
27 fvex 5862 . . . . . 6  |-  (Unit `  L )  e.  _V
28 eqid 2441 . . . . . . . 8  |-  (mulGrp `  L )  =  (mulGrp `  L )
29 eqid 2441 . . . . . . . 8  |-  ( .r
`  L )  =  ( .r `  L
)
3028, 29mgpplusg 17013 . . . . . . 7  |-  ( .r
`  L )  =  ( +g  `  (mulGrp `  L ) )
3110, 30ressplusg 14611 . . . . . 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 6290 . . . 4  |-  ( x ( .r `  L
) y )  =  ( x ( +g  `  ( (mulGrp `  L
)s  (Unit `  L )
) ) y )
3419, 26, 333eqtr3g 2505 . . 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 15982 . 2  |-  ( ph  ->  ( invg `  ( (mulGrp `  K )s  (Unit `  K ) ) )  =  ( invg `  ( (mulGrp `  L
)s  (Unit `  L )
) ) )
36 eqid 2441 . . 3  |-  ( invr `  K )  =  (
invr `  K )
371, 2, 36invrfval 17190 . 2  |-  ( invr `  K )  =  ( invg `  (
(mulGrp `  K )s  (Unit `  K ) ) )
38 eqid 2441 . . 3  |-  ( invr `  L )  =  (
invr `  L )
399, 10, 38invrfval 17190 . 2  |-  ( invr `  L )  =  ( invg `  (
(mulGrp `  L )s  (Unit `  L ) ) )
4035, 37, 393eqtr4g 2507 1  |-  ( ph  ->  ( invr `  K
)  =  ( invr `  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   _Vcvv 3093   ` cfv 5574  (class class class)co 6277   Basecbs 14504   ↾s cress 14505   +g cplusg 14569   .rcmulr 14570   invgcminusg 15923  mulGrpcmgp 17009  Unitcui 17156   invrcinvr 17188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-tpos 6953  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-0g 14711  df-minusg 15927  df-mgp 17010  df-ur 17022  df-oppr 17140  df-dvdsr 17158  df-unit 17159  df-invr 17189
This theorem is referenced by: (None)
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