MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prdsinvlem Structured version   Unicode version

Theorem prdsinvlem 16736
Description: Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
Hypotheses
Ref Expression
prdsinvlem.y  |-  Y  =  ( S X_s R )
prdsinvlem.b  |-  B  =  ( Base `  Y
)
prdsinvlem.p  |-  .+  =  ( +g  `  Y )
prdsinvlem.s  |-  ( ph  ->  S  e.  V )
prdsinvlem.i  |-  ( ph  ->  I  e.  W )
prdsinvlem.r  |-  ( ph  ->  R : I --> Grp )
prdsinvlem.f  |-  ( ph  ->  F  e.  B )
prdsinvlem.z  |-  .0.  =  ( 0g  o.  R
)
prdsinvlem.n  |-  N  =  ( y  e.  I  |->  ( ( invg `  ( R `  y
) ) `  ( F `  y )
) )
Assertion
Ref Expression
prdsinvlem  |-  ( ph  ->  ( N  e.  B  /\  ( N  .+  F
)  =  .0.  )
)
Distinct variable groups:    y, B    y, F    y, I    ph, y    y, R    y, S    y, V    y, W    y, Y
Allowed substitution hints:    .+ ( y)    N( y)    .0. ( y)

Proof of Theorem prdsinvlem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 prdsinvlem.n . . 3  |-  N  =  ( y  e.  I  |->  ( ( invg `  ( R `  y
) ) `  ( F `  y )
) )
2 prdsinvlem.r . . . . . . 7  |-  ( ph  ->  R : I --> Grp )
32ffvelrnda 6037 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  ( R `  y )  e.  Grp )
4 prdsinvlem.y . . . . . . 7  |-  Y  =  ( S X_s R )
5 prdsinvlem.b . . . . . . 7  |-  B  =  ( Base `  Y
)
6 prdsinvlem.s . . . . . . . 8  |-  ( ph  ->  S  e.  V )
76adantr 466 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  S  e.  V )
8 prdsinvlem.i . . . . . . . 8  |-  ( ph  ->  I  e.  W )
98adantr 466 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  I  e.  W )
10 ffn 5746 . . . . . . . . 9  |-  ( R : I --> Grp  ->  R  Fn  I )
112, 10syl 17 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
1211adantr 466 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  R  Fn  I )
13 prdsinvlem.f . . . . . . . 8  |-  ( ph  ->  F  e.  B )
1413adantr 466 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  F  e.  B )
15 simpr 462 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  y  e.  I )
164, 5, 7, 9, 12, 14, 15prdsbasprj 15320 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( Base `  ( R `  y )
) )
17 eqid 2429 . . . . . . 7  |-  ( Base `  ( R `  y
) )  =  (
Base `  ( R `  y ) )
18 eqid 2429 . . . . . . 7  |-  ( invg `  ( R `
 y ) )  =  ( invg `  ( R `  y
) )
1917, 18grpinvcl 16653 . . . . . 6  |-  ( ( ( R `  y
)  e.  Grp  /\  ( F `  y )  e.  ( Base `  ( R `  y )
) )  ->  (
( invg `  ( R `  y ) ) `  ( F `
 y ) )  e.  ( Base `  ( R `  y )
) )
203, 16, 19syl2anc 665 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  (
( invg `  ( R `  y ) ) `  ( F `
 y ) )  e.  ( Base `  ( R `  y )
) )
2120ralrimiva 2846 . . . 4  |-  ( ph  ->  A. y  e.  I 
( ( invg `  ( R `  y
) ) `  ( F `  y )
)  e.  ( Base `  ( R `  y
) ) )
224, 5, 6, 8, 11prdsbasmpt 15318 . . . 4  |-  ( ph  ->  ( ( y  e.  I  |->  ( ( invg `  ( R `
 y ) ) `
 ( F `  y ) ) )  e.  B  <->  A. y  e.  I  ( ( invg `  ( R `
 y ) ) `
 ( F `  y ) )  e.  ( Base `  ( R `  y )
) ) )
2321, 22mpbird 235 . . 3  |-  ( ph  ->  ( y  e.  I  |->  ( ( invg `  ( R `  y
) ) `  ( F `  y )
) )  e.  B
)
241, 23syl5eqel 2521 . 2  |-  ( ph  ->  N  e.  B )
252ffvelrnda 6037 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( R `  x )  e.  Grp )
266adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  S  e.  V )
278adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  W )
2811adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  R  Fn  I )
2913adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  F  e.  B )
30 simpr 462 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
314, 5, 26, 27, 28, 29, 30prdsbasprj 15320 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  ( R `  x )
) )
32 eqid 2429 . . . . . . 7  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
33 eqid 2429 . . . . . . 7  |-  ( +g  `  ( R `  x
) )  =  ( +g  `  ( R `
 x ) )
34 eqid 2429 . . . . . . 7  |-  ( 0g
`  ( R `  x ) )  =  ( 0g `  ( R `  x )
)
35 eqid 2429 . . . . . . 7  |-  ( invg `  ( R `
 x ) )  =  ( invg `  ( R `  x
) )
3632, 33, 34, 35grplinv 16654 . . . . . 6  |-  ( ( ( R `  x
)  e.  Grp  /\  ( F `  x )  e.  ( Base `  ( R `  x )
) )  ->  (
( ( invg `  ( R `  x
) ) `  ( F `  x )
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  ( 0g `  ( R `  x )
) )
3725, 31, 36syl2anc 665 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( ( invg `  ( R `  x
) ) `  ( F `  x )
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  ( 0g `  ( R `  x )
) )
38 fveq2 5881 . . . . . . . . . 10  |-  ( y  =  x  ->  ( R `  y )  =  ( R `  x ) )
3938fveq2d 5885 . . . . . . . . 9  |-  ( y  =  x  ->  ( invg `  ( R `
 y ) )  =  ( invg `  ( R `  x
) ) )
40 fveq2 5881 . . . . . . . . 9  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
4139, 40fveq12d 5887 . . . . . . . 8  |-  ( y  =  x  ->  (
( invg `  ( R `  y ) ) `  ( F `
 y ) )  =  ( ( invg `  ( R `
 x ) ) `
 ( F `  x ) ) )
42 fvex 5891 . . . . . . . 8  |-  ( ( invg `  ( R `  x )
) `  ( F `  x ) )  e. 
_V
4341, 1, 42fvmpt 5964 . . . . . . 7  |-  ( x  e.  I  ->  ( N `  x )  =  ( ( invg `  ( R `
 x ) ) `
 ( F `  x ) ) )
4443adantl 467 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( N `  x )  =  ( ( invg `  ( R `
 x ) ) `
 ( F `  x ) ) )
4544oveq1d 6320 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( N `  x
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  ( ( ( invg `  ( R `
 x ) ) `
 ( F `  x ) ) ( +g  `  ( R `
 x ) ) ( F `  x
) ) )
46 prdsinvlem.z . . . . . . 7  |-  .0.  =  ( 0g  o.  R
)
4746fveq1i 5882 . . . . . 6  |-  (  .0.  `  x )  =  ( ( 0g  o.  R
) `  x )
48 fvco2 5956 . . . . . . 7  |-  ( ( R  Fn  I  /\  x  e.  I )  ->  ( ( 0g  o.  R ) `  x
)  =  ( 0g
`  ( R `  x ) ) )
4911, 48sylan 473 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( 0g  o.  R
) `  x )  =  ( 0g `  ( R `  x ) ) )
5047, 49syl5eq 2482 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (  .0.  `  x )  =  ( 0g `  ( R `  x )
) )
5137, 45, 503eqtr4d 2480 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( N `  x
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  (  .0.  `  x
) )
5251mpteq2dva 4512 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( ( N `  x ) ( +g  `  ( R `  x
) ) ( F `
 x ) ) )  =  ( x  e.  I  |->  (  .0.  `  x ) ) )
53 prdsinvlem.p . . . 4  |-  .+  =  ( +g  `  Y )
544, 5, 6, 8, 11, 24, 13, 53prdsplusgval 15321 . . 3  |-  ( ph  ->  ( N  .+  F
)  =  ( x  e.  I  |->  ( ( N `  x ) ( +g  `  ( R `  x )
) ( F `  x ) ) ) )
55 fn0g 16447 . . . . . . 7  |-  0g  Fn  _V
5655a1i 11 . . . . . 6  |-  ( ph  ->  0g  Fn  _V )
57 ssv 3490 . . . . . . 7  |-  ran  R  C_ 
_V
5857a1i 11 . . . . . 6  |-  ( ph  ->  ran  R  C_  _V )
59 fnco 5702 . . . . . 6  |-  ( ( 0g  Fn  _V  /\  R  Fn  I  /\  ran  R  C_  _V )  ->  ( 0g  o.  R
)  Fn  I )
6056, 11, 58, 59syl3anc 1264 . . . . 5  |-  ( ph  ->  ( 0g  o.  R
)  Fn  I )
6146fneq1i 5688 . . . . 5  |-  (  .0. 
Fn  I  <->  ( 0g  o.  R )  Fn  I
)
6260, 61sylibr 215 . . . 4  |-  ( ph  ->  .0.  Fn  I )
63 dffn5 5926 . . . 4  |-  (  .0. 
Fn  I  <->  .0.  =  ( x  e.  I  |->  (  .0.  `  x
) ) )
6462, 63sylib 199 . . 3  |-  ( ph  ->  .0.  =  ( x  e.  I  |->  (  .0.  `  x ) ) )
6552, 54, 643eqtr4d 2480 . 2  |-  ( ph  ->  ( N  .+  F
)  =  .0.  )
6624, 65jca 534 1  |-  ( ph  ->  ( N  e.  B  /\  ( N  .+  F
)  =  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087    C_ wss 3442    |-> cmpt 4484   ran crn 4855    o. ccom 4858    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   Basecbs 15075   +g cplusg 15143   0gc0g 15288   X_scprds 15294   Grpcgrp 16611   invgcminusg 16612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15077  df-ndx 15078  df-slot 15079  df-base 15080  df-plusg 15156  df-mulr 15157  df-sca 15159  df-vsca 15160  df-ip 15161  df-tset 15162  df-ple 15163  df-ds 15165  df-hom 15167  df-cco 15168  df-0g 15290  df-prds 15296  df-mgm 16430  df-sgrp 16469  df-mnd 16479  df-grp 16615  df-minusg 16616
This theorem is referenced by:  prdsgrpd  16737  prdsinvgd  16738
  Copyright terms: Public domain W3C validator