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Theorem prdsinvlem 14652
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  |->  ( ( inv g `  ( 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  |->  ( ( inv g `  ( R `  y
) ) `  ( F `  y )
) )
2 prdsinvlem.r . . . . . . 7  |-  ( ph  ->  R : I --> Grp )
3 ffvelrn 5701 . . . . . . 7  |-  ( ( R : I --> Grp  /\  y  e.  I )  ->  ( R `  y
)  e.  Grp )
42, 3sylan 457 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  ( R `  y )  e.  Grp )
5 prdsinvlem.y . . . . . . 7  |-  Y  =  ( S X_s R )
6 prdsinvlem.b . . . . . . 7  |-  B  =  ( Base `  Y
)
7 prdsinvlem.s . . . . . . . 8  |-  ( ph  ->  S  e.  V )
87adantr 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  S  e.  V )
9 prdsinvlem.i . . . . . . . 8  |-  ( ph  ->  I  e.  W )
109adantr 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  I  e.  W )
11 ffn 5427 . . . . . . . . 9  |-  ( R : I --> Grp  ->  R  Fn  I )
122, 11syl 15 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
1312adantr 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  R  Fn  I )
14 prdsinvlem.f . . . . . . . 8  |-  ( ph  ->  F  e.  B )
1514adantr 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  F  e.  B )
16 simpr 447 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  y  e.  I )
175, 6, 8, 10, 13, 15, 16prdsbasprj 13420 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( Base `  ( R `  y )
) )
18 eqid 2316 . . . . . . 7  |-  ( Base `  ( R `  y
) )  =  (
Base `  ( R `  y ) )
19 eqid 2316 . . . . . . 7  |-  ( inv g `  ( R `
 y ) )  =  ( inv g `  ( R `  y
) )
2018, 19grpinvcl 14576 . . . . . 6  |-  ( ( ( R `  y
)  e.  Grp  /\  ( F `  y )  e.  ( Base `  ( R `  y )
) )  ->  (
( inv g `  ( R `  y ) ) `  ( F `
 y ) )  e.  ( Base `  ( R `  y )
) )
214, 17, 20syl2anc 642 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  (
( inv g `  ( R `  y ) ) `  ( F `
 y ) )  e.  ( Base `  ( R `  y )
) )
2221ralrimiva 2660 . . . 4  |-  ( ph  ->  A. y  e.  I 
( ( inv g `  ( R `  y
) ) `  ( F `  y )
)  e.  ( Base `  ( R `  y
) ) )
235, 6, 7, 9, 12prdsbasmpt 13418 . . . 4  |-  ( ph  ->  ( ( y  e.  I  |->  ( ( inv g `  ( R `
 y ) ) `
 ( F `  y ) ) )  e.  B  <->  A. y  e.  I  ( ( inv g `  ( R `
 y ) ) `
 ( F `  y ) )  e.  ( Base `  ( R `  y )
) ) )
2422, 23mpbird 223 . . 3  |-  ( ph  ->  ( y  e.  I  |->  ( ( inv g `  ( R `  y
) ) `  ( F `  y )
) )  e.  B
)
251, 24syl5eqel 2400 . 2  |-  ( ph  ->  N  e.  B )
26 ffvelrn 5701 . . . . . . 7  |-  ( ( R : I --> Grp  /\  x  e.  I )  ->  ( R `  x
)  e.  Grp )
272, 26sylan 457 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( R `  x )  e.  Grp )
287adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  S  e.  V )
299adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  W )
3012adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  R  Fn  I )
3114adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  F  e.  B )
32 simpr 447 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
335, 6, 28, 29, 30, 31, 32prdsbasprj 13420 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  ( R `  x )
) )
34 eqid 2316 . . . . . . 7  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
35 eqid 2316 . . . . . . 7  |-  ( +g  `  ( R `  x
) )  =  ( +g  `  ( R `
 x ) )
36 eqid 2316 . . . . . . 7  |-  ( 0g
`  ( R `  x ) )  =  ( 0g `  ( R `  x )
)
37 eqid 2316 . . . . . . 7  |-  ( inv g `  ( R `
 x ) )  =  ( inv g `  ( R `  x
) )
3834, 35, 36, 37grplinv 14577 . . . . . 6  |-  ( ( ( R `  x
)  e.  Grp  /\  ( F `  x )  e.  ( Base `  ( R `  x )
) )  ->  (
( ( inv g `  ( R `  x
) ) `  ( F `  x )
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  ( 0g `  ( R `  x )
) )
3927, 33, 38syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( ( inv g `  ( R `  x
) ) `  ( F `  x )
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  ( 0g `  ( R `  x )
) )
40 fveq2 5563 . . . . . . . . . 10  |-  ( y  =  x  ->  ( R `  y )  =  ( R `  x ) )
4140fveq2d 5567 . . . . . . . . 9  |-  ( y  =  x  ->  ( inv g `  ( R `
 y ) )  =  ( inv g `  ( R `  x
) ) )
42 fveq2 5563 . . . . . . . . 9  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
4341, 42fveq12d 5569 . . . . . . . 8  |-  ( y  =  x  ->  (
( inv g `  ( R `  y ) ) `  ( F `
 y ) )  =  ( ( inv g `  ( R `
 x ) ) `
 ( F `  x ) ) )
44 fvex 5577 . . . . . . . 8  |-  ( ( inv g `  ( R `  x )
) `  ( F `  x ) )  e. 
_V
4543, 1, 44fvmpt 5640 . . . . . . 7  |-  ( x  e.  I  ->  ( N `  x )  =  ( ( inv g `  ( R `
 x ) ) `
 ( F `  x ) ) )
4645adantl 452 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( N `  x )  =  ( ( inv g `  ( R `
 x ) ) `
 ( F `  x ) ) )
4746oveq1d 5915 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( N `  x
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  ( ( ( inv g `  ( R `
 x ) ) `
 ( F `  x ) ) ( +g  `  ( R `
 x ) ) ( F `  x
) ) )
48 prdsinvlem.z . . . . . . 7  |-  .0.  =  ( 0g  o.  R
)
4948fveq1i 5564 . . . . . 6  |-  (  .0.  `  x )  =  ( ( 0g  o.  R
) `  x )
50 fvco2 5632 . . . . . . 7  |-  ( ( R  Fn  I  /\  x  e.  I )  ->  ( ( 0g  o.  R ) `  x
)  =  ( 0g
`  ( R `  x ) ) )
5112, 50sylan 457 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( 0g  o.  R
) `  x )  =  ( 0g `  ( R `  x ) ) )
5249, 51syl5eq 2360 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (  .0.  `  x )  =  ( 0g `  ( R `  x )
) )
5339, 47, 523eqtr4d 2358 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( N `  x
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  (  .0.  `  x
) )
5453mpteq2dva 4143 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( ( N `  x ) ( +g  `  ( R `  x
) ) ( F `
 x ) ) )  =  ( x  e.  I  |->  (  .0.  `  x ) ) )
55 prdsinvlem.p . . . 4  |-  .+  =  ( +g  `  Y )
565, 6, 7, 9, 12, 25, 14, 55prdsplusgval 13421 . . 3  |-  ( ph  ->  ( N  .+  F
)  =  ( x  e.  I  |->  ( ( N `  x ) ( +g  `  ( R `  x )
) ( F `  x ) ) ) )
57 fn0g 14434 . . . . . . 7  |-  0g  Fn  _V
5857a1i 10 . . . . . 6  |-  ( ph  ->  0g  Fn  _V )
59 ssv 3232 . . . . . . 7  |-  ran  R  C_ 
_V
6059a1i 10 . . . . . 6  |-  ( ph  ->  ran  R  C_  _V )
61 fnco 5389 . . . . . 6  |-  ( ( 0g  Fn  _V  /\  R  Fn  I  /\  ran  R  C_  _V )  ->  ( 0g  o.  R
)  Fn  I )
6258, 12, 60, 61syl3anc 1182 . . . . 5  |-  ( ph  ->  ( 0g  o.  R
)  Fn  I )
6348fneq1i 5375 . . . . 5  |-  (  .0. 
Fn  I  <->  ( 0g  o.  R )  Fn  I
)
6462, 63sylibr 203 . . . 4  |-  ( ph  ->  .0.  Fn  I )
65 dffn5 5606 . . . 4  |-  (  .0. 
Fn  I  <->  .0.  =  ( x  e.  I  |->  (  .0.  `  x
) ) )
6664, 65sylib 188 . . 3  |-  ( ph  ->  .0.  =  ( x  e.  I  |->  (  .0.  `  x ) ) )
6754, 56, 663eqtr4d 2358 . 2  |-  ( ph  ->  ( N  .+  F
)  =  .0.  )
6825, 67jca 518 1  |-  ( ph  ->  ( N  e.  B  /\  ( N  .+  F
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577   _Vcvv 2822    C_ wss 3186    e. cmpt 4114   ran crn 4727    o. ccom 4730    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900   Basecbs 13195   +g cplusg 13255   X_scprds 13395   0gc0g 13449   Grpcgrp 14411   inv gcminusg 14412
This theorem is referenced by:  prdsgrpd  14653  prdsinvgd  14654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-plusg 13268  df-mulr 13269  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-hom 13279  df-cco 13280  df-prds 13397  df-0g 13453  df-mnd 14416  df-grp 14538  df-minusg 14539
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