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Theorem prdsinvlem 14603
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 5663 . . . . . . 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 5389 . . . . . . . . 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 13371 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( Base `  ( R `  y )
) )
18 eqid 2283 . . . . . . 7  |-  ( Base `  ( R `  y
) )  =  (
Base `  ( R `  y ) )
19 eqid 2283 . . . . . . 7  |-  ( inv g `  ( R `
 y ) )  =  ( inv g `  ( R `  y
) )
2018, 19grpinvcl 14527 . . . . . 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 2626 . . . 4  |-  ( ph  ->  A. y  e.  I 
( ( inv g `  ( R `  y
) ) `  ( F `  y )
)  e.  ( Base `  ( R `  y
) ) )
235, 6, 7, 9, 12prdsbasmpt 13369 . . . 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 2367 . 2  |-  ( ph  ->  N  e.  B )
26 ffvelrn 5663 . . . . . . 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 13371 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  ( R `  x )
) )
34 eqid 2283 . . . . . . 7  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
35 eqid 2283 . . . . . . 7  |-  ( +g  `  ( R `  x
) )  =  ( +g  `  ( R `
 x ) )
36 eqid 2283 . . . . . . 7  |-  ( 0g
`  ( R `  x ) )  =  ( 0g `  ( R `  x )
)
37 eqid 2283 . . . . . . 7  |-  ( inv g `  ( R `
 x ) )  =  ( inv g `  ( R `  x
) )
3834, 35, 36, 37grplinv 14528 . . . . . 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 5525 . . . . . . . . . 10  |-  ( y  =  x  ->  ( R `  y )  =  ( R `  x ) )
4140fveq2d 5529 . . . . . . . . 9  |-  ( y  =  x  ->  ( inv g `  ( R `
 y ) )  =  ( inv g `  ( R `  x
) ) )
42 fveq2 5525 . . . . . . . . 9  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
4341, 42fveq12d 5531 . . . . . . . 8  |-  ( y  =  x  ->  (
( inv g `  ( R `  y ) ) `  ( F `
 y ) )  =  ( ( inv g `  ( R `
 x ) ) `
 ( F `  x ) ) )
44 fvex 5539 . . . . . . . 8  |-  ( ( inv g `  ( R `  x )
) `  ( F `  x ) )  e. 
_V
4543, 1, 44fvmpt 5602 . . . . . . 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 5873 . . . . 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 5526 . . . . . 6  |-  (  .0.  `  x )  =  ( ( 0g  o.  R
) `  x )
50 fvco2 5594 . . . . . . 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 2327 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (  .0.  `  x )  =  ( 0g `  ( R `  x )
) )
5339, 47, 523eqtr4d 2325 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( N `  x
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  (  .0.  `  x
) )
5453mpteq2dva 4106 . . 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 13372 . . 3  |-  ( ph  ->  ( N  .+  F
)  =  ( x  e.  I  |->  ( ( N `  x ) ( +g  `  ( R `  x )
) ( F `  x ) ) ) )
57 fn0g 14385 . . . . . . 7  |-  0g  Fn  _V
5857a1i 10 . . . . . 6  |-  ( ph  ->  0g  Fn  _V )
59 ssv 3198 . . . . . . 7  |-  ran  R  C_ 
_V
6059a1i 10 . . . . . 6  |-  ( ph  ->  ran  R  C_  _V )
61 fnco 5352 . . . . . 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 5338 . . . . 5  |-  (  .0. 
Fn  I  <->  ( 0g  o.  R )  Fn  I
)
6462, 63sylibr 203 . . . 4  |-  ( ph  ->  .0.  Fn  I )
65 dffn5 5568 . . . 4  |-  (  .0. 
Fn  I  <->  .0.  =  ( x  e.  I  |->  (  .0.  `  x
) ) )
6664, 65sylib 188 . . 3  |-  ( ph  ->  .0.  =  ( x  e.  I  |->  (  .0.  `  x ) ) )
6754, 56, 663eqtr4d 2325 . 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152    e. cmpt 4077   ran crn 4690    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   X_scprds 13346   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363
This theorem is referenced by:  prdsgrpd  14604  prdsinvgd  14605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490
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