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Theorem prdsinvlem 14926
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 )
32ffvelrnda 5870 . . . . . 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 452 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  S  e.  V )
8 prdsinvlem.i . . . . . . . 8  |-  ( ph  ->  I  e.  W )
98adantr 452 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  I  e.  W )
10 ffn 5591 . . . . . . . . 9  |-  ( R : I --> Grp  ->  R  Fn  I )
112, 10syl 16 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
1211adantr 452 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  R  Fn  I )
13 prdsinvlem.f . . . . . . . 8  |-  ( ph  ->  F  e.  B )
1413adantr 452 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  F  e.  B )
15 simpr 448 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  y  e.  I )
164, 5, 7, 9, 12, 14, 15prdsbasprj 13694 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( Base `  ( R `  y )
) )
17 eqid 2436 . . . . . . 7  |-  ( Base `  ( R `  y
) )  =  (
Base `  ( R `  y ) )
18 eqid 2436 . . . . . . 7  |-  ( inv g `  ( R `
 y ) )  =  ( inv g `  ( R `  y
) )
1917, 18grpinvcl 14850 . . . . . 6  |-  ( ( ( R `  y
)  e.  Grp  /\  ( F `  y )  e.  ( Base `  ( R `  y )
) )  ->  (
( inv g `  ( R `  y ) ) `  ( F `
 y ) )  e.  ( Base `  ( R `  y )
) )
203, 16, 19syl2anc 643 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  (
( inv g `  ( R `  y ) ) `  ( F `
 y ) )  e.  ( Base `  ( R `  y )
) )
2120ralrimiva 2789 . . . 4  |-  ( ph  ->  A. y  e.  I 
( ( inv g `  ( R `  y
) ) `  ( F `  y )
)  e.  ( Base `  ( R `  y
) ) )
224, 5, 6, 8, 11prdsbasmpt 13692 . . . 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 )
) ) )
2321, 22mpbird 224 . . 3  |-  ( ph  ->  ( y  e.  I  |->  ( ( inv g `  ( R `  y
) ) `  ( F `  y )
) )  e.  B
)
241, 23syl5eqel 2520 . 2  |-  ( ph  ->  N  e.  B )
252ffvelrnda 5870 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( R `  x )  e.  Grp )
266adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  S  e.  V )
278adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  W )
2811adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  R  Fn  I )
2913adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  F  e.  B )
30 simpr 448 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
314, 5, 26, 27, 28, 29, 30prdsbasprj 13694 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  ( R `  x )
) )
32 eqid 2436 . . . . . . 7  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
33 eqid 2436 . . . . . . 7  |-  ( +g  `  ( R `  x
) )  =  ( +g  `  ( R `
 x ) )
34 eqid 2436 . . . . . . 7  |-  ( 0g
`  ( R `  x ) )  =  ( 0g `  ( R `  x )
)
35 eqid 2436 . . . . . . 7  |-  ( inv g `  ( R `
 x ) )  =  ( inv g `  ( R `  x
) )
3632, 33, 34, 35grplinv 14851 . . . . . 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 )
) )
3725, 31, 36syl2anc 643 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( ( inv g `  ( R `  x
) ) `  ( F `  x )
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  ( 0g `  ( R `  x )
) )
38 fveq2 5728 . . . . . . . . . 10  |-  ( y  =  x  ->  ( R `  y )  =  ( R `  x ) )
3938fveq2d 5732 . . . . . . . . 9  |-  ( y  =  x  ->  ( inv g `  ( R `
 y ) )  =  ( inv g `  ( R `  x
) ) )
40 fveq2 5728 . . . . . . . . 9  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
4139, 40fveq12d 5734 . . . . . . . 8  |-  ( y  =  x  ->  (
( inv g `  ( R `  y ) ) `  ( F `
 y ) )  =  ( ( inv g `  ( R `
 x ) ) `
 ( F `  x ) ) )
42 fvex 5742 . . . . . . . 8  |-  ( ( inv g `  ( R `  x )
) `  ( F `  x ) )  e. 
_V
4341, 1, 42fvmpt 5806 . . . . . . 7  |-  ( x  e.  I  ->  ( N `  x )  =  ( ( inv g `  ( R `
 x ) ) `
 ( F `  x ) ) )
4443adantl 453 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( N `  x )  =  ( ( inv g `  ( R `
 x ) ) `
 ( F `  x ) ) )
4544oveq1d 6096 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( N `  x
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  ( ( ( inv g `  ( R `
 x ) ) `
 ( F `  x ) ) ( +g  `  ( R `
 x ) ) ( F `  x
) ) )
46 prdsinvlem.z . . . . . . 7  |-  .0.  =  ( 0g  o.  R
)
4746fveq1i 5729 . . . . . 6  |-  (  .0.  `  x )  =  ( ( 0g  o.  R
) `  x )
48 fvco2 5798 . . . . . . 7  |-  ( ( R  Fn  I  /\  x  e.  I )  ->  ( ( 0g  o.  R ) `  x
)  =  ( 0g
`  ( R `  x ) ) )
4911, 48sylan 458 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( 0g  o.  R
) `  x )  =  ( 0g `  ( R `  x ) ) )
5047, 49syl5eq 2480 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (  .0.  `  x )  =  ( 0g `  ( R `  x )
) )
5137, 45, 503eqtr4d 2478 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( N `  x
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  (  .0.  `  x
) )
5251mpteq2dva 4295 . . 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 13695 . . 3  |-  ( ph  ->  ( N  .+  F
)  =  ( x  e.  I  |->  ( ( N `  x ) ( +g  `  ( R `  x )
) ( F `  x ) ) ) )
55 fn0g 14708 . . . . . . 7  |-  0g  Fn  _V
5655a1i 11 . . . . . 6  |-  ( ph  ->  0g  Fn  _V )
57 ssv 3368 . . . . . . 7  |-  ran  R  C_ 
_V
5857a1i 11 . . . . . 6  |-  ( ph  ->  ran  R  C_  _V )
59 fnco 5553 . . . . . 6  |-  ( ( 0g  Fn  _V  /\  R  Fn  I  /\  ran  R  C_  _V )  ->  ( 0g  o.  R
)  Fn  I )
6056, 11, 58, 59syl3anc 1184 . . . . 5  |-  ( ph  ->  ( 0g  o.  R
)  Fn  I )
6146fneq1i 5539 . . . . 5  |-  (  .0. 
Fn  I  <->  ( 0g  o.  R )  Fn  I
)
6260, 61sylibr 204 . . . 4  |-  ( ph  ->  .0.  Fn  I )
63 dffn5 5772 . . . 4  |-  (  .0. 
Fn  I  <->  .0.  =  ( x  e.  I  |->  (  .0.  `  x
) ) )
6462, 63sylib 189 . . 3  |-  ( ph  ->  .0.  =  ( x  e.  I  |->  (  .0.  `  x ) ) )
6552, 54, 643eqtr4d 2478 . 2  |-  ( ph  ->  ( N  .+  F
)  =  .0.  )
6624, 65jca 519 1  |-  ( ph  ->  ( N  e.  B  /\  ( N  .+  F
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    C_ wss 3320    e. cmpt 4266   ran crn 4879    o. ccom 4882    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   X_scprds 13669   0gc0g 13723   Grpcgrp 14685   inv gcminusg 14686
This theorem is referenced by:  prdsgrpd  14927  prdsinvgd  14928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-hom 13553  df-cco 13554  df-prds 13671  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813
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