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Theorem prdsinvlem 16872
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 472 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  S  e.  V )
8 prdsinvlem.i . . . . . . . 8  |-  ( ph  ->  I  e.  W )
98adantr 472 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  I  e.  W )
10 ffn 5739 . . . . . . . . 9  |-  ( R : I --> Grp  ->  R  Fn  I )
112, 10syl 17 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
1211adantr 472 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  R  Fn  I )
13 prdsinvlem.f . . . . . . . 8  |-  ( ph  ->  F  e.  B )
1413adantr 472 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  F  e.  B )
15 simpr 468 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  y  e.  I )
164, 5, 7, 9, 12, 14, 15prdsbasprj 15448 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( Base `  ( R `  y )
) )
17 eqid 2471 . . . . . . 7  |-  ( Base `  ( R `  y
) )  =  (
Base `  ( R `  y ) )
18 eqid 2471 . . . . . . 7  |-  ( invg `  ( R `
 y ) )  =  ( invg `  ( R `  y
) )
1917, 18grpinvcl 16789 . . . . . 6  |-  ( ( ( R `  y
)  e.  Grp  /\  ( F `  y )  e.  ( Base `  ( R `  y )
) )  ->  (
( invg `  ( R `  y ) ) `  ( F `
 y ) )  e.  ( Base `  ( R `  y )
) )
203, 16, 19syl2anc 673 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  (
( invg `  ( R `  y ) ) `  ( F `
 y ) )  e.  ( Base `  ( R `  y )
) )
2120ralrimiva 2809 . . . 4  |-  ( ph  ->  A. y  e.  I 
( ( invg `  ( R `  y
) ) `  ( F `  y )
)  e.  ( Base `  ( R `  y
) ) )
224, 5, 6, 8, 11prdsbasmpt 15446 . . . 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 240 . . 3  |-  ( ph  ->  ( y  e.  I  |->  ( ( invg `  ( R `  y
) ) `  ( F `  y )
) )  e.  B
)
241, 23syl5eqel 2553 . 2  |-  ( ph  ->  N  e.  B )
252ffvelrnda 6037 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( R `  x )  e.  Grp )
266adantr 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  S  e.  V )
278adantr 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  W )
2811adantr 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  R  Fn  I )
2913adantr 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  F  e.  B )
30 simpr 468 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
314, 5, 26, 27, 28, 29, 30prdsbasprj 15448 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  ( R `  x )
) )
32 eqid 2471 . . . . . . 7  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
33 eqid 2471 . . . . . . 7  |-  ( +g  `  ( R `  x
) )  =  ( +g  `  ( R `
 x ) )
34 eqid 2471 . . . . . . 7  |-  ( 0g
`  ( R `  x ) )  =  ( 0g `  ( R `  x )
)
35 eqid 2471 . . . . . . 7  |-  ( invg `  ( R `
 x ) )  =  ( invg `  ( R `  x
) )
3632, 33, 34, 35grplinv 16790 . . . . . 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 673 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( ( invg `  ( R `  x
) ) `  ( F `  x )
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  ( 0g `  ( R `  x )
) )
38 fveq2 5879 . . . . . . . . . 10  |-  ( y  =  x  ->  ( R `  y )  =  ( R `  x ) )
3938fveq2d 5883 . . . . . . . . 9  |-  ( y  =  x  ->  ( invg `  ( R `
 y ) )  =  ( invg `  ( R `  x
) ) )
40 fveq2 5879 . . . . . . . . 9  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
4139, 40fveq12d 5885 . . . . . . . 8  |-  ( y  =  x  ->  (
( invg `  ( R `  y ) ) `  ( F `
 y ) )  =  ( ( invg `  ( R `
 x ) ) `
 ( F `  x ) ) )
42 fvex 5889 . . . . . . . 8  |-  ( ( invg `  ( R `  x )
) `  ( F `  x ) )  e. 
_V
4341, 1, 42fvmpt 5963 . . . . . . 7  |-  ( x  e.  I  ->  ( N `  x )  =  ( ( invg `  ( R `
 x ) ) `
 ( F `  x ) ) )
4443adantl 473 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( N `  x )  =  ( ( invg `  ( R `
 x ) ) `
 ( F `  x ) ) )
4544oveq1d 6323 . . . . 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 5880 . . . . . 6  |-  (  .0.  `  x )  =  ( ( 0g  o.  R
) `  x )
48 fvco2 5955 . . . . . . 7  |-  ( ( R  Fn  I  /\  x  e.  I )  ->  ( ( 0g  o.  R ) `  x
)  =  ( 0g
`  ( R `  x ) ) )
4911, 48sylan 479 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( 0g  o.  R
) `  x )  =  ( 0g `  ( R `  x ) ) )
5047, 49syl5eq 2517 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (  .0.  `  x )  =  ( 0g `  ( R `  x )
) )
5137, 45, 503eqtr4d 2515 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( N `  x
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  (  .0.  `  x
) )
5251mpteq2dva 4482 . . 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 15449 . . 3  |-  ( ph  ->  ( N  .+  F
)  =  ( x  e.  I  |->  ( ( N `  x ) ( +g  `  ( R `  x )
) ( F `  x ) ) ) )
55 fn0g 16583 . . . . . . 7  |-  0g  Fn  _V
5655a1i 11 . . . . . 6  |-  ( ph  ->  0g  Fn  _V )
57 ssv 3438 . . . . . . 7  |-  ran  R  C_ 
_V
5857a1i 11 . . . . . 6  |-  ( ph  ->  ran  R  C_  _V )
59 fnco 5694 . . . . . 6  |-  ( ( 0g  Fn  _V  /\  R  Fn  I  /\  ran  R  C_  _V )  ->  ( 0g  o.  R
)  Fn  I )
6056, 11, 58, 59syl3anc 1292 . . . . 5  |-  ( ph  ->  ( 0g  o.  R
)  Fn  I )
6146fneq1i 5680 . . . . 5  |-  (  .0. 
Fn  I  <->  ( 0g  o.  R )  Fn  I
)
6260, 61sylibr 217 . . . 4  |-  ( ph  ->  .0.  Fn  I )
63 dffn5 5924 . . . 4  |-  (  .0. 
Fn  I  <->  .0.  =  ( x  e.  I  |->  (  .0.  `  x
) ) )
6462, 63sylib 201 . . 3  |-  ( ph  ->  .0.  =  ( x  e.  I  |->  (  .0.  `  x ) ) )
6552, 54, 643eqtr4d 2515 . 2  |-  ( ph  ->  ( N  .+  F
)  =  .0.  )
6624, 65jca 541 1  |-  ( ph  ->  ( N  e.  B  /\  ( N  .+  F
)  =  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    C_ wss 3390    |-> cmpt 4454   ran crn 4840    o. ccom 4843    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Basecbs 15199   +g cplusg 15268   0gc0g 15416   X_scprds 15422   Grpcgrp 16747   invgcminusg 16748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-hom 15292  df-cco 15293  df-0g 15418  df-prds 15424  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752
This theorem is referenced by:  prdsgrpd  16873  prdsinvgd  16874
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