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Theorem prdsinvlem 15668
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 5848 . . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  S  e.  V )
8 prdsinvlem.i . . . . . . . 8  |-  ( ph  ->  I  e.  W )
98adantr 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  I  e.  W )
10 ffn 5564 . . . . . . . . 9  |-  ( R : I --> Grp  ->  R  Fn  I )
112, 10syl 16 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
1211adantr 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  R  Fn  I )
13 prdsinvlem.f . . . . . . . 8  |-  ( ph  ->  F  e.  B )
1413adantr 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  F  e.  B )
15 simpr 461 . . . . . . 7  |-  ( (
ph  /\  y  e.  I )  ->  y  e.  I )
164, 5, 7, 9, 12, 14, 15prdsbasprj 14415 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( Base `  ( R `  y )
) )
17 eqid 2443 . . . . . . 7  |-  ( Base `  ( R `  y
) )  =  (
Base `  ( R `  y ) )
18 eqid 2443 . . . . . . 7  |-  ( invg `  ( R `
 y ) )  =  ( invg `  ( R `  y
) )
1917, 18grpinvcl 15588 . . . . . 6  |-  ( ( ( R `  y
)  e.  Grp  /\  ( F `  y )  e.  ( Base `  ( R `  y )
) )  ->  (
( invg `  ( R `  y ) ) `  ( F `
 y ) )  e.  ( Base `  ( R `  y )
) )
203, 16, 19syl2anc 661 . . . . 5  |-  ( (
ph  /\  y  e.  I )  ->  (
( invg `  ( R `  y ) ) `  ( F `
 y ) )  e.  ( Base `  ( R `  y )
) )
2120ralrimiva 2804 . . . 4  |-  ( ph  ->  A. y  e.  I 
( ( invg `  ( R `  y
) ) `  ( F `  y )
)  e.  ( Base `  ( R `  y
) ) )
224, 5, 6, 8, 11prdsbasmpt 14413 . . . 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 232 . . 3  |-  ( ph  ->  ( y  e.  I  |->  ( ( invg `  ( R `  y
) ) `  ( F `  y )
) )  e.  B
)
241, 23syl5eqel 2527 . 2  |-  ( ph  ->  N  e.  B )
252ffvelrnda 5848 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( R `  x )  e.  Grp )
266adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  S  e.  V )
278adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  W )
2811adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  R  Fn  I )
2913adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  F  e.  B )
30 simpr 461 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
314, 5, 26, 27, 28, 29, 30prdsbasprj 14415 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  ( R `  x )
) )
32 eqid 2443 . . . . . . 7  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
33 eqid 2443 . . . . . . 7  |-  ( +g  `  ( R `  x
) )  =  ( +g  `  ( R `
 x ) )
34 eqid 2443 . . . . . . 7  |-  ( 0g
`  ( R `  x ) )  =  ( 0g `  ( R `  x )
)
35 eqid 2443 . . . . . . 7  |-  ( invg `  ( R `
 x ) )  =  ( invg `  ( R `  x
) )
3632, 33, 34, 35grplinv 15589 . . . . . 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 661 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( ( invg `  ( R `  x
) ) `  ( F `  x )
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  ( 0g `  ( R `  x )
) )
38 fveq2 5696 . . . . . . . . . 10  |-  ( y  =  x  ->  ( R `  y )  =  ( R `  x ) )
3938fveq2d 5700 . . . . . . . . 9  |-  ( y  =  x  ->  ( invg `  ( R `
 y ) )  =  ( invg `  ( R `  x
) ) )
40 fveq2 5696 . . . . . . . . 9  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
4139, 40fveq12d 5702 . . . . . . . 8  |-  ( y  =  x  ->  (
( invg `  ( R `  y ) ) `  ( F `
 y ) )  =  ( ( invg `  ( R `
 x ) ) `
 ( F `  x ) ) )
42 fvex 5706 . . . . . . . 8  |-  ( ( invg `  ( R `  x )
) `  ( F `  x ) )  e. 
_V
4341, 1, 42fvmpt 5779 . . . . . . 7  |-  ( x  e.  I  ->  ( N `  x )  =  ( ( invg `  ( R `
 x ) ) `
 ( F `  x ) ) )
4443adantl 466 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( N `  x )  =  ( ( invg `  ( R `
 x ) ) `
 ( F `  x ) ) )
4544oveq1d 6111 . . . . 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 5697 . . . . . 6  |-  (  .0.  `  x )  =  ( ( 0g  o.  R
) `  x )
48 fvco2 5771 . . . . . . 7  |-  ( ( R  Fn  I  /\  x  e.  I )  ->  ( ( 0g  o.  R ) `  x
)  =  ( 0g
`  ( R `  x ) ) )
4911, 48sylan 471 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( 0g  o.  R
) `  x )  =  ( 0g `  ( R `  x ) ) )
5047, 49syl5eq 2487 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (  .0.  `  x )  =  ( 0g `  ( R `  x )
) )
5137, 45, 503eqtr4d 2485 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( N `  x
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  (  .0.  `  x
) )
5251mpteq2dva 4383 . . 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 14416 . . 3  |-  ( ph  ->  ( N  .+  F
)  =  ( x  e.  I  |->  ( ( N `  x ) ( +g  `  ( R `  x )
) ( F `  x ) ) ) )
55 fn0g 15438 . . . . . . 7  |-  0g  Fn  _V
5655a1i 11 . . . . . 6  |-  ( ph  ->  0g  Fn  _V )
57 ssv 3381 . . . . . . 7  |-  ran  R  C_ 
_V
5857a1i 11 . . . . . 6  |-  ( ph  ->  ran  R  C_  _V )
59 fnco 5524 . . . . . 6  |-  ( ( 0g  Fn  _V  /\  R  Fn  I  /\  ran  R  C_  _V )  ->  ( 0g  o.  R
)  Fn  I )
6056, 11, 58, 59syl3anc 1218 . . . . 5  |-  ( ph  ->  ( 0g  o.  R
)  Fn  I )
6146fneq1i 5510 . . . . 5  |-  (  .0. 
Fn  I  <->  ( 0g  o.  R )  Fn  I
)
6260, 61sylibr 212 . . . 4  |-  ( ph  ->  .0.  Fn  I )
63 dffn5 5742 . . . 4  |-  (  .0. 
Fn  I  <->  .0.  =  ( x  e.  I  |->  (  .0.  `  x
) ) )
6462, 63sylib 196 . . 3  |-  ( ph  ->  .0.  =  ( x  e.  I  |->  (  .0.  `  x ) ) )
6552, 54, 643eqtr4d 2485 . 2  |-  ( ph  ->  ( N  .+  F
)  =  .0.  )
6624, 65jca 532 1  |-  ( ph  ->  ( N  e.  B  /\  ( N  .+  F
)  =  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   _Vcvv 2977    C_ wss 3333    e. cmpt 4355   ran crn 4846    o. ccom 4849    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   Basecbs 14179   +g cplusg 14243   0gc0g 14383   X_scprds 14389   Grpcgrp 15415   invgcminusg 15416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-hom 14267  df-cco 14268  df-0g 14385  df-prds 14391  df-mnd 15420  df-grp 15550  df-minusg 15551
This theorem is referenced by:  prdsgrpd  15669  prdsinvgd  15670
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