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Theorem prdsinvlem 15978
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 6019 . . . . . 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 5729 . . . . . . . . 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 14723 . . . . . 6  |-  ( (
ph  /\  y  e.  I )  ->  ( F `  y )  e.  ( Base `  ( R `  y )
) )
17 eqid 2467 . . . . . . 7  |-  ( Base `  ( R `  y
) )  =  (
Base `  ( R `  y ) )
18 eqid 2467 . . . . . . 7  |-  ( invg `  ( R `
 y ) )  =  ( invg `  ( R `  y
) )
1917, 18grpinvcl 15896 . . . . . 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 2878 . . . 4  |-  ( ph  ->  A. y  e.  I 
( ( invg `  ( R `  y
) ) `  ( F `  y )
)  e.  ( Base `  ( R `  y
) ) )
224, 5, 6, 8, 11prdsbasmpt 14721 . . . 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 2559 . 2  |-  ( ph  ->  N  e.  B )
252ffvelrnda 6019 . . . . . 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 14723 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  ( R `  x )
) )
32 eqid 2467 . . . . . . 7  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
33 eqid 2467 . . . . . . 7  |-  ( +g  `  ( R `  x
) )  =  ( +g  `  ( R `
 x ) )
34 eqid 2467 . . . . . . 7  |-  ( 0g
`  ( R `  x ) )  =  ( 0g `  ( R `  x )
)
35 eqid 2467 . . . . . . 7  |-  ( invg `  ( R `
 x ) )  =  ( invg `  ( R `  x
) )
3632, 33, 34, 35grplinv 15897 . . . . . 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 5864 . . . . . . . . . 10  |-  ( y  =  x  ->  ( R `  y )  =  ( R `  x ) )
3938fveq2d 5868 . . . . . . . . 9  |-  ( y  =  x  ->  ( invg `  ( R `
 y ) )  =  ( invg `  ( R `  x
) ) )
40 fveq2 5864 . . . . . . . . 9  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
4139, 40fveq12d 5870 . . . . . . . 8  |-  ( y  =  x  ->  (
( invg `  ( R `  y ) ) `  ( F `
 y ) )  =  ( ( invg `  ( R `
 x ) ) `
 ( F `  x ) ) )
42 fvex 5874 . . . . . . . 8  |-  ( ( invg `  ( R `  x )
) `  ( F `  x ) )  e. 
_V
4341, 1, 42fvmpt 5948 . . . . . . 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 6297 . . . . 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 5865 . . . . . 6  |-  (  .0.  `  x )  =  ( ( 0g  o.  R
) `  x )
48 fvco2 5940 . . . . . . 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 2520 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (  .0.  `  x )  =  ( 0g `  ( R `  x )
) )
5137, 45, 503eqtr4d 2518 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( N `  x
) ( +g  `  ( R `  x )
) ( F `  x ) )  =  (  .0.  `  x
) )
5251mpteq2dva 4533 . . 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 14724 . . 3  |-  ( ph  ->  ( N  .+  F
)  =  ( x  e.  I  |->  ( ( N `  x ) ( +g  `  ( R `  x )
) ( F `  x ) ) ) )
55 fn0g 15746 . . . . . . 7  |-  0g  Fn  _V
5655a1i 11 . . . . . 6  |-  ( ph  ->  0g  Fn  _V )
57 ssv 3524 . . . . . . 7  |-  ran  R  C_ 
_V
5857a1i 11 . . . . . 6  |-  ( ph  ->  ran  R  C_  _V )
59 fnco 5687 . . . . . 6  |-  ( ( 0g  Fn  _V  /\  R  Fn  I  /\  ran  R  C_  _V )  ->  ( 0g  o.  R
)  Fn  I )
6056, 11, 58, 59syl3anc 1228 . . . . 5  |-  ( ph  ->  ( 0g  o.  R
)  Fn  I )
6146fneq1i 5673 . . . . 5  |-  (  .0. 
Fn  I  <->  ( 0g  o.  R )  Fn  I
)
6260, 61sylibr 212 . . . 4  |-  ( ph  ->  .0.  Fn  I )
63 dffn5 5911 . . . 4  |-  (  .0. 
Fn  I  <->  .0.  =  ( x  e.  I  |->  (  .0.  `  x
) ) )
6462, 63sylib 196 . . 3  |-  ( ph  ->  .0.  =  ( x  e.  I  |->  (  .0.  `  x ) ) )
6552, 54, 643eqtr4d 2518 . 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 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476    |-> cmpt 4505   ran crn 5000    o. ccom 5003    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   0gc0g 14691   X_scprds 14697   Grpcgrp 15723   invgcminusg 15724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-hom 14575  df-cco 14576  df-0g 14693  df-prds 14699  df-mnd 15728  df-grp 15858  df-minusg 15859
This theorem is referenced by:  prdsgrpd  15979  prdsinvgd  15980
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