Theorem ajval 9863

Description: Value of the adjoint function.

Hypotheses

Ref	Expression
ajval.1	|- X = (BaseSet` U)
ajval.2	|- Y = (BaseSet` W)
ajval.3	|- P = (.i` U)
ajval.4	|- Q = (.i` W)
ajval.5	|- A = (UadjW)

Assertion

Ref	Expression
ajval	|- ((U e. CPreHil /\ W e. NrmCVec /\ T:X-->Y) -> (A` T) = U.{s | (s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))})

Distinct variable groups:   x,s,y,T   U,s,x,y   W,s,x,y   X,s,x,y   Y,s,y

Proof of Theorem ajval

Step	Hyp	Ref	Expression
1		ajval.1	. . . . . 6 |- X = (BaseSet` U)
2		ajval.2	. . . . . 6 |- Y = (BaseSet` W)
3		ajval.3	. . . . . 6 |- P = (.i` U)
4		ajval.4	. . . . . 6 |- Q = (.i` W)
5		ajval.5	. . . . . 6 |- A = (UadjW)
6	1, 2, 3, 4, 5	ajfval 9809	. . . . 5 |- ((U e. NrmCVec /\ W e. NrmCVec) -> A = {<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))})
7		phnv 9814	. . . . 5 |- (U e. CPreHil -> U e. NrmCVec)
8	6, 7	sylan 497	. . . 4 |- ((U e. CPreHil /\ W e. NrmCVec) -> A = {<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))})
9	8	fveq1d 4683	. . 3 |- ((U e. CPreHil /\ W e. NrmCVec) -> (A` T) = ({<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))}` T))
10	9	3adant3 896	. 2 |- ((U e. CPreHil /\ W e. NrmCVec /\ T:X-->Y) -> (A` T) = ({<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))}` T))
11		eqid 1884	. . . . 5 |- {<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))} = {<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))}
12		feq1 4551	. . . . . 6 |- (t = T -> (t:X-->Y <-> T:X-->Y))
13		fveq1 4680	. . . . . . . . 9 |- (t = T -> (t` x) = (T` x))
14	13	opreq1d 4897	. . . . . . . 8 |- (t = T -> ((t` x)Qy) = ((T` x)Qy))
15	14	eqeq1d 1892	. . . . . . 7 |- (t = T -> (((t` x)Qy) = (xP(s` y)) <-> ((T` x)Qy) = (xP(s` y))))
16	15	2ralbidv 2140	. . . . . 6 |- (t = T -> (A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)) <-> A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y))))
17	12, 16	3anbi13d 1170	. . . . 5 |- (t = T -> ((t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y))) <-> (T:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))))
18	11, 17	fvopab5 4756	. . . 4 |- ((Fun {<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))} /\ T e. _V) -> ({<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))}` T) = U.{s | (T:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))})
19	5	ajfun 9862	. . . . 5 |- ((U e. CPreHil /\ W e. NrmCVec) -> Fun A)
20		funeq 4441	. . . . . 6 |- (A = {<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))} -> (Fun A <-> Fun {<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))}))
21	8, 20	syl 12	. . . . 5 |- ((U e. CPreHil /\ W e. NrmCVec) -> (Fun A <-> Fun {<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))}))
22	19, 21	mpbid 212	. . . 4 |- ((U e. CPreHil /\ W e. NrmCVec) -> Fun {<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))})
23		fvex 4689	. . . . . 6 |- (BaseSet` U) e. _V
24	1, 23	eqeltri 1967	. . . . 5 |- X e. _V
25		fex 4595	. . . . 5 |- ((T:X-->Y /\ X e. _V) -> T e. _V)
26	24, 25	mpan2 760	. . . 4 |- (T:X-->Y -> T e. _V)
27	18, 22, 26	syl2an 503	. . 3 |- (((U e. CPreHil /\ W e. NrmCVec) /\ T:X-->Y) -> ({<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))}` T) = U.{s | (T:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))})
28	27	3impa 1062	. 2 |- ((U e. CPreHil /\ W e. NrmCVec /\ T:X-->Y) -> ({<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))}` T) = U.{s | (T:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))})
29		3anass 862	. . . . . 6 |- ((T:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y))) <-> (T:X-->Y /\ (s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))))
30	29	baib 749	. . . . 5 |- (T:X-->Y -> ((T:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y))) <-> (s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))))
31	30	abbidv 2008	. . . 4 |- (T:X-->Y -> {s | (T:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))} = {s | (s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))})
32	31	unieqd 3188	. . 3 |- (T:X-->Y -> U.{s | (T:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))} = U.{s | (s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))})
33	32	3ad2ant3 899	. 2 |- ((U e. CPreHil /\ W e. NrmCVec /\ T:X-->Y) -> U.{s | (T:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))} = U.{s | (s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))})
34	10, 28, 33	3eqtrd 1929	1 |- ((U e. CPreHil /\ W e. NrmCVec /\ T:X-->Y) -> (A` T) = U.{s | (s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  _Vcvv 2292  U.cuni 3177  {copab 3395  Fun wfun 3992  -->wf 3994  ` cfv 3998  (class class class)co 4884  NrmCVeccnv 9535  BaseSetcba 9537  .icip 9688  adjcaj 9748  CPreHilcphl 9812

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240  df-top 8861  df-bases 8863  df-topgen 8864  df-cld 8939  df-ntr 8940  df-cls 8941  df-cn 9030  df-cnp 9031  df-haus 9059  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-vs 9550  df-nm 9551  df-ims 9552  df-ip 9689  df-aj 9750  df-ph 9813

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