Theorem ajfval 9809

Description: The adjoint function.

Hypotheses

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

Assertion

Ref	Expression
ajfval	|- ((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)))})

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

Proof of Theorem ajfval

Step	Hyp	Ref	Expression
1		df-xp 4000	. . . . . 6 |- ((Y ^m X) X. (X ^m Y)) = {<.t, s>. | (t e. (Y ^m X) /\ s e. (X ^m Y))}
2		ajfval.2	. . . . . . . . . 10 |- Y = (BaseSet` W)
3		fvex 4689	. . . . . . . . . 10 |- (BaseSet` W) e. _V
4	2, 3	eqeltri 1967	. . . . . . . . 9 |- Y e. _V
5		ajfval.1	. . . . . . . . . 10 |- X = (BaseSet` U)
6		fvex 4689	. . . . . . . . . 10 |- (BaseSet` U) e. _V
7	5, 6	eqeltri 1967	. . . . . . . . 9 |- X e. _V
8	4, 7	elmap 5393	. . . . . . . 8 |- (t e. (Y ^m X) <-> t:X-->Y)
9	7, 4	elmap 5393	. . . . . . . 8 |- (s e. (X ^m Y) <-> s:Y-->X)
10	8, 9	anbi12i 540	. . . . . . 7 |- ((t e. (Y ^m X) /\ s e. (X ^m Y)) <-> (t:X-->Y /\ s:Y-->X))
11	10	opabbii 3402	. . . . . 6 |- {<.t, s>. | (t e. (Y ^m X) /\ s e. (X ^m Y))} = {<.t, s>. | (t:X-->Y /\ s:Y-->X)}
12	1, 11	eqtr2i 1909	. . . . 5 |- {<.t, s>. | (t:X-->Y /\ s:Y-->X)} = ((Y ^m X) X. (X ^m Y))
13		oprex 4907	. . . . . 6 |- (Y ^m X) e. _V
14		oprex 4907	. . . . . 6 |- (X ^m Y) e. _V
15	13, 14	xpex 4096	. . . . 5 |- ((Y ^m X) X. (X ^m Y)) e. _V
16	12, 15	eqeltri 1967	. . . 4 |- {<.t, s>. | (t:X-->Y /\ s:Y-->X)} e. _V
17		3simpa 872	. . . . 5 |- ((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))
18	17	ssopab2i 3574	. . . 4 |- {<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))} C_ {<.t, s>. | (t:X-->Y /\ s:Y-->X)}
19	16, 18	ssexi 3456	. . 3 |- {<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))} e. _V
20		fveq2 4681	. . . . . . 7 |- (u = U -> (BaseSet` u) = (BaseSet` U))
21	20, 5	syl6eqr 1946	. . . . . 6 |- (u = U -> (BaseSet` u) = X)
22	21	feq2d 4557	. . . . 5 |- (u = U -> (t:(BaseSet` u)-->(BaseSet` w) <-> t:X-->(BaseSet` w)))
23		feq3 4553	. . . . . 6 |- ((BaseSet` u) = X -> (s:(BaseSet` w)-->(BaseSet` u) <-> s:(BaseSet` w)-->X))
24	21, 23	syl 12	. . . . 5 |- (u = U -> (s:(BaseSet` w)-->(BaseSet` u) <-> s:(BaseSet` w)-->X))
25		fveq2 4681	. . . . . . . . . 10 |- (u = U -> (.i` u) = (.i` U))
26		ajfval.3	. . . . . . . . . 10 |- P = (.i` U)
27	25, 26	syl6eqr 1946	. . . . . . . . 9 |- (u = U -> (.i` u) = P)
28	27	opreqd 4899	. . . . . . . 8 |- (u = U -> (x(.i` u)(s` y)) = (xP(s` y)))
29	28	eqeq2d 1895	. . . . . . 7 |- (u = U -> (((t` x)(.i` w)y) = (x(.i` u)(s` y)) <-> ((t` x)(.i` w)y) = (xP(s` y))))
30	29	ralbidv 2123	. . . . . 6 |- (u = U -> (A.y e. (BaseSet` w)((t` x)(.i` w)y) = (x(.i` u)(s` y)) <-> A.y e. (BaseSet` w)((t` x)(.i` w)y) = (xP(s` y))))
31	21, 30	raleqbidv 2274	. . . . 5 |- (u = U -> (A.x e. (BaseSet` u)A.y e. (BaseSet` w)((t` x)(.i` w)y) = (x(.i` u)(s` y)) <-> A.x e. X A.y e. (BaseSet` w)((t` x)(.i` w)y) = (xP(s` y))))
32	22, 24, 31	3anbi123d 1168	. . . 4 |- (u = U -> ((t:(BaseSet` u)-->(BaseSet` w) /\ s:(BaseSet` w)-->(BaseSet` u) /\ A.x e. (BaseSet` u)A.y e. (BaseSet` w)((t` x)(.i` w)y) = (x(.i` u)(s` y))) <-> (t:X-->(BaseSet` w) /\ s:(BaseSet` w)-->X /\ A.x e. X A.y e. (BaseSet` w)((t` x)(.i` w)y) = (xP(s` y)))))
33	32	opabbidv 3401	. . 3 |- (u = U -> {<.t, s>. | (t:(BaseSet` u)-->(BaseSet` w) /\ s:(BaseSet` w)-->(BaseSet` u) /\ A.x e. (BaseSet` u)A.y e. (BaseSet` w)((t` x)(.i` w)y) = (x(.i` u)(s` y)))} = {<.t, s>. | (t:X-->(BaseSet` w) /\ s:(BaseSet` w)-->X /\ A.x e. X A.y e. (BaseSet` w)((t` x)(.i` w)y) = (xP(s` y)))})
34		fveq2 4681	. . . . . . 7 |- (w = W -> (BaseSet` w) = (BaseSet` W))
35	34, 2	syl6eqr 1946	. . . . . 6 |- (w = W -> (BaseSet` w) = Y)
36		feq3 4553	. . . . . 6 |- ((BaseSet` w) = Y -> (t:X-->(BaseSet` w) <-> t:X-->Y))
37	35, 36	syl 12	. . . . 5 |- (w = W -> (t:X-->(BaseSet` w) <-> t:X-->Y))
38	35	feq2d 4557	. . . . 5 |- (w = W -> (s:(BaseSet` w)-->X <-> s:Y-->X))
39		fveq2 4681	. . . . . . . . . 10 |- (w = W -> (.i` w) = (.i` W))
40		ajfval.4	. . . . . . . . . 10 |- Q = (.i` W)
41	39, 40	syl6eqr 1946	. . . . . . . . 9 |- (w = W -> (.i` w) = Q)
42	41	opreqd 4899	. . . . . . . 8 |- (w = W -> ((t` x)(.i` w)y) = ((t` x)Qy))
43	42	eqeq1d 1892	. . . . . . 7 |- (w = W -> (((t` x)(.i` w)y) = (xP(s` y)) <-> ((t` x)Qy) = (xP(s` y))))
44	35, 43	raleqbidv 2274	. . . . . 6 |- (w = W -> (A.y e. (BaseSet` w)((t` x)(.i` w)y) = (xP(s` y)) <-> A.y e. Y ((t` x)Qy) = (xP(s` y))))
45	44	ralbidv 2123	. . . . 5 |- (w = W -> (A.x e. X A.y e. (BaseSet` w)((t` x)(.i` w)y) = (xP(s` y)) <-> A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y))))
46	37, 38, 45	3anbi123d 1168	. . . 4 |- (w = W -> ((t:X-->(BaseSet` w) /\ s:(BaseSet` w)-->X /\ A.x e. X A.y e. (BaseSet` w)((t` x)(.i` w)y) = (xP(s` y))) <-> (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))))
47	46	opabbidv 3401	. . 3 |- (w = W -> {<.t, s>. | (t:X-->(BaseSet` w) /\ s:(BaseSet` w)-->X /\ A.x e. X A.y e. (BaseSet` w)((t` x)(.i` w)y) = (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)))})
48		df-aj 9750	. . 3 |- adj = {<.<.u, w>., a>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ a = {<.t, s>. | (t:(BaseSet` u)-->(BaseSet` w) /\ s:(BaseSet` w)-->(BaseSet` u) /\ A.x e. (BaseSet` u)A.y e. (BaseSet` w)((t` x)(.i` w)y) = (x(.i` u)(s` y)))})}
49	19, 33, 47, 48	oprabval2 4957	. 2 |- ((U e. NrmCVec /\ W e. NrmCVec) -> (UadjW) = {<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))})
50		ajfval.5	. 2 |- A = (UadjW)
51	49, 50	syl5eq 1940	1 |- ((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)))})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  {copab 3395   X. cxp 3984  -->wf 3994  ` cfv 3998  (class class class)co 4884   ^m cmap 5381  NrmCVeccnv 9535  BaseSetcba 9537  .icip 9688  adjcaj 9748

This theorem is referenced by:  ajfuni 9861  ajval 9863

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  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-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-aj 9750

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