Theorem trnfset 18500

Description: The mapping from designated atom to set of translations.

Hypotheses

Ref	Expression
trnset.a	|- A = (Atoms` K)
trnset.s	|- S = (PSubSp` K)
trnset.p	|- P = (+P` K)
trnset.o	|- O = (_|_P` K)
trnset.w	|- W = (WAtoms` K)
trnset.m	|- M = (PAut` K)
trnset.l	|- L = (Dil` K)
trnset.t	|- T = (Trn` K)

Assertion

Ref	Expression
trnfset	|- (K e. C -> T = (d e. A |-> {f e. (L` d) | A.q e. (W` d)A.r e. (W` d)((qP(f` q)) i^i (O` {d})) = ((rP(f` r)) i^i (O` {d}))}))

Distinct variable groups:   A,d   f,d,q,r,K   f,L   W,q,r

Proof of Theorem trnfset

Step	Hyp	Ref	Expression
1		elisset 2547	. 2 |- (K e. C -> K e. _V)
2		trnset.t	. . 3 |- T = (Trn` K)
3		fveq2 4765	. . . . . 6 |- (k = K -> (Atoms` k) = (Atoms` K))
4		trnset.a	. . . . . 6 |- A = (Atoms` K)
5	3, 4	syl6eqr 2195	. . . . 5 |- (k = K -> (Atoms` k) = A)
6		fveq2 4765	. . . . . . . 8 |- (k = K -> (Dil` k) = (Dil` K))
7		trnset.l	. . . . . . . 8 |- L = (Dil` K)
8	6, 7	syl6eqr 2195	. . . . . . 7 |- (k = K -> (Dil` k) = L)
9	8	fveq1d 4767	. . . . . 6 |- (k = K -> ((Dil` k)` d) = (L` d))
10		fveq2 4765	. . . . . . . . 9 |- (k = K -> (WAtoms` k) = (WAtoms` K))
11		trnset.w	. . . . . . . . 9 |- W = (WAtoms` K)
12	10, 11	syl6eqr 2195	. . . . . . . 8 |- (k = K -> (WAtoms` k) = W)
13	12	fveq1d 4767	. . . . . . 7 |- (k = K -> ((WAtoms` k)` d) = (W` d))
14		fveq2 4765	. . . . . . . . . . . 12 |- (k = K -> (+P` k) = (+P` K))
15		trnset.p	. . . . . . . . . . . 12 |- P = (+P` K)
16	14, 15	syl6eqr 2195	. . . . . . . . . . 11 |- (k = K -> (+P` k) = P)
17	16	opreqd 4995	. . . . . . . . . 10 |- (k = K -> (q(+P` k)(f` q)) = (qP(f` q)))
18		fveq2 4765	. . . . . . . . . . . 12 |- (k = K -> (_|_P` k) = (_|_P` K))
19		trnset.o	. . . . . . . . . . . 12 |- O = (_|_P` K)
20	18, 19	syl6eqr 2195	. . . . . . . . . . 11 |- (k = K -> (_|_P` k) = O)
21	20	fveq1d 4767	. . . . . . . . . 10 |- (k = K -> ((_|_P` k)` {d}) = (O` {d}))
22	17, 21	ineq12d 3010	. . . . . . . . 9 |- (k = K -> ((q(+P` k)(f` q)) i^i ((_|_P` k)` {d})) = ((qP(f` q)) i^i (O` {d})))
23	16	opreqd 4995	. . . . . . . . . 10 |- (k = K -> (r(+P` k)(f` r)) = (rP(f` r)))
24	23, 21	ineq12d 3010	. . . . . . . . 9 |- (k = K -> ((r(+P` k)(f` r)) i^i ((_|_P` k)` {d})) = ((rP(f` r)) i^i (O` {d})))
25	22, 24	eqeq12d 2155	. . . . . . . 8 |- (k = K -> (((q(+P` k)(f` q)) i^i ((_|_P` k)` {d})) = ((r(+P` k)(f` r)) i^i ((_|_P` k)` {d})) <-> ((qP(f` q)) i^i (O` {d})) = ((rP(f` r)) i^i (O` {d}))))
26	13, 25	raleqbidv 2520	. . . . . . 7 |- (k = K -> (A.r e. ((WAtoms` k)` d)((q(+P` k)(f` q)) i^i ((_|_P` k)` {d})) = ((r(+P` k)(f` r)) i^i ((_|_P` k)` {d})) <-> A.r e. (W` d)((qP(f` q)) i^i (O` {d})) = ((rP(f` r)) i^i (O` {d}))))
27	13, 26	raleqbidv 2520	. . . . . 6 |- (k = K -> (A.q e. ((WAtoms` k)` d)A.r e. ((WAtoms` k)` d)((q(+P` k)(f` q)) i^i ((_|_P` k)` {d})) = ((r(+P` k)(f` r)) i^i ((_|_P` k)` {d})) <-> A.q e. (W` d)A.r e. (W` d)((qP(f` q)) i^i (O` {d})) = ((rP(f` r)) i^i (O` {d}))))
28	9, 27	rabeqbidv 2536	. . . . 5 |- (k = K -> {f e. ((Dil` k)` d) | A.q e. ((WAtoms` k)` d)A.r e. ((WAtoms` k)` d)((q(+P` k)(f` q)) i^i ((_|_P` k)` {d})) = ((r(+P` k)(f` r)) i^i ((_|_P` k)` {d}))} = {f e. (L` d) | A.q e. (W` d)A.r e. (W` d)((qP(f` q)) i^i (O` {d})) = ((rP(f` r)) i^i (O` {d}))})
29	5, 28	mpteq12dv 5101	. . . 4 |- (k = K -> (d e. (Atoms` k) |-> {f e. ((Dil` k)` d) | A.q e. ((WAtoms` k)` d)A.r e. ((WAtoms` k)` d)((q(+P` k)(f` q)) i^i ((_|_P` k)` {d})) = ((r(+P` k)(f` r)) i^i ((_|_P` k)` {d}))}) = (d e. A |-> {f e. (L` d) | A.q e. (W` d)A.r e. (W` d)((qP(f` q)) i^i (O` {d})) = ((rP(f` r)) i^i (O` {d}))}))
30		df-trn 18461	. . . 4 |- Trn = (k e. _V |-> (d e. (Atoms` k) |-> {f e. ((Dil` k)` d) | A.q e. ((WAtoms` k)` d)A.r e. ((WAtoms` k)` d)((q(+P` k)(f` q)) i^i ((_|_P` k)` {d})) = ((r(+P` k)(f` r)) i^i ((_|_P` k)` {d}))}))
31		fvex 4778	. . . . . 6 |- (Atoms` K) e. _V
32	4, 31	eqeltri 2214	. . . . 5 |- A e. _V
33		mptexg 5109	. . . . 5 |- (A e. _V -> (d e. A |-> {f e. (L` d) | A.q e. (W` d)A.r e. (W` d)((qP(f` q)) i^i (O` {d})) = ((rP(f` r)) i^i (O` {d}))}) e. _V)
34	32, 33	ax-mp 7	. . . 4 |- (d e. A |-> {f e. (L` d) | A.q e. (W` d)A.r e. (W` d)((qP(f` q)) i^i (O` {d})) = ((rP(f` r)) i^i (O` {d}))}) e. _V
35	29, 30, 34	fvmpt 5119	. . 3 |- (K e. _V -> (Trn` K) = (d e. A |-> {f e. (L` d) | A.q e. (W` d)A.r e. (W` d)((qP(f` q)) i^i (O` {d})) = ((rP(f` r)) i^i (O` {d}))}))
36	2, 35	syl5eq 2185	. 2 |- (K e. _V -> T = (d e. A |-> {f e. (L` d) | A.q e. (W` d)A.r e. (W` d)((qP(f` q)) i^i (O` {d})) = ((rP(f` r)) i^i (O` {d}))}))
37	1, 36	syl 13	1 |- (K e. C -> T = (d e. A |-> {f e. (L` d) | A.q e. (W` d)A.r e. (W` d)((qP(f` q)) i^i (O` {d})) = ((rP(f` r)) i^i (O` {d}))}))

Syntax hints:   -> wi 3   = wceq 1586   e. wcel 1588  A.wral 2355  {crab 2358  _Vcvv 2538   i^i cin 2826  {csn 3238  ` cfv 4131  (class class class)co 4981   e. cmpt 5097  Atomscatm 17654  PSubSpcpsubsp 17928  +Pcpadd 18227  _|_Pcpol 18319  WAtomscwpoints 18399  PAutcpaut 18400  Dilcdil 18456  Trnctrn 18457

This theorem is referenced by:  trnset 18501

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1592  ax-gen 1593  ax-8 1594  ax-9 1595  ax-10 1596  ax-11 1597  ax-12 1598  ax-13 1599  ax-14 1600  ax-17 1605  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763  ax-10o 1781  ax-16 1854  ax-11o 1864  ax-ext 2123  ax-rep 3596  ax-sep 3606  ax-nul 3613  ax-pow 3649  ax-pr 3687  ax-un 3929

This theorem depends on definitions:  df-bi 220  df-or 338  df-an 339  df-ex 1616  df-sb 1816  df-eu 2041  df-mo 2042  df-clab 2129  df-cleq 2134  df-clel 2137  df-ne 2268  df-ral 2359  df-rex 2360  df-rab 2362  df-v 2540  df-dif 2830  df-un 2832  df-in 2834  df-ss 2836  df-nul 3083  df-pw 3229  df-sn 3242  df-pr 3243  df-op 3246  df-uni 3367  df-br 3508  df-opab 3566  df-id 3747  df-xp 4133  df-rel 4134  df-cnv 4135  df-co 4136  df-dm 4137  df-rn 4138  df-res 4139  df-ima 4140  df-fun 4141  df-fn 4142  df-fv 4147  df-opr 4983  df-mpt 5099  df-trn 18461

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