Theorem isnvlem 9561

Description: Lemma for isnv 9563.

Hypotheses

Ref	Expression
isnvlem.1	|- X = ran G
isnvlem.2	|- Z = (Id` G)

Assertion

Ref	Expression
isnvlem	|- ((G e. _V /\ S e. _V /\ N e. _V) -> (<.<.G, S>., N>. e. NrmCVec <-> (<.G, S>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y))))))

Distinct variable groups:   x,y,G   x,N,y   x,S,y   x,X,y

Proof of Theorem isnvlem

Step	Hyp	Ref	Expression
1		opeq1 3158	. . . . 5 |- (g = G -> <.g, s>. = <.G, s>.)
2	1	eleq1d 1963	. . . 4 |- (g = G -> (<.g, s>. e. CVec <-> <.G, s>. e. CVec))
3		rneq 4186	. . . . . 6 |- (g = G -> ran g = ran G)
4		isnvlem.1	. . . . . 6 |- X = ran G
5	3, 4	syl6eqr 1946	. . . . 5 |- (g = G -> ran g = X)
6	5	feq2d 4557	. . . 4 |- (g = G -> (n:ran g-->RR <-> n:X-->RR))
7		fveq2 4681	. . . . . . . . 9 |- (g = G -> (Id` g) = (Id` G))
8		isnvlem.2	. . . . . . . . 9 |- Z = (Id` G)
9	7, 8	syl6eqr 1946	. . . . . . . 8 |- (g = G -> (Id` g) = Z)
10	9	eqeq2d 1895	. . . . . . 7 |- (g = G -> (x = (Id` g) <-> x = Z))
11	10	imbi2d 674	. . . . . 6 |- (g = G -> (((n` x) = 0 -> x = (Id` g)) <-> ((n` x) = 0 -> x = Z)))
12		opreq 4888	. . . . . . . . 9 |- (g = G -> (xgy) = (xGy))
13	12	fveq2d 4685	. . . . . . . 8 |- (g = G -> (n` (xgy)) = (n` (xGy)))
14	13	breq1d 3348	. . . . . . 7 |- (g = G -> ((n` (xgy)) <_ ((n` x) + (n` y)) <-> (n` (xGy)) <_ ((n` x) + (n` y))))
15	5, 14	raleqbidv 2274	. . . . . 6 |- (g = G -> (A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y)) <-> A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))))
16	11, 15	3anbi13d 1170	. . . . 5 |- (g = G -> ((((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y))) <-> (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))))
17	5, 16	raleqbidv 2274	. . . 4 |- (g = G -> (A.x e. ran g(((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y))) <-> A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))))
18	2, 6, 17	3anbi123d 1168	. . 3 |- (g = G -> ((<.g, s>. e. CVec /\ n:ran g-->RR /\ A.x e. ran g(((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y)))) <-> (<.G, s>. e. CVec /\ n:X-->RR /\ A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))))))
19		opeq2 3159	. . . . 5 |- (s = S -> <.G, s>. = <.G, S>.)
20	19	eleq1d 1963	. . . 4 |- (s = S -> (<.G, s>. e. CVec <-> <.G, S>. e. CVec))
21		opreq 4888	. . . . . . . . 9 |- (s = S -> (ysx) = (ySx))
22	21	fveq2d 4685	. . . . . . . 8 |- (s = S -> (n` (ysx)) = (n` (ySx)))
23	22	eqeq1d 1892	. . . . . . 7 |- (s = S -> ((n` (ysx)) = ((abs` y) x. (n` x)) <-> (n` (ySx)) = ((abs` y) x. (n` x))))
24	23	ralbidv 2123	. . . . . 6 |- (s = S -> (A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) <-> A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x))))
25	24	3anbi2d 1173	. . . . 5 |- (s = S -> ((((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))) <-> (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))))
26	25	ralbidv 2123	. . . 4 |- (s = S -> (A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))) <-> A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))))
27	20, 26	3anbi13d 1170	. . 3 |- (s = S -> ((<.G, s>. e. CVec /\ n:X-->RR /\ A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))) <-> (<.G, S>. e. CVec /\ n:X-->RR /\ A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))))))
28		feq1 4551	. . . 4 |- (n = N -> (n:X-->RR <-> N:X-->RR))
29		fveq1 4680	. . . . . . . 8 |- (n = N -> (n` x) = (N` x))
30	29	eqeq1d 1892	. . . . . . 7 |- (n = N -> ((n` x) = 0 <-> (N` x) = 0))
31	30	imbi1d 675	. . . . . 6 |- (n = N -> (((n` x) = 0 -> x = Z) <-> ((N` x) = 0 -> x = Z)))
32		fveq1 4680	. . . . . . . 8 |- (n = N -> (n` (ySx)) = (N` (ySx)))
33	29	opreq2d 4898	. . . . . . . 8 |- (n = N -> ((abs` y) x. (n` x)) = ((abs` y) x. (N` x)))
34	32, 33	eqeq12d 1899	. . . . . . 7 |- (n = N -> ((n` (ySx)) = ((abs` y) x. (n` x)) <-> (N` (ySx)) = ((abs` y) x. (N` x))))
35	34	ralbidv 2123	. . . . . 6 |- (n = N -> (A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) <-> A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x))))
36		fveq1 4680	. . . . . . . 8 |- (n = N -> (n` (xGy)) = (N` (xGy)))
37		fveq1 4680	. . . . . . . . 9 |- (n = N -> (n` y) = (N` y))
38	29, 37	opreq12d 4900	. . . . . . . 8 |- (n = N -> ((n` x) + (n` y)) = ((N` x) + (N` y)))
39	36, 38	breq12d 3351	. . . . . . 7 |- (n = N -> ((n` (xGy)) <_ ((n` x) + (n` y)) <-> (N` (xGy)) <_ ((N` x) + (N` y))))
40	39	ralbidv 2123	. . . . . 6 |- (n = N -> (A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)) <-> A.y e. X (N` (xGy)) <_ ((N` x) + (N` y))))
41	31, 35, 40	3anbi123d 1168	. . . . 5 |- (n = N -> ((((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))) <-> (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)))))
42	41	ralbidv 2123	. . . 4 |- (n = N -> (A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y))) <-> A.x e. X (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)))))
43	28, 42	3anbi23d 1171	. . 3 |- (n = N -> ((<.G, S>. e. CVec /\ n:X-->RR /\ A.x e. X (((n` x) = 0 -> x = Z) /\ A.y e. CC (n` (ySx)) = ((abs` y) x. (n` x)) /\ A.y e. X (n` (xGy)) <_ ((n` x) + (n` y)))) <-> (<.G, S>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y))))))
44	18, 27, 43	eloprabg 4936	. 2 |- ((G e. _V /\ S e. _V /\ N e. _V) -> (<.<.G, S>., N>. e. {<.<.g, s>., n>. | (<.g, s>. e. CVec /\ n:ran g-->RR /\ A.x e. ran g(((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y))))} <-> (<.G, S>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y))))))
45		df-nv 9543	. . 3 |- NrmCVec = {<.<.g, s>., n>. | (<.g, s>. e. CVec /\ n:ran g-->RR /\ A.x e. ran g(((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y))))}
46	45	eleq2i 1961	. 2 |- (<.<.G, S>., N>. e. NrmCVec <-> <.<.G, S>., N>. e. {<.<.g, s>., n>. | (<.g, s>. e. CVec /\ n:ran g-->RR /\ A.x e. ran g(((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y))))})
47	44, 46	syl5bb 591	1 |- ((G e. _V /\ S e. _V /\ N e. _V) -> (<.<.G, S>., N>. e. NrmCVec <-> (<.G, S>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y))))))

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  <.cop 3046   class class class wbr 3338  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  CCcc 6384  RRcr 6385  0cc0 6386   + caddc 6389   x. cmul 6391   <_ cle 6448  abscabs 8000  Idcgi 9312  CVeccvc 9496  NrmCVeccnv 9535

This theorem is referenced by:  nvex 9562  isnv 9563

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-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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

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-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-nv 9543

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