Theorem minveclem8 8552

Description: Lemma for minvecex 8578.

Hypothesis

Ref	Expression
minvec8.hf	|- (j e. NN -> (F` j) = (N` (AM(f` j))))

Assertion

Ref	Expression
minveclem8	|- (n e. NN -> (F` n) = (N` (AM(f` n))))

Distinct variable groups:   A,j   j,F   j,M   j,N   f,j,n

Proof of Theorem minveclem8

Step	Hyp	Ref	Expression
1		fveq2 3724	. . 3 |- (j = n -> (F` j) = (F` n))
2		fveq2 3724	. . . . 5 |- (j = n -> (f` j) = (f` n))
3	2	opreq2d 3976	. . . 4 |- (j = n -> (AM(f` j)) = (AM(f` n)))
4	3	fveq2d 3728	. . 3 |- (j = n -> (N` (AM(f` j))) = (N` (AM(f` n))))
5	1, 4	eqeq12d 1489	. 2 |- (j = n -> ((F` j) = (N` (AM(f` j))) <-> (F` n) = (N` (AM(f` n)))))
6		minvec8.hf	. 2 |- (j e. NN -> (F` j) = (N` (AM(f` j))))
7	5, 6	vtoclga 1852	1 |- (n e. NN -> (F` n) = (N` (AM(f` n))))

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  ` cfv 3182  (class class class)co 3963  NNcn 5296

This theorem is referenced by:  minveclem22 8566  minveclem31 8575

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965

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