Theorem ipval2lem1 8347

Description: Lemma for ipval3 8355.

Hypothesis

Ref	Expression
ipval2lem.1	|- M e. V

Assertion

Ref	Expression
ipval2lem1	|- [_M / k]_((i^k) x. ((N` (AG((i^k)SB)))^2)) = ((i^M) x. ((N` (AG((i^M)SB)))^2))

Distinct variable groups:   A,k   B,k   k,G   k,N   S,k

Proof of Theorem ipval2lem1

Step	Hyp	Ref	Expression
1		ipval2lem.1	. . 3 |- M e. V
2		csbopr12g 3993	. . 3 |- (M e. V -> [_M / k]_((i^k) x. ((N` (AG((i^k)SB)))^2)) = ([_M / k]_(i^k) x. [_M / k]_((N` (AG((i^k)SB)))^2)))
3	1, 2	ax-mp 7	. 2 |- [_M / k]_((i^k) x. ((N` (AG((i^k)SB)))^2)) = ([_M / k]_(i^k) x. [_M / k]_((N` (AG((i^k)SB)))^2))
4		csbopr2g 3995	. . . . 5 |- (M e. V -> [_M / k]_(i^k) = (i^[_M / k]_k))
5	1, 4	ax-mp 7	. . . 4 |- [_M / k]_(i^k) = (i^[_M / k]_k)
6		csbvarg 2024	. . . . . 6 |- (M e. V -> [_M / k]_k = M)
7	1, 6	ax-mp 7	. . . . 5 |- [_M / k]_k = M
8	7	opreq2i 3978	. . . 4 |- (i^[_M / k]_k) = (i^M)
9	5, 8	eqtr 1498	. . 3 |- [_M / k]_(i^k) = (i^M)
10		csbopr1g 3994	. . . . 5 |- (M e. V -> [_M / k]_((N` (AG((i^k)SB)))^2) = ([_M / k]_(N` (AG((i^k)SB)))^2))
11	1, 10	ax-mp 7	. . . 4 |- [_M / k]_((N` (AG((i^k)SB)))^2) = ([_M / k]_(N` (AG((i^k)SB)))^2)
12		csbfv2g 3749	. . . . . . 7 |- (M e. V -> [_M / k]_(N` (AG((i^k)SB))) = (N` [_M / k]_(AG((i^k)SB))))
13	1, 12	ax-mp 7	. . . . . 6 |- [_M / k]_(N` (AG((i^k)SB))) = (N` [_M / k]_(AG((i^k)SB)))
14		csbopr2g 3995	. . . . . . . . 9 |- (M e. V -> [_M / k]_(AG((i^k)SB)) = (AG[_M / k]_((i^k)SB)))
15	1, 14	ax-mp 7	. . . . . . . 8 |- [_M / k]_(AG((i^k)SB)) = (AG[_M / k]_((i^k)SB))
16		csbopr1g 3994	. . . . . . . . . . 11 |- (M e. V -> [_M / k]_((i^k)SB) = ([_M / k]_(i^k)SB))
17	1, 16	ax-mp 7	. . . . . . . . . 10 |- [_M / k]_((i^k)SB) = ([_M / k]_(i^k)SB)
18	9	opreq1i 3977	. . . . . . . . . 10 |- ([_M / k]_(i^k)SB) = ((i^M)SB)
19	17, 18	eqtr 1498	. . . . . . . . 9 |- [_M / k]_((i^k)SB) = ((i^M)SB)
20	19	opreq2i 3978	. . . . . . . 8 |- (AG[_M / k]_((i^k)SB)) = (AG((i^M)SB))
21	15, 20	eqtr 1498	. . . . . . 7 |- [_M / k]_(AG((i^k)SB)) = (AG((i^M)SB))
22	21	fveq2i 3733	. . . . . 6 |- (N` [_M / k]_(AG((i^k)SB))) = (N` (AG((i^M)SB)))
23	13, 22	eqtr 1498	. . . . 5 |- [_M / k]_(N` (AG((i^k)SB))) = (N` (AG((i^M)SB)))
24	23	opreq1i 3977	. . . 4 |- ([_M / k]_(N` (AG((i^k)SB)))^2) = ((N` (AG((i^M)SB)))^2)
25	11, 24	eqtr 1498	. . 3 |- [_M / k]_((N` (AG((i^k)SB)))^2) = ((N` (AG((i^M)SB)))^2)
26	9, 25	opreq12i 3979	. 2 |- ([_M / k]_(i^k) x. [_M / k]_((N` (AG((i^k)SB)))^2)) = ((i^M) x. ((N` (AG((i^M)SB)))^2))
27	3, 26	eqtr 1498	1 |- [_M / k]_((i^k) x. ((N` (AG((i^k)SB)))^2)) = ((i^M) x. ((N` (AG((i^M)SB)))^2))

Syntax hints:   = wceq 958   e. wcel 960  Vcvv 1814  [_csb 2004  ` cfv 3188  (class class class)co 3969  ici 5248   x. cmul 5251  2c2 5963  ^cexp 6569

This theorem is referenced by:  ipval2 8353

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971

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