Theorem csbfsumlem 8286

Description: Lemma for csbfsum 8287.

Hypotheses

Ref	Expression
csbfsumlem.1	|- A e. _V
csbfsumlem.2	|- B e. _V

Assertion

Ref	Expression
csbfsumlem	|- (N e. (ZZ>=` M) -> [_A / x]_sum_k e. (M...N)B = sum_k e. (M...N)[_A / x]_B)

Distinct variable groups:   x,k,A   x,M,k   x,N

Proof of Theorem csbfsumlem

Step	Hyp	Ref	Expression
1		csbfsumlem.1	. . . 4 |- A e. _V
2		opreq2 4890	. . . . . 6 |- (n = M -> (M...n) = (M...M))
3	2	sumeq1d 8250	. . . . 5 |- (n = M -> sum_k e. (M...n)B = sum_k e. (M...M)B)
4	3	csbeq2dv 2562	. . . 4 |- ((n = M /\ A e. _V) -> [_A / x]_sum_k e. (M...n)B = [_A / x]_sum_k e. (M...M)B)
5	1, 4	mpan2 760	. . 3 |- (n = M -> [_A / x]_sum_k e. (M...n)B = [_A / x]_sum_k e. (M...M)B)
6	2	sumeq1d 8250	. . 3 |- (n = M -> sum_k e. (M...n)[_A / x]_B = sum_k e. (M...M)[_A / x]_B)
7	5, 6	eqeq12d 1899	. 2 |- (n = M -> ([_A / x]_sum_k e. (M...n)B = sum_k e. (M...n)[_A / x]_B <-> [_A / x]_sum_k e. (M...M)B = sum_k e. (M...M)[_A / x]_B))
8		opreq2 4890	. . . . . 6 |- (n = m -> (M...n) = (M...m))
9	8	sumeq1d 8250	. . . . 5 |- (n = m -> sum_k e. (M...n)B = sum_k e. (M...m)B)
10	9	csbeq2dv 2562	. . . 4 |- ((n = m /\ A e. _V) -> [_A / x]_sum_k e. (M...n)B = [_A / x]_sum_k e. (M...m)B)
11	1, 10	mpan2 760	. . 3 |- (n = m -> [_A / x]_sum_k e. (M...n)B = [_A / x]_sum_k e. (M...m)B)
12	8	sumeq1d 8250	. . 3 |- (n = m -> sum_k e. (M...n)[_A / x]_B = sum_k e. (M...m)[_A / x]_B)
13	11, 12	eqeq12d 1899	. 2 |- (n = m -> ([_A / x]_sum_k e. (M...n)B = sum_k e. (M...n)[_A / x]_B <-> [_A / x]_sum_k e. (M...m)B = sum_k e. (M...m)[_A / x]_B))
14		opreq2 4890	. . . . . 6 |- (n = (m + 1) -> (M...n) = (M...(m + 1)))
15	14	sumeq1d 8250	. . . . 5 |- (n = (m + 1) -> sum_k e. (M...n)B = sum_k e. (M...(m + 1))B)
16	15	csbeq2dv 2562	. . . 4 |- ((n = (m + 1) /\ A e. _V) -> [_A / x]_sum_k e. (M...n)B = [_A / x]_sum_k e. (M...(m + 1))B)
17	1, 16	mpan2 760	. . 3 |- (n = (m + 1) -> [_A / x]_sum_k e. (M...n)B = [_A / x]_sum_k e. (M...(m + 1))B)
18	14	sumeq1d 8250	. . 3 |- (n = (m + 1) -> sum_k e. (M...n)[_A / x]_B = sum_k e. (M...(m + 1))[_A / x]_B)
19	17, 18	eqeq12d 1899	. 2 |- (n = (m + 1) -> ([_A / x]_sum_k e. (M...n)B = sum_k e. (M...n)[_A / x]_B <-> [_A / x]_sum_k e. (M...(m + 1))B = sum_k e. (M...(m + 1))[_A / x]_B))
20		opreq2 4890	. . . . . 6 |- (n = N -> (M...n) = (M...N))
21	20	sumeq1d 8250	. . . . 5 |- (n = N -> sum_k e. (M...n)B = sum_k e. (M...N)B)
22	21	csbeq2dv 2562	. . . 4 |- ((n = N /\ A e. _V) -> [_A / x]_sum_k e. (M...n)B = [_A / x]_sum_k e. (M...N)B)
23	1, 22	mpan2 760	. . 3 |- (n = N -> [_A / x]_sum_k e. (M...n)B = [_A / x]_sum_k e. (M...N)B)
24	20	sumeq1d 8250	. . 3 |- (n = N -> sum_k e. (M...n)[_A / x]_B = sum_k e. (M...N)[_A / x]_B)
25	23, 24	eqeq12d 1899	. 2 |- (n = N -> ([_A / x]_sum_k e. (M...n)B = sum_k e. (M...n)[_A / x]_B <-> [_A / x]_sum_k e. (M...N)B = sum_k e. (M...N)[_A / x]_B))
26		csbcomg 2560	. . . 4 |- ((A e. _V /\ M e. ZZ) -> [_A / x]_[_M / k]_B = [_M / k]_[_A / x]_B)
27	1, 26	mpan 759	. . 3 |- (M e. ZZ -> [_A / x]_[_M / k]_B = [_M / k]_[_A / x]_B)
28		csbfsumlem.2	. . . . . 6 |- B e. _V
29	28	fsum1slem 8268	. . . . 5 |- (M e. ZZ -> sum_k e. (M...M)B = [_M / k]_B)
30	29	csbeq2dv 2562	. . . 4 |- ((M e. ZZ /\ A e. _V) -> [_A / x]_sum_k e. (M...M)B = [_A / x]_[_M / k]_B)
31	1, 30	mpan2 760	. . 3 |- (M e. ZZ -> [_A / x]_sum_k e. (M...M)B = [_A / x]_[_M / k]_B)
32	1, 28	csbex 2549	. . . 4 |- [_A / x]_B e. _V
33	32	fsum1slem 8268	. . 3 |- (M e. ZZ -> sum_k e. (M...M)[_A / x]_B = [_M / k]_[_A / x]_B)
34	27, 31, 33	3eqtr4d 1937	. 2 |- (M e. ZZ -> [_A / x]_sum_k e. (M...M)B = sum_k e. (M...M)[_A / x]_B)
35		simpr 350	. . . . 5 |- ((m e. (ZZ>=` M) /\ [_A / x]_sum_k e. (M...m)B = sum_k e. (M...m)[_A / x]_B) -> [_A / x]_sum_k e. (M...m)B = sum_k e. (M...m)[_A / x]_B)
36	35	opreq1d 4897	. . . 4 |- ((m e. (ZZ>=` M) /\ [_A / x]_sum_k e. (M...m)B = sum_k e. (M...m)[_A / x]_B) -> ([_A / x]_sum_k e. (M...m)B + [_(m + 1) / k]_[_A / x]_B) = (sum_k e. (M...m)[_A / x]_B + [_(m + 1) / k]_[_A / x]_B))
37	28	fsump1slem 8272	. . . . . . . 8 |- (m e. (ZZ>=` M) -> sum_k e. (M...(m + 1))B = (sum_k e. (M...m)B + [_(m + 1) / k]_B))
38	37	csbeq2dv 2562	. . . . . . 7 |- ((m e. (ZZ>=` M) /\ A e. _V) -> [_A / x]_sum_k e. (M...(m + 1))B = [_A / x]_(sum_k e. (M...m)B + [_(m + 1) / k]_B))
39	1, 38	mpan2 760	. . . . . 6 |- (m e. (ZZ>=` M) -> [_A / x]_sum_k e. (M...(m + 1))B = [_A / x]_(sum_k e. (M...m)B + [_(m + 1) / k]_B))
40		csbopr12g 4912	. . . . . . . 8 |- (A e. _V -> [_A / x]_(sum_k e. (M...m)B + [_(m + 1) / k]_B) = ([_A / x]_sum_k e. (M...m)B + [_A / x]_[_(m + 1) / k]_B))
41	1, 40	ax-mp 7	. . . . . . 7 |- [_A / x]_(sum_k e. (M...m)B + [_(m + 1) / k]_B) = ([_A / x]_sum_k e. (M...m)B + [_A / x]_[_(m + 1) / k]_B)
42		oprex 4907	. . . . . . . . 9 |- (m + 1) e. _V
43		csbcomg 2560	. . . . . . . . 9 |- ((A e. _V /\ (m + 1) e. _V) -> [_A / x]_[_(m + 1) / k]_B = [_(m + 1) / k]_[_A / x]_B)
44	1, 42, 43	mp2an 761	. . . . . . . 8 |- [_A / x]_[_(m + 1) / k]_B = [_(m + 1) / k]_[_A / x]_B
45	44	opreq2i 4893	. . . . . . 7 |- ([_A / x]_sum_k e. (M...m)B + [_A / x]_[_(m + 1) / k]_B) = ([_A / x]_sum_k e. (M...m)B + [_(m + 1) / k]_[_A / x]_B)
46	41, 45	eqtri 1908	. . . . . 6 |- [_A / x]_(sum_k e. (M...m)B + [_(m + 1) / k]_B) = ([_A / x]_sum_k e. (M...m)B + [_(m + 1) / k]_[_A / x]_B)
47	39, 46	syl6eq 1944	. . . . 5 |- (m e. (ZZ>=` M) -> [_A / x]_sum_k e. (M...(m + 1))B = ([_A / x]_sum_k e. (M...m)B + [_(m + 1) / k]_[_A / x]_B))
48	47	adantr 425	. . . 4 |- ((m e. (ZZ>=` M) /\ [_A / x]_sum_k e. (M...m)B = sum_k e. (M...m)[_A / x]_B) -> [_A / x]_sum_k e. (M...(m + 1))B = ([_A / x]_sum_k e. (M...m)B + [_(m + 1) / k]_[_A / x]_B))
49	32	fsump1slem 8272	. . . . 5 |- (m e. (ZZ>=` M) -> sum_k e. (M...(m + 1))[_A / x]_B = (sum_k e. (M...m)[_A / x]_B + [_(m + 1) / k]_[_A / x]_B))
50	49	adantr 425	. . . 4 |- ((m e. (ZZ>=` M) /\ [_A / x]_sum_k e. (M...m)B = sum_k e. (M...m)[_A / x]_B) -> sum_k e. (M...(m + 1))[_A / x]_B = (sum_k e. (M...m)[_A / x]_B + [_(m + 1) / k]_[_A / x]_B))
51	36, 48, 50	3eqtr4d 1937	. . 3 |- ((m e. (ZZ>=` M) /\ [_A / x]_sum_k e. (M...m)B = sum_k e. (M...m)[_A / x]_B) -> [_A / x]_sum_k e. (M...(m + 1))B = sum_k e. (M...(m + 1))[_A / x]_B)
52	51	ex 402	. 2 |- (m e. (ZZ>=` M) -> ([_A / x]_sum_k e. (M...m)B = sum_k e. (M...m)[_A / x]_B -> [_A / x]_sum_k e. (M...(m + 1))B = sum_k e. (M...(m + 1))[_A / x]_B))
53	7, 13, 19, 25, 34, 52	uzind4 7619	1 |- (N e. (ZZ>=` M) -> [_A / x]_sum_k e. (M...N)B = sum_k e. (M...N)[_A / x]_B)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  [_csb 2540  ` cfv 3998  (class class class)co 4884  1c1 6387   + caddc 6389  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637  sum_csu 8239

This theorem is referenced by:  csbfsum 8287

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  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-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-sum 8240

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