Theorem fprodp1s 14682

Description: The composite of the next term in a finite sum of A(k) is the previous term composed with A(N + 1).

Hypothesis

Ref	Expression
fprodp1s.1	|- G e. D

Assertion

Ref	Expression
fprodp1s	|- ((N e. (ZZ>=` M) /\ A.k e. (M...(N + 1))A e. B) -> prod_k e. (M...(N + 1))GA = (prod_k e. (M...N)GAG[_(N + 1) / k]_A))

Distinct variable groups:   k,M   k,N

Proof of Theorem fprodp1s

Step	Hyp	Ref	Expression
1		class2set 3471	. . . . 5 |- {x e. A | A e. _V} e. _V
2		fprodp1s.1	. . . . 5 |- G e. D
3	1, 2	fprodp1slem 14681	. . . 4 |- (N e. (ZZ>=` M) -> prod_k e. (M...(N + 1))G{x e. A | A e. _V} = (prod_k e. (M...N)G{x e. A | A e. _V}G[_(N + 1) / k]_{x e. A | A e. _V}))
4	3	adantr 425	. . 3 |- ((N e. (ZZ>=` M) /\ A.k e. (M...(N + 1))A e. _V) -> prod_k e. (M...(N + 1))G{x e. A | A e. _V} = (prod_k e. (M...N)G{x e. A | A e. _V}G[_(N + 1) / k]_{x e. A | A e. _V}))
5		class2seteq 3472	. . . . . 6 |- (A e. _V -> {x e. A | A e. _V} = A)
6	5	ralimi 2168	. . . . 5 |- (A.k e. (M...(N + 1))A e. _V -> A.k e. (M...(N + 1)){x e. A | A e. _V} = A)
7	6, 2	prodeq3d 14668	. . . 4 |- (A.k e. (M...(N + 1))A e. _V -> prod_k e. (M...(N + 1))G{x e. A | A e. _V} = prod_k e. (M...(N + 1))GA)
8	7	adantl 424	. . 3 |- ((N e. (ZZ>=` M) /\ A.k e. (M...(N + 1))A e. _V) -> prod_k e. (M...(N + 1))G{x e. A | A e. _V} = prod_k e. (M...(N + 1))GA)
9		eluzel2 7593	. . . . . . . . . 10 |- (N e. (ZZ>=` M) -> M e. ZZ)
10		eluzelz 7592	. . . . . . . . . 10 |- (N e. (ZZ>=` M) -> N e. ZZ)
11		fzssp1 7679	. . . . . . . . . 10 |- ((M e. ZZ /\ N e. ZZ) -> (M...N) C_ (M...(N + 1)))
12	9, 10, 11	syl11anc 524	. . . . . . . . 9 |- (N e. (ZZ>=` M) -> (M...N) C_ (M...(N + 1)))
13	12	sseld 2619	. . . . . . . 8 |- (N e. (ZZ>=` M) -> (k e. (M...N) -> k e. (M...(N + 1))))
14	5	a1i 8	. . . . . . . 8 |- (N e. (ZZ>=` M) -> (A e. _V -> {x e. A | A e. _V} = A))
15	13, 14	imim12d 69	. . . . . . 7 |- (N e. (ZZ>=` M) -> ((k e. (M...(N + 1)) -> A e. _V) -> (k e. (M...N) -> {x e. A | A e. _V} = A)))
16	15	ralimdv2 2173	. . . . . 6 |- (N e. (ZZ>=` M) -> (A.k e. (M...(N + 1))A e. _V -> A.k e. (M...N){x e. A | A e. _V} = A))
17	16	imp 377	. . . . 5 |- ((N e. (ZZ>=` M) /\ A.k e. (M...(N + 1))A e. _V) -> A.k e. (M...N){x e. A | A e. _V} = A)
18	17, 2	prodeq3d 14668	. . . 4 |- ((N e. (ZZ>=` M) /\ A.k e. (M...(N + 1))A e. _V) -> prod_k e. (M...N)G{x e. A | A e. _V} = prod_k e. (M...N)GA)
19		ra4sbca 2537	. . . . . . 7 |- (((N + 1) e. (M...(N + 1)) /\ A.k e. (M...(N + 1))A e. _V) -> [(N + 1) / k]A e. _V)
20		peano2uz 7616	. . . . . . . 8 |- (N e. (ZZ>=` M) -> (N + 1) e. (ZZ>=` M))
21		eluzfz2 7659	. . . . . . . 8 |- ((N + 1) e. (ZZ>=` M) -> (N + 1) e. (M...(N + 1)))
22	20, 21	syl 12	. . . . . . 7 |- (N e. (ZZ>=` M) -> (N + 1) e. (M...(N + 1)))
23	19, 22	sylan 497	. . . . . 6 |- ((N e. (ZZ>=` M) /\ A.k e. (M...(N + 1))A e. _V) -> [(N + 1) / k]A e. _V)
24		equid 1484	. . . . . . 7 |- x = x
25		oprex 4907	. . . . . . 7 |- (N + 1) e. _V
26	5	a1i 8	. . . . . . . 8 |- (x = x -> (A e. _V -> {x e. A | A e. _V} = A))
27	26	sbcimdv 2519	. . . . . . 7 |- ((x = x /\ (N + 1) e. _V) -> ([(N + 1) / k]A e. _V -> [(N + 1) / k]{x e. A | A e. _V} = A))
28	24, 25, 27	mp2an 761	. . . . . 6 |- ([(N + 1) / k]A e. _V -> [(N + 1) / k]{x e. A | A e. _V} = A)
29	23, 28	syl 12	. . . . 5 |- ((N e. (ZZ>=` M) /\ A.k e. (M...(N + 1))A e. _V) -> [(N + 1) / k]{x e. A | A e. _V} = A)
30		sbceqdig 2554	. . . . . 6 |- ((N + 1) e. _V -> ([(N + 1) / k]{x e. A | A e. _V} = A <-> [_(N + 1) / k]_{x e. A | A e. _V} = [_(N + 1) / k]_A))
31	25, 30	ax-mp 7	. . . . 5 |- ([(N + 1) / k]{x e. A | A e. _V} = A <-> [_(N + 1) / k]_{x e. A | A e. _V} = [_(N + 1) / k]_A)
32	29, 31	sylib 215	. . . 4 |- ((N e. (ZZ>=` M) /\ A.k e. (M...(N + 1))A e. _V) -> [_(N + 1) / k]_{x e. A | A e. _V} = [_(N + 1) / k]_A)
33	18, 32	opreq12d 4900	. . 3 |- ((N e. (ZZ>=` M) /\ A.k e. (M...(N + 1))A e. _V) -> (prod_k e. (M...N)G{x e. A | A e. _V}G[_(N + 1) / k]_{x e. A | A e. _V}) = (prod_k e. (M...N)GAG[_(N + 1) / k]_A))
34	4, 8, 33	3eqtr3d 1934	. 2 |- ((N e. (ZZ>=` M) /\ A.k e. (M...(N + 1))A e. _V) -> prod_k e. (M...(N + 1))GA = (prod_k e. (M...N)GAG[_(N + 1) / k]_A))
35		elisset 2299	. . 3 |- (A e. B -> A e. _V)
36	35	ralimi 2168	. 2 |- (A.k e. (M...(N + 1))A e. B -> A.k e. (M...(N + 1))A e. _V)
37	34, 36	sylan2 500	1 |- ((N e. (ZZ>=` M) /\ A.k e. (M...(N + 1))A e. B) -> prod_k e. (M...(N + 1))GA = (prod_k e. (M...N)GAG[_(N + 1) / k]_A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  {crab 2108  _Vcvv 2292  [_csb 2540   C_ wss 2593  ` cfv 3998  (class class class)co 4884  1c1 6387   + caddc 6389  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637  prod_cprd2 14654

This theorem is referenced by:  fprodp1s1 14683  clfsebs 14707  fincmpzer 14711  fprodneg 14741

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-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-prod 14653  df-prod2 14655

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