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Theorem fsumrev 8289
Description: Reversal of a finite sum. Warning: The HTML proof page is 0.6 MB in size.
Assertion
Ref Expression
fsumrev |- ((N e. (ZZ>=` M) /\ K e. ZZ /\ A.j e. (M...N)A e. CC) -> sum_j e. (M...N)A = sum_k e. ((K - N)...(K - M))[_(K - k) / j]_A)
Distinct variable groups:   A,k   j,k,K   j,M,k   j,N,k
Proof of Theorem fsumrev
StepHypRef Expression
1 nncan 6635 . . . . . . . . . 10 |- ((K e. CC /\ M e. CC) -> (K - (K - M)) = M)
2 zcn 7349 . . . . . . . . . 10 |- (K e. ZZ -> K e. CC)
3 zcn 7349 . . . . . . . . . 10 |- (M e. ZZ -> M e. CC)
41, 2, 3syl2an 503 . . . . . . . . 9 |- ((K e. ZZ /\ M e. ZZ) -> (K - (K - M)) = M)
54csbeq1d 2544 . . . . . . . 8 |- ((K e. ZZ /\ M e. ZZ) -> [_(K - (K - M)) / j]_A = [_M / j]_A)
65adantr 425 . . . . . . 7 |- (((K e. ZZ /\ M e. ZZ) /\ A.j e. (M...M)A e. CC) -> [_(K - (K - M)) / j]_A = [_M / j]_A)
7 simpr 350 . . . . . . . . . . 11 |- ((K e. ZZ /\ M e. ZZ) -> M e. ZZ)
8 simpl 346 . . . . . . . . . . 11 |- ((K e. ZZ /\ M e. ZZ) -> K e. ZZ)
9 fzrevral 7698 . . . . . . . . . . 11 |- ((M e. ZZ /\ M e. ZZ /\ K e. ZZ) -> (A.j e. (M...M)A e. CC <-> A.k e. ((K - M)...(K - M))[(K - k) / j]A e. CC))
107, 7, 8, 9syl111anc 1100 . . . . . . . . . 10 |- ((K e. ZZ /\ M e. ZZ) -> (A.j e. (M...M)A e. CC <-> A.k e. ((K - M)...(K - M))[(K - k) / j]A e. CC))
11 oprex 4907 . . . . . . . . . . . 12 |- (K - k) e. _V
12 sbcel1g 2556 . . . . . . . . . . . 12 |- ((K - k) e. _V -> ([(K - k) / j]A e. CC <-> [_(K - k) / j]_A e. CC))
1311, 12ax-mp 7 . . . . . . . . . . 11 |- ([(K - k) / j]A e. CC <-> [_(K - k) / j]_A e. CC)
1413ralbii 2127 . . . . . . . . . 10 |- (A.k e. ((K - M)...(K - M))[(K - k) / j]A e. CC <-> A.k e. ((K - M)...(K - M))[_(K - k) / j]_A e. CC)
1510, 14syl6bb 595 . . . . . . . . 9 |- ((K e. ZZ /\ M e. ZZ) -> (A.j e. (M...M)A e. CC <-> A.k e. ((K - M)...(K - M))[_(K - k) / j]_A e. CC))
1615biimpa 460 . . . . . . . 8 |- (((K e. ZZ /\ M e. ZZ) /\ A.j e. (M...M)A e. CC) -> A.k e. ((K - M)...(K - M))[_(K - k) / j]_A e. CC)
17 fsum1s 8269 . . . . . . . . . 10 |- (((K - M) e. ZZ /\ A.k e. ((K - M)...(K - M))[_(K - k) / j]_A e. CC) -> sum_k e. ((K - M)...(K - M))[_(K - k) / j]_A = [_(K - M) / k]_[_(K - k) / j]_A)
18 oprex 4907 . . . . . . . . . . 11 |- (K - M) e. _V
1911ax-gen 1305 . . . . . . . . . . 11 |- A.k(K - k) e. _V
20 opreq2 4890 . . . . . . . . . . . 12 |- (k = (K - M) -> (K - k) = (K - (K - M)))
2120csbco3g 2585 . . . . . . . . . . 11 |- (((K - M) e. _V /\ A.k(K - k) e. _V) -> [_(K - M) / k]_[_(K - k) / j]_A = [_(K - (K - M)) / j]_A)
2218, 19, 21mp2an 761 . . . . . . . . . 10 |- [_(K - M) / k]_[_(K - k) / j]_A = [_(K - (K - M)) / j]_A
2317, 22syl6eq 1944 . . . . . . . . 9 |- (((K - M) e. ZZ /\ A.k e. ((K - M)...(K - M))[_(K - k) / j]_A e. CC) -> sum_k e. ((K - M)...(K - M))[_(K - k) / j]_A = [_(K - (K - M)) / j]_A)
24 zsubcl 7377 . . . . . . . . 9 |- ((K e. ZZ /\ M e. ZZ) -> (K - M) e. ZZ)
2523, 24sylan 497 . . . . . . . 8 |- (((K e. ZZ /\ M e. ZZ) /\ A.k e. ((K - M)...(K - M))[_(K - k) / j]_A e. CC) -> sum_k e. ((K - M)...(K - M))[_(K - k) / j]_A = [_(K - (K - M)) / j]_A)
2616, 25syldan 516 . . . . . . 7 |- (((K e. ZZ /\ M e. ZZ) /\ A.j e. (M...M)A e. CC) -> sum_k e. ((K - M)...(K - M))[_(K - k) / j]_A = [_(K - (K - M)) / j]_A)
27 fsum1s 8269 . . . . . . . 8 |- ((M e. ZZ /\ A.j e. (M...M)A e. CC) -> sum_j e. (M...M)A = [_M / j]_A)
2827adantll 428 . . . . . . 7 |- (((K e. ZZ /\ M e. ZZ) /\ A.j e. (M...M)A e. CC) -> sum_j e. (M...M)A = [_M / j]_A)
296, 26, 283eqtr4rd 1939 . . . . . 6 |- (((K e. ZZ /\ M e. ZZ) /\ A.j e. (M...M)A e. CC) -> sum_j e. (M...M)A = sum_k e. ((K - M)...(K - M))[_(K - k) / j]_A)
3029anasss 488 . . . . 5 |- ((K e. ZZ /\ (M e. ZZ /\ A.j e. (M...M)A e. CC)) -> sum_j e. (M...M)A = sum_k e. ((K - M)...(K - M))[_(K - k) / j]_A)
3130an1s 544 . . . 4 |- ((M e. ZZ /\ (K e. ZZ /\ A.j e. (M...M)A e. CC)) -> sum_j e. (M...M)A = sum_k e. ((K - M)...(K - M))[_(K - k) / j]_A)
3231ex 402 . . 3 |- (M e. ZZ -> ((K e. ZZ /\ A.j e. (M...M)A e. CC) -> sum_j e. (M...M)A = sum_k e. ((K - M)...(K - M))[_(K - k) / j]_A))
33 eluzel2 7593 . . . . . . . . . 10 |- (n e. (ZZ>=` M) -> M e. ZZ)
34 eluzelz 7592 . . . . . . . . . 10 |- (n e. (ZZ>=` M) -> n e. ZZ)
35 fzssp1 7679 . . . . . . . . . 10 |- ((M e. ZZ /\ n e. ZZ) -> (M...n) C_ (M...(n + 1)))
3633, 34, 35syl11anc 524 . . . . . . . . 9 |- (n e. (ZZ>=` M) -> (M...n) C_ (M...(n + 1)))
3736sseld 2619 . . . . . . . 8 |- (n e. (ZZ>=` M) -> (j e. (M...n) -> j e. (M...(n + 1))))
3837imim1d 33 . . . . . . 7 |- (n e. (ZZ>=` M) -> ((j e. (M...(n + 1)) -> A e. CC) -> (j e. (M...n) -> A e. CC)))
3938ralimdv2 2173 . . . . . 6 |- (n e. (ZZ>=` M) -> (A.j e. (M...(n + 1))A e. CC -> A.j e. (M...n)A e. CC))
4039anim2d 620 . . . . 5 |- (n e. (ZZ>=` M) -> ((K e. ZZ /\ A.j e. (M...(n + 1))A e. CC) -> (K e. ZZ /\ A.j e. (M...n)A e. CC)))
4140imim1d 33 . . . 4 |- (n e. (ZZ>=` M) -> (((K e. ZZ /\ A.j e. (M...n)A e. CC) -> sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A) -> ((K e. ZZ /\ A.j e. (M...(n + 1))A e. CC) -> sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A)))
42 nncan 6635 . . . . . . . . . . . . . 14 |- ((K e. CC /\ (n + 1) e. CC) -> (K - (K - (n + 1))) = (n + 1))
43 zcn 7349 . . . . . . . . . . . . . . 15 |- (n e. ZZ -> n e. CC)
44 peano2cn 6498 . . . . . . . . . . . . . . 15 |- (n e. CC -> (n + 1) e. CC)
4534, 43, 443syl 24 . . . . . . . . . . . . . 14 |- (n e. (ZZ>=` M) -> (n + 1) e. CC)
4642, 2, 45syl2an 503 . . . . . . . . . . . . 13 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> (K - (K - (n + 1))) = (n + 1))
4746csbeq1d 2544 . . . . . . . . . . . 12 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> [_(K - (K - (n + 1))) / j]_A = [_(n + 1) / j]_A)
48 oprex 4907 . . . . . . . . . . . . 13 |- (K - (n + 1)) e. _V
49 opreq2 4890 . . . . . . . . . . . . . 14 |- (k = (K - (n + 1)) -> (K - k) = (K - (K - (n + 1))))
5049csbco3g 2585 . . . . . . . . . . . . 13 |- (((K - (n + 1)) e. _V /\ A.k(K - k) e. _V) -> [_(K - (n + 1)) / k]_[_(K - k) / j]_A = [_(K - (K - (n + 1))) / j]_A)
5148, 19, 50mp2an 761 . . . . . . . . . . . 12 |- [_(K - (n + 1)) / k]_[_(K - k) / j]_A = [_(K - (K - (n + 1))) / j]_A
5247, 51syl5req 1941 . . . . . . . . . . 11 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> [_(n + 1) / j]_A = [_(K - (n + 1)) / k]_[_(K - k) / j]_A)
5352adantr 425 . . . . . . . . . 10 |- (((K e. ZZ /\ n e. (ZZ>=` M)) /\ A.j e. (M...(n + 1))A e. CC) -> [_(n + 1) / j]_A = [_(K - (n + 1)) / k]_[_(K - k) / j]_A)
54 id 73 . . . . . . . . . 10 |- (sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A -> sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A)
5553, 54opreqan12d 4902 . . . . . . . . 9 |- ((((K e. ZZ /\ n e. (ZZ>=` M)) /\ A.j e. (M...(n + 1))A e. CC) /\ sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A) -> ([_(n + 1) / j]_A + sum_j e. (M...n)A) = ([_(K - (n + 1)) / k]_[_(K - k) / j]_A + sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A))
56 fsump1s 8273 . . . . . . . . . . . 12 |- ((n e. (ZZ>=` M) /\ A.j e. (M...(n + 1))A e. CC) -> sum_j e. (M...(n + 1))A = (sum_j e. (M...n)A + [_(n + 1) / j]_A))
5739imp 377 . . . . . . . . . . . . . 14 |- ((n e. (ZZ>=` M) /\ A.j e. (M...(n + 1))A e. CC) -> A.j e. (M...n)A e. CC)
58 fsumcl 8275 . . . . . . . . . . . . . 14 |- ((n e. (ZZ>=` M) /\ A.j e. (M...n)A e. CC) -> sum_j e. (M...n)A e. CC)
5957, 58syldan 516 . . . . . . . . . . . . 13 |- ((n e. (ZZ>=` M) /\ A.j e. (M...(n + 1))A e. CC) -> sum_j e. (M...n)A e. CC)
60 ra4csbela 2587 . . . . . . . . . . . . . 14 |- (((n + 1) e. (M...(n + 1)) /\ A.j e. (M...(n + 1))A e. CC) -> [_(n + 1) / j]_A e. CC)
61 peano2uz 7616 . . . . . . . . . . . . . . 15 |- (n e. (ZZ>=` M) -> (n + 1) e. (ZZ>=` M))
62 eluzfz2 7659 . . . . . . . . . . . . . . 15 |- ((n + 1) e. (ZZ>=` M) -> (n + 1) e. (M...(n + 1)))
6361, 62syl 12 . . . . . . . . . . . . . 14 |- (n e. (ZZ>=` M) -> (n + 1) e. (M...(n + 1)))
6460, 63sylan 497 . . . . . . . . . . . . 13 |- ((n e. (ZZ>=` M) /\ A.j e. (M...(n + 1))A e. CC) -> [_(n + 1) / j]_A e. CC)
65 addcom 6458 . . . . . . . . . . . . 13 |- ((sum_j e. (M...n)A e. CC /\ [_(n + 1) / j]_A e. CC) -> (sum_j e. (M...n)A + [_(n + 1) / j]_A) = ([_(n + 1) / j]_A + sum_j e. (M...n)A))
6659, 64, 65syl11anc 524 . . . . . . . . . . . 12 |- ((n e. (ZZ>=` M) /\ A.j e. (M...(n + 1))A e. CC) -> (sum_j e. (M...n)A + [_(n + 1) / j]_A) = ([_(n + 1) / j]_A + sum_j e. (M...n)A))
6756, 66eqtrd 1925 . . . . . . . . . . 11 |- ((n e. (ZZ>=` M) /\ A.j e. (M...(n + 1))A e. CC) -> sum_j e. (M...(n + 1))A = ([_(n + 1) / j]_A + sum_j e. (M...n)A))
6867adantll 428 . . . . . . . . . 10 |- (((K e. ZZ /\ n e. (ZZ>=` M)) /\ A.j e. (M...(n + 1))A e. CC) -> sum_j e. (M...(n + 1))A = ([_(n + 1) / j]_A + sum_j e. (M...n)A))
6968adantr 425 . . . . . . . . 9 |- ((((K e. ZZ /\ n e. (ZZ>=` M)) /\ A.j e. (M...(n + 1))A e. CC) /\ sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A) -> sum_j e. (M...(n + 1))A = ([_(n + 1) / j]_A + sum_j e. (M...n)A))
70 zre 7348 . . . . . . . . . . . . . . . . . 18 |- (M e. ZZ -> M e. RR)
7133, 70syl 12 . . . . . . . . . . . . . . . . 17 |- (n e. (ZZ>=` M) -> M e. RR)
72 zre 7348 . . . . . . . . . . . . . . . . . 18 |- (n e. ZZ -> n e. RR)
7334, 72syl 12 . . . . . . . . . . . . . . . . 17 |- (n e. (ZZ>=` M) -> n e. RR)
74 eluzle 7594 . . . . . . . . . . . . . . . . 17 |- (n e. (ZZ>=` M) -> M <_ n)
75 letrp1 6994 . . . . . . . . . . . . . . . . 17 |- ((M e. RR /\ n e. RR /\ M <_ n) -> M <_ (n + 1))
7671, 73, 74, 75syl111anc 1100 . . . . . . . . . . . . . . . 16 |- (n e. (ZZ>=` M) -> M <_ (n + 1))
7776adantl 424 . . . . . . . . . . . . . . 15 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> M <_ (n + 1))
7833adantl 424 . . . . . . . . . . . . . . . 16 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> M e. ZZ)
7934peano2zdi 7376 . . . . . . . . . . . . . . . . 17 |- (n e. (ZZ>=` M) -> (n + 1) e. ZZ)
8079adantl 424 . . . . . . . . . . . . . . . 16 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> (n + 1) e. ZZ)
81 simpl 346 . . . . . . . . . . . . . . . 16 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> K e. ZZ)
82 lesub2 6850 . . . . . . . . . . . . . . . . 17 |- ((M e. RR /\ (n + 1) e. RR /\ K e. RR) -> (M <_ (n + 1) <-> (K - (n + 1)) <_ (K - M)))
83 zre 7348 . . . . . . . . . . . . . . . . 17 |- ((n + 1) e. ZZ -> (n + 1) e. RR)
84 zre 7348 . . . . . . . . . . . . . . . . 17 |- (K e. ZZ -> K e. RR)
8582, 70, 83, 84syl3an 1139 . . . . . . . . . . . . . . . 16 |- ((M e. ZZ /\ (n + 1) e. ZZ /\ K e. ZZ) -> (M <_ (n + 1) <-> (K - (n + 1)) <_ (K - M)))
8678, 80, 81, 85syl111anc 1100 . . . . . . . . . . . . . . 15 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> (M <_ (n + 1) <-> (K - (n + 1)) <_ (K - M)))
8777, 86mpbid 212 . . . . . . . . . . . . . 14 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> (K - (n + 1)) <_ (K - M))
88 zsubcl 7377 . . . . . . . . . . . . . . . 16 |- ((K e. ZZ /\ (n + 1) e. ZZ) -> (K - (n + 1)) e. ZZ)
8988, 79sylan2 500 . . . . . . . . . . . . . . 15 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> (K - (n + 1)) e. ZZ)
9024, 33sylan2 500 . . . . . . . . . . . . . . 15 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> (K - M) e. ZZ)
91 eluz 7595 . . . . . . . . . . . . . . 15 |- (((K - (n + 1)) e. ZZ /\ (K - M) e. ZZ) -> ((K - M) e. (ZZ>=` (K - (n + 1))) <-> (K - (n + 1)) <_ (K - M)))
9289, 90, 91syl11anc 524 . . . . . . . . . . . . . 14 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> ((K - M) e. (ZZ>=` (K - (n + 1))) <-> (K - (n + 1)) <_ (K - M)))
9387, 92mpbird 213 . . . . . . . . . . . . 13 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> (K - M) e. (ZZ>=` (K - (n + 1))))
9493adantr 425 . . . . . . . . . . . 12 |- (((K e. ZZ /\ n e. (ZZ>=` M)) /\ A.j e. (M...(n + 1))A e. CC) -> (K - M) e. (ZZ>=` (K - (n + 1))))
95 zleltp1 7391 . . . . . . . . . . . . . . . . 17 |- ((M e. ZZ /\ n e. ZZ) -> (M <_ n <-> M < (n + 1)))
9633, 34, 95syl11anc 524 . . . . . . . . . . . . . . . 16 |- (n e. (ZZ>=` M) -> (M <_ n <-> M < (n + 1)))
9774, 96mpbid 212 . . . . . . . . . . . . . . 15 |- (n e. (ZZ>=` M) -> M < (n + 1))
9897adantl 424 . . . . . . . . . . . . . 14 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> M < (n + 1))
99 ltsub2 6852 . . . . . . . . . . . . . . . 16 |- ((M e. RR /\ (n + 1) e. RR /\ K e. RR) -> (M < (n + 1) <-> (K - (n + 1)) < (K - M)))
10099, 70, 83, 84syl3an 1139 . . . . . . . . . . . . . . 15 |- ((M e. ZZ /\ (n + 1) e. ZZ /\ K e. ZZ) -> (M < (n + 1) <-> (K - (n + 1)) < (K - M)))
10178, 80, 81, 100syl111anc 1100 . . . . . . . . . . . . . 14 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> (M < (n + 1) <-> (K - (n + 1)) < (K - M)))
10298, 101mpbid 212 . . . . . . . . . . . . 13 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> (K - (n + 1)) < (K - M))
103102adantr 425 . . . . . . . . . . . 12 |- (((K e. ZZ /\ n e. (ZZ>=` M)) /\ A.j e. (M...(n + 1))A e. CC) -> (K - (n + 1)) < (K - M))
104 fzrevral 7698 . . . . . . . . . . . . . . 15 |- ((M e. ZZ /\ (n + 1) e. ZZ /\ K e. ZZ) -> (A.j e. (M...(n + 1))A e. CC <-> A.k e. ((K - (n + 1))...(K - M))[(K - k) / j]A e. CC))
10578, 80, 81, 104syl111anc 1100 . . . . . . . . . . . . . 14 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> (A.j e. (M...(n + 1))A e. CC <-> A.k e. ((K - (n + 1))...(K - M))[(K - k) / j]A e. CC))
10613ralbii 2127 . . . . . . . . . . . . . 14 |- (A.k e. ((K - (n + 1))...(K - M))[(K - k) / j]A e. CC <-> A.k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A e. CC)
107105, 106syl6bb 595 . . . . . . . . . . . . 13 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> (A.j e. (M...(n + 1))A e. CC <-> A.k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A e. CC))
108107biimpa 460 . . . . . . . . . . . 12 |- (((K e. ZZ /\ n e. (ZZ>=` M)) /\ A.j e. (M...(n + 1))A e. CC) -> A.k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A e. CC)
109 fsum1ps 8278 . . . . . . . . . . . 12 |- (((K - M) e. (ZZ>=` (K - (n + 1))) /\ (K - (n + 1)) < (K - M) /\ A.k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A e. CC) -> sum_k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A = ([_(K - (n + 1)) / k]_[_(K - k) / j]_A + sum_k e. (((K - (n + 1)) + 1)...(K - M))[_(K - k) / j]_A))
11094, 103, 108, 109syl111anc 1100 . . . . . . . . . . 11 |- (((K e. ZZ /\ n e. (ZZ>=` M)) /\ A.j e. (M...(n + 1))A e. CC) -> sum_k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A = ([_(K - (n + 1)) / k]_[_(K - k) / j]_A + sum_k e. (((K - (n + 1)) + 1)...(K - M))[_(K - k) / j]_A))
111 ax1cn 6422 . . . . . . . . . . . . . . . . 17 |- 1 e. CC
112 nppcan2 6637 . . . . . . . . . . . . . . . . 17 |- ((K e. CC /\ n e. CC /\ 1 e. CC) -> ((K - (n + 1)) + 1) = (K - n))
113111, 112mp3an3 1180 . . . . . . . . . . . . . . . 16 |- ((K e. CC /\ n e. CC) -> ((K - (n + 1)) + 1) = (K - n))
11434, 43syl 12 . . . . . . . . . . . . . . . 16 |- (n e. (ZZ>=` M) -> n e. CC)
115113, 2, 114syl2an 503 . . . . . . . . . . . . . . 15 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> ((K - (n + 1)) + 1) = (K - n))
116115opreq1d 4897 . . . . . . . . . . . . . 14 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> (((K - (n + 1)) + 1)...(K - M)) = ((K - n)...(K - M)))
117116sumeq1d 8250 . . . . . . . . . . . . 13 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> sum_k e. (((K - (n + 1)) + 1)...(K - M))[_(K - k) / j]_A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A)
118117opreq2d 4898 . . . . . . . . . . . 12 |- ((K e. ZZ /\ n e. (ZZ>=` M)) -> ([_(K - (n + 1)) / k]_[_(K - k) / j]_A + sum_k e. (((K - (n + 1)) + 1)...(K - M))[_(K - k) / j]_A) = ([_(K - (n + 1)) / k]_[_(K - k) / j]_A + sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A))
119118adantr 425 . . . . . . . . . . 11 |- (((K e. ZZ /\ n e. (ZZ>=` M)) /\ A.j e. (M...(n + 1))A e. CC) -> ([_(K - (n + 1)) / k]_[_(K - k) / j]_A + sum_k e. (((K - (n + 1)) + 1)...(K - M))[_(K - k) / j]_A) = ([_(K - (n + 1)) / k]_[_(K - k) / j]_A + sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A))
120110, 119eqtrd 1925 . . . . . . . . . 10 |- (((K e. ZZ /\ n e. (ZZ>=` M)) /\ A.j e. (M...(n + 1))A e. CC) -> sum_k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A = ([_(K - (n + 1)) / k]_[_(K - k) / j]_A + sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A))
121120adantr 425 . . . . . . . . 9 |- ((((K e. ZZ /\ n e. (ZZ>=` M)) /\ A.j e. (M...(n + 1))A e. CC) /\ sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A) -> sum_k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A = ([_(K - (n + 1)) / k]_[_(K - k) / j]_A + sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A))
12255, 69, 1213eqtr4d 1937 . . . . . . . 8 |- ((((K e. ZZ /\ n e. (ZZ>=` M)) /\ A.j e. (M...(n + 1))A e. CC) /\ sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A) -> sum_j e. (M...(n + 1))A = sum_k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A)
123122exp41 413 . . . . . . 7 |- (K e. ZZ -> (n e. (ZZ>=` M) -> (A.j e. (M...(n + 1))A e. CC -> (sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A -> sum_j e. (M...(n + 1))A = sum_k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A))))
124123com12 14 . . . . . 6 |- (n e. (ZZ>=` M) -> (K e. ZZ -> (A.j e. (M...(n + 1))A e. CC -> (sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A -> sum_j e. (M...(n + 1))A = sum_k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A))))
125124imp3a 388 . . . . 5 |- (n e. (ZZ>=` M) -> ((K e. ZZ /\ A.j e. (M...(n + 1))A e. CC) -> (sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A -> sum_j e. (M...(n + 1))A = sum_k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A)))
126125a2d 16 . . . 4 |- (n e. (ZZ>=` M) -> (((K e. ZZ /\ A.j e. (M...(n + 1))A e. CC) -> sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A) -> ((K e. ZZ /\ A.j e. (M...(n + 1))A e. CC) -> sum_j e. (M...(n + 1))A = sum_k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A)))
12741, 126syld 30 . . 3 |- (n e. (ZZ>=` M) -> (((K e. ZZ /\ A.j e. (M...n)A e. CC) -> sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A) -> ((K e. ZZ /\ A.j e. (M...(n + 1))A e. CC) -> sum_j e. (M...(n + 1))A = sum_k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A)))
128 opreq2 4890 . . . . . 6 |- (m = M -> (M...m) = (M...M))
129128raleqdv 2269 . . . . 5 |- (m = M -> (A.j e. (M...m)A e. CC <-> A.j e. (M...M)A e. CC))
130129anbi2d 678 . . . 4 |- (m = M -> ((K e. ZZ /\ A.j e. (M...m)A e. CC) <-> (K e. ZZ /\ A.j e. (M...M)A e. CC)))
131128sumeq1d 8250 . . . . 5 |- (m = M -> sum_j e. (M...m)A = sum_j e. (M...M)A)
132 opreq2 4890 . . . . . . 7 |- (m = M -> (K - m) = (K - M))
133132opreq1d 4897 . . . . . 6 |- (m = M -> ((K - m)...(K - M)) = ((K - M)...(K - M)))
134133sumeq1d 8250 . . . . 5 |- (m = M -> sum_k e. ((K - m)...(K - M))[_(K - k) / j]_A = sum_k e. ((K - M)...(K - M))[_(K - k) / j]_A)
135131, 134eqeq12d 1899 . . . 4 |- (m = M -> (sum_j e. (M...m)A = sum_k e. ((K - m)...(K - M))[_(K - k) / j]_A <-> sum_j e. (M...M)A = sum_k e. ((K - M)...(K - M))[_(K - k) / j]_A))
136130, 135imbi12d 688 . . 3 |- (m = M -> (((K e. ZZ /\ A.j e. (M...m)A e. CC) -> sum_j e. (M...m)A = sum_k e. ((K - m)...(K - M))[_(K - k) / j]_A) <-> ((K e. ZZ /\ A.j e. (M...M)A e. CC) -> sum_j e. (M...M)A = sum_k e. ((K - M)...(K - M))[_(K - k) / j]_A)))
137 opreq2 4890 . . . . . 6 |- (m = n -> (M...m) = (M...n))
138137raleqdv 2269 . . . . 5 |- (m = n -> (A.j e. (M...m)A e. CC <-> A.j e. (M...n)A e. CC))
139138anbi2d 678 . . . 4 |- (m = n -> ((K e. ZZ /\ A.j e. (M...m)A e. CC) <-> (K e. ZZ /\ A.j e. (M...n)A e. CC)))
140137sumeq1d 8250 . . . . 5 |- (m = n -> sum_j e. (M...m)A = sum_j e. (M...n)A)
141 opreq2 4890 . . . . . . 7 |- (m = n -> (K - m) = (K - n))
142141opreq1d 4897 . . . . . 6 |- (m = n -> ((K - m)...(K - M)) = ((K - n)...(K - M)))
143142sumeq1d 8250 . . . . 5 |- (m = n -> sum_k e. ((K - m)...(K - M))[_(K - k) / j]_A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A)
144140, 143eqeq12d 1899 . . . 4 |- (m = n -> (sum_j e. (M...m)A = sum_k e. ((K - m)...(K - M))[_(K - k) / j]_A <-> sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A))
145139, 144imbi12d 688 . . 3 |- (m = n -> (((K e. ZZ /\ A.j e. (M...m)A e. CC) -> sum_j e. (M...m)A = sum_k e. ((K - m)...(K - M))[_(K - k) / j]_A) <-> ((K e. ZZ /\ A.j e. (M...n)A e. CC) -> sum_j e. (M...n)A = sum_k e. ((K - n)...(K - M))[_(K - k) / j]_A)))
146 opreq2 4890 . . . . . 6 |- (m = (n + 1) -> (M...m) = (M...(n + 1)))
147146raleqdv 2269 . . . . 5 |- (m = (n + 1) -> (A.j e. (M...m)A e. CC <-> A.j e. (M...(n + 1))A e. CC))
148147anbi2d 678 . . . 4 |- (m = (n + 1) -> ((K e. ZZ /\ A.j e. (M...m)A e. CC) <-> (K e. ZZ /\ A.j e. (M...(n + 1))A e. CC)))
149146sumeq1d 8250 . . . . 5 |- (m = (n + 1) -> sum_j e. (M...m)A = sum_j e. (M...(n + 1))A)
150 opreq2 4890 . . . . . . 7 |- (m = (n + 1) -> (K - m) = (K - (n + 1)))
151150opreq1d 4897 . . . . . 6 |- (m = (n + 1) -> ((K - m)...(K - M)) = ((K - (n + 1))...(K - M)))
152151sumeq1d 8250 . . . . 5 |- (m = (n + 1) -> sum_k e. ((K - m)...(K - M))[_(K - k) / j]_A = sum_k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A)
153149, 152eqeq12d 1899 . . . 4 |- (m = (n + 1) -> (sum_j e. (M...m)A = sum_k e. ((K - m)...(K - M))[_(K - k) / j]_A <-> sum_j e. (M...(n + 1))A = sum_k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A))
154148, 153imbi12d 688 . . 3 |- (m = (n + 1) -> (((K e. ZZ /\ A.j e. (M...m)A e. CC) -> sum_j e. (M...m)A = sum_k e. ((K - m)...(K - M))[_(K - k) / j]_A) <-> ((K e. ZZ /\ A.j e. (M...(n + 1))A e. CC) -> sum_j e. (M...(n + 1))A = sum_k e. ((K - (n + 1))...(K - M))[_(K - k) / j]_A)))
155 opreq2 4890 . . . . . 6 |- (m = N -> (M...m) = (M...N))
156155raleqdv 2269 . . . . 5 |- (m = N -> (A.j e. (M...m)A e. CC <-> A.j e. (M...N)A e. CC))
157156anbi2d 678 . . . 4 |- (m = N -> ((K e. ZZ /\ A.j e. (M...m)A e. CC) <-> (K e. ZZ /\ A.j e. (M...N)A e. CC)))
158155sumeq1d 8250 . . . . 5 |- (m = N -> sum_j e. (M...m)A = sum_j e. (M...N)A)
159 opreq2 4890 . . . . . . 7 |- (m = N -> (K - m) = (K - N))
160159opreq1d 4897 . . . . . 6 |- (m = N -> ((K - m)...(K - M)) = ((K - N)...(K - M)))
161160sumeq1d 8250 . . . . 5 |- (m = N -> sum_k e. ((K - m)...(K - M))[_(K - k) / j]_A = sum_k e. ((K - N)...(K - M))[_(K - k) / j]_A)
162158, 161eqeq12d 1899 . . . 4 |- (m = N -> (sum_j e. (M...m)A = sum_k e. ((K - m)...(K - M))[_(K - k) / j]_A <-> sum_j e. (M...N)A = sum_k e. ((K - N)...(K - M))[_(K - k) / j]_A))
163157, 162imbi12d 688 . . 3 |- (m = N -> (((K e. ZZ /\ A.j e. (M...m)A e. CC) -> sum_j e. (M...m)A = sum_k e. ((K - m)...(K - M))[_(K - k) / j]_A) <-> ((K e. ZZ /\ A.j e. (M...N)A e. CC) -> sum_j e. (M...N)A = sum_k e. ((K - N)...(K - M))[_(K - k) / j]_A)))
16432, 127, 136, 145, 154, 163uzind4ALT 7620 . 2 |- (N e. (ZZ>=` M) -> ((K e. ZZ /\ A.j e. (M...N)A e. CC) -> sum_j e. (M...N)A = sum_k e. ((K - N)...(K - M))[_(K - k) / j]_A))
1651643impib 1065 1 |- ((N e. (ZZ>=` M) /\ K e. ZZ /\ A.j e. (M...N)A e. CC) -> sum_j e. (M...N)A = sum_k e. ((K - N)...(K - M))[_(K - k) / j]_A)
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  _Vcvv 2292  [_csb 2540   C_ wss 2593   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  1c1 6387   + caddc 6389   - cmin 6445   <_ cle 6448  ZZcz 6451   < clt 6653  ZZ>=cuz 7586  ...cfz 7637  sum_csu 8239
This theorem is referenced by:  fsumrev2 8290  fsumshft 8291  fsum0diag2 8521
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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