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Theorem fsumadd 8282
Description: The sum of two finite sums.
Assertion
Ref Expression
fsumadd |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. CC /\ B e. CC)) -> sum_k e. (M...N)(A + B) = (sum_k e. (M...N)A + sum_k e. (M...N)B))
Distinct variable groups:   k,M   k,N
Proof of Theorem fsumadd
StepHypRef Expression
1 opreq2 4890 . . . . 5 |- (j = M -> (M...j) = (M...M))
21raleqdv 2269 . . . 4 |- (j = M -> (A.k e. (M...j)(A e. CC /\ B e. CC) <-> A.k e. (M...M)(A e. CC /\ B e. CC)))
31sumeq1d 8250 . . . . 5 |- (j = M -> sum_k e. (M...j)(A + B) = sum_k e. (M...M)(A + B))
41sumeq1d 8250 . . . . . 6 |- (j = M -> sum_k e. (M...j)A = sum_k e. (M...M)A)
51sumeq1d 8250 . . . . . 6 |- (j = M -> sum_k e. (M...j)B = sum_k e. (M...M)B)
64, 5opreq12d 4900 . . . . 5 |- (j = M -> (sum_k e. (M...j)A + sum_k e. (M...j)B) = (sum_k e. (M...M)A + sum_k e. (M...M)B))
73, 6eqeq12d 1899 . . . 4 |- (j = M -> (sum_k e. (M...j)(A + B) = (sum_k e. (M...j)A + sum_k e. (M...j)B) <-> sum_k e. (M...M)(A + B) = (sum_k e. (M...M)A + sum_k e. (M...M)B)))
82, 7imbi12d 688 . . 3 |- (j = M -> ((A.k e. (M...j)(A e. CC /\ B e. CC) -> sum_k e. (M...j)(A + B) = (sum_k e. (M...j)A + sum_k e. (M...j)B)) <-> (A.k e. (M...M)(A e. CC /\ B e. CC) -> sum_k e. (M...M)(A + B) = (sum_k e. (M...M)A + sum_k e. (M...M)B))))
9 opreq2 4890 . . . . 5 |- (j = m -> (M...j) = (M...m))
109raleqdv 2269 . . . 4 |- (j = m -> (A.k e. (M...j)(A e. CC /\ B e. CC) <-> A.k e. (M...m)(A e. CC /\ B e. CC)))
119sumeq1d 8250 . . . . 5 |- (j = m -> sum_k e. (M...j)(A + B) = sum_k e. (M...m)(A + B))
129sumeq1d 8250 . . . . . 6 |- (j = m -> sum_k e. (M...j)A = sum_k e. (M...m)A)
139sumeq1d 8250 . . . . . 6 |- (j = m -> sum_k e. (M...j)B = sum_k e. (M...m)B)
1412, 13opreq12d 4900 . . . . 5 |- (j = m -> (sum_k e. (M...j)A + sum_k e. (M...j)B) = (sum_k e. (M...m)A + sum_k e. (M...m)B))
1511, 14eqeq12d 1899 . . . 4 |- (j = m -> (sum_k e. (M...j)(A + B) = (sum_k e. (M...j)A + sum_k e. (M...j)B) <-> sum_k e. (M...m)(A + B) = (sum_k e. (M...m)A + sum_k e. (M...m)B)))
1610, 15imbi12d 688 . . 3 |- (j = m -> ((A.k e. (M...j)(A e. CC /\ B e. CC) -> sum_k e. (M...j)(A + B) = (sum_k e. (M...j)A + sum_k e. (M...j)B)) <-> (A.k e. (M...m)(A e. CC /\ B e. CC) -> sum_k e. (M...m)(A + B) = (sum_k e. (M...m)A + sum_k e. (M...m)B))))
17 opreq2 4890 . . . . 5 |- (j = (m + 1) -> (M...j) = (M...(m + 1)))
1817raleqdv 2269 . . . 4 |- (j = (m + 1) -> (A.k e. (M...j)(A e. CC /\ B e. CC) <-> A.k e. (M...(m + 1))(A e. CC /\ B e. CC)))
1917sumeq1d 8250 . . . . 5 |- (j = (m + 1) -> sum_k e. (M...j)(A + B) = sum_k e. (M...(m + 1))(A + B))
2017sumeq1d 8250 . . . . . 6 |- (j = (m + 1) -> sum_k e. (M...j)A = sum_k e. (M...(m + 1))A)
2117sumeq1d 8250 . . . . . 6 |- (j = (m + 1) -> sum_k e. (M...j)B = sum_k e. (M...(m + 1))B)
2220, 21opreq12d 4900 . . . . 5 |- (j = (m + 1) -> (sum_k e. (M...j)A + sum_k e. (M...j)B) = (sum_k e. (M...(m + 1))A + sum_k e. (M...(m + 1))B))
2319, 22eqeq12d 1899 . . . 4 |- (j = (m + 1) -> (sum_k e. (M...j)(A + B) = (sum_k e. (M...j)A + sum_k e. (M...j)B) <-> sum_k e. (M...(m + 1))(A + B) = (sum_k e. (M...(m + 1))A + sum_k e. (M...(m + 1))B)))
2418, 23imbi12d 688 . . 3 |- (j = (m + 1) -> ((A.k e. (M...j)(A e. CC /\ B e. CC) -> sum_k e. (M...j)(A + B) = (sum_k e. (M...j)A + sum_k e. (M...j)B)) <-> (A.k e. (M...(m + 1))(A e. CC /\ B e. CC) -> sum_k e. (M...(m + 1))(A + B) = (sum_k e. (M...(m + 1))A + sum_k e. (M...(m + 1))B))))
25 opreq2 4890 . . . . 5 |- (j = N -> (M...j) = (M...N))
2625raleqdv 2269 . . . 4 |- (j = N -> (A.k e. (M...j)(A e. CC /\ B e. CC) <-> A.k e. (M...N)(A e. CC /\ B e. CC)))
2725sumeq1d 8250 . . . . 5 |- (j = N -> sum_k e. (M...j)(A + B) = sum_k e. (M...N)(A + B))
2825sumeq1d 8250 . . . . . 6 |- (j = N -> sum_k e. (M...j)A = sum_k e. (M...N)A)
2925sumeq1d 8250 . . . . . 6 |- (j = N -> sum_k e. (M...j)B = sum_k e. (M...N)B)
3028, 29opreq12d 4900 . . . . 5 |- (j = N -> (sum_k e. (M...j)A + sum_k e. (M...j)B) = (sum_k e. (M...N)A + sum_k e. (M...N)B))
3127, 30eqeq12d 1899 . . . 4 |- (j = N -> (sum_k e. (M...j)(A + B) = (sum_k e. (M...j)A + sum_k e. (M...j)B) <-> sum_k e. (M...N)(A + B) = (sum_k e. (M...N)A + sum_k e. (M...N)B)))
3226, 31imbi12d 688 . . 3 |- (j = N -> ((A.k e. (M...j)(A e. CC /\ B e. CC) -> sum_k e. (M...j)(A + B) = (sum_k e. (M...j)A + sum_k e. (M...j)B)) <-> (A.k e. (M...N)(A e. CC /\ B e. CC) -> sum_k e. (M...N)(A + B) = (sum_k e. (M...N)A + sum_k e. (M...N)B))))
33 csbopr12g 4912 . . . . . 6 |- (M e. ZZ -> [_M / k]_(A + B) = ([_M / k]_A + [_M / k]_B))
3433adantr 425 . . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)(A e. CC /\ B e. CC)) -> [_M / k]_(A + B) = ([_M / k]_A + [_M / k]_B))
35 fsum1s 8269 . . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)(A + B) e. CC) -> sum_k e. (M...M)(A + B) = [_M / k]_(A + B))
36 addcl 6454 . . . . . . 7 |- ((A e. CC /\ B e. CC) -> (A + B) e. CC)
3736ralimi 2168 . . . . . 6 |- (A.k e. (M...M)(A e. CC /\ B e. CC) -> A.k e. (M...M)(A + B) e. CC)
3835, 37sylan2 500 . . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)(A e. CC /\ B e. CC)) -> sum_k e. (M...M)(A + B) = [_M / k]_(A + B))
39 fsum1s 8269 . . . . . . 7 |- ((M e. ZZ /\ A.k e. (M...M)A e. CC) -> sum_k e. (M...M)A = [_M / k]_A)
40 simpl 346 . . . . . . . 8 |- ((A e. CC /\ B e. CC) -> A e. CC)
4140ralimi 2168 . . . . . . 7 |- (A.k e. (M...M)(A e. CC /\ B e. CC) -> A.k e. (M...M)A e. CC)
4239, 41sylan2 500 . . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)(A e. CC /\ B e. CC)) -> sum_k e. (M...M)A = [_M / k]_A)
43 fsum1s 8269 . . . . . . 7 |- ((M e. ZZ /\ A.k e. (M...M)B e. CC) -> sum_k e. (M...M)B = [_M / k]_B)
44 simpr 350 . . . . . . . 8 |- ((A e. CC /\ B e. CC) -> B e. CC)
4544ralimi 2168 . . . . . . 7 |- (A.k e. (M...M)(A e. CC /\ B e. CC) -> A.k e. (M...M)B e. CC)
4643, 45sylan2 500 . . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)(A e. CC /\ B e. CC)) -> sum_k e. (M...M)B = [_M / k]_B)
4742, 46opreq12d 4900 . . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)(A e. CC /\ B e. CC)) -> (sum_k e. (M...M)A + sum_k e. (M...M)B) = ([_M / k]_A + [_M / k]_B))
4834, 38, 473eqtr4d 1937 . . . 4 |- ((M e. ZZ /\ A.k e. (M...M)(A e. CC /\ B e. CC)) -> sum_k e. (M...M)(A + B) = (sum_k e. (M...M)A + sum_k e. (M...M)B))
4948ex 402 . . 3 |- (M e. ZZ -> (A.k e. (M...M)(A e. CC /\ B e. CC) -> sum_k e. (M...M)(A + B) = (sum_k e. (M...M)A + sum_k e. (M...M)B)))
50 eluzel2 7593 . . . . . . . . 9 |- (m e. (ZZ>=` M) -> M e. ZZ)
51 eluzelz 7592 . . . . . . . . 9 |- (m e. (ZZ>=` M) -> m e. ZZ)
52 fzssp1 7679 . . . . . . . . 9 |- ((M e. ZZ /\ m e. ZZ) -> (M...m) C_ (M...(m + 1)))
5350, 51, 52syl11anc 524 . . . . . . . 8 |- (m e. (ZZ>=` M) -> (M...m) C_ (M...(m + 1)))
5453sseld 2619 . . . . . . 7 |- (m e. (ZZ>=` M) -> (k e. (M...m) -> k e. (M...(m + 1))))
5554imim1d 33 . . . . . 6 |- (m e. (ZZ>=` M) -> ((k e. (M...(m + 1)) -> (A e. CC /\ B e. CC)) -> (k e. (M...m) -> (A e. CC /\ B e. CC))))
5655ralimdv2 2173 . . . . 5 |- (m e. (ZZ>=` M) -> (A.k e. (M...(m + 1))(A e. CC /\ B e. CC) -> A.k e. (M...m)(A e. CC /\ B e. CC)))
5756imim1d 33 . . . 4 |- (m e. (ZZ>=` M) -> ((A.k e. (M...m)(A e. CC /\ B e. CC) -> sum_k e. (M...m)(A + B) = (sum_k e. (M...m)A + sum_k e. (M...m)B)) -> (A.k e. (M...(m + 1))(A e. CC /\ B e. CC) -> sum_k e. (M...m)(A + B) = (sum_k e. (M...m)A + sum_k e. (M...m)B))))
58 opreq1 4889 . . . . . . . 8 |- (sum_k e. (M...m)(A + B) = (sum_k e. (M...m)A + sum_k e. (M...m)B) -> (sum_k e. (M...m)(A + B) + [_(m + 1) / k]_(A + B)) = ((sum_k e. (M...m)A + sum_k e. (M...m)B) + [_(m + 1) / k]_(A + B)))
5956imp 377 . . . . . . . . . . 11 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) -> A.k e. (M...m)(A e. CC /\ B e. CC))
60 fsumcl 8275 . . . . . . . . . . . . 13 |- ((m e. (ZZ>=` M) /\ A.k e. (M...m)A e. CC) -> sum_k e. (M...m)A e. CC)
6140ralimi 2168 . . . . . . . . . . . . 13 |- (A.k e. (M...m)(A e. CC /\ B e. CC) -> A.k e. (M...m)A e. CC)
6260, 61sylan2 500 . . . . . . . . . . . 12 |- ((m e. (ZZ>=` M) /\ A.k e. (M...m)(A e. CC /\ B e. CC)) -> sum_k e. (M...m)A e. CC)
63 fsumcl 8275 . . . . . . . . . . . . 13 |- ((m e. (ZZ>=` M) /\ A.k e. (M...m)B e. CC) -> sum_k e. (M...m)B e. CC)
6444ralimi 2168 . . . . . . . . . . . . 13 |- (A.k e. (M...m)(A e. CC /\ B e. CC) -> A.k e. (M...m)B e. CC)
6563, 64sylan2 500 . . . . . . . . . . . 12 |- ((m e. (ZZ>=` M) /\ A.k e. (M...m)(A e. CC /\ B e. CC)) -> sum_k e. (M...m)B e. CC)
6662, 65jca 310 . . . . . . . . . . 11 |- ((m e. (ZZ>=` M) /\ A.k e. (M...m)(A e. CC /\ B e. CC)) -> (sum_k e. (M...m)A e. CC /\ sum_k e. (M...m)B e. CC))
6759, 66syldan 516 . . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) -> (sum_k e. (M...m)A e. CC /\ sum_k e. (M...m)B e. CC))
68 ra4csbela 2587 . . . . . . . . . . 11 |- (((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. CC) -> [_(m + 1) / k]_A e. CC)
69 peano2uz 7616 . . . . . . . . . . . 12 |- (m e. (ZZ>=` M) -> (m + 1) e. (ZZ>=` M))
70 eluzfz2 7659 . . . . . . . . . . . 12 |- ((m + 1) e. (ZZ>=` M) -> (m + 1) e. (M...(m + 1)))
7169, 70syl 12 . . . . . . . . . . 11 |- (m e. (ZZ>=` M) -> (m + 1) e. (M...(m + 1)))
7240ralimi 2168 . . . . . . . . . . 11 |- (A.k e. (M...(m + 1))(A e. CC /\ B e. CC) -> A.k e. (M...(m + 1))A e. CC)
7368, 71, 72syl2an 503 . . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) -> [_(m + 1) / k]_A e. CC)
74 ra4csbela 2587 . . . . . . . . . . 11 |- (((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))B e. CC) -> [_(m + 1) / k]_B e. CC)
7544ralimi 2168 . . . . . . . . . . 11 |- (A.k e. (M...(m + 1))(A e. CC /\ B e. CC) -> A.k e. (M...(m + 1))B e. CC)
7674, 71, 75syl2an 503 . . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) -> [_(m + 1) / k]_B e. CC)
77 add4 6491 . . . . . . . . . 10 |- (((sum_k e. (M...m)A e. CC /\ sum_k e. (M...m)B e. CC) /\ ([_(m + 1) / k]_A e. CC /\ [_(m + 1) / k]_B e. CC)) -> ((sum_k e. (M...m)A + sum_k e. (M...m)B) + ([_(m + 1) / k]_A + [_(m + 1) / k]_B)) = ((sum_k e. (M...m)A + [_(m + 1) / k]_A) + (sum_k e. (M...m)B + [_(m + 1) / k]_B)))
7867, 73, 76, 77syl12anc 1098 . . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) -> ((sum_k e. (M...m)A + sum_k e. (M...m)B) + ([_(m + 1) / k]_A + [_(m + 1) / k]_B)) = ((sum_k e. (M...m)A + [_(m + 1) / k]_A) + (sum_k e. (M...m)B + [_(m + 1) / k]_B)))
79 oprex 4907 . . . . . . . . . . 11 |- (m + 1) e. _V
80 csbopr12g 4912 . . . . . . . . . . 11 |- ((m + 1) e. _V -> [_(m + 1) / k]_(A + B) = ([_(m + 1) / k]_A + [_(m + 1) / k]_B))
8179, 80ax-mp 7 . . . . . . . . . 10 |- [_(m + 1) / k]_(A + B) = ([_(m + 1) / k]_A + [_(m + 1) / k]_B)
8281opreq2i 4893 . . . . . . . . 9 |- ((sum_k e. (M...m)A + sum_k e. (M...m)B) + [_(m + 1) / k]_(A + B)) = ((sum_k e. (M...m)A + sum_k e. (M...m)B) + ([_(m + 1) / k]_A + [_(m + 1) / k]_B))
8378, 82syl5eq 1940 . . . . . . . 8 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) -> ((sum_k e. (M...m)A + sum_k e. (M...m)B) + [_(m + 1) / k]_(A + B)) = ((sum_k e. (M...m)A + [_(m + 1) / k]_A) + (sum_k e. (M...m)B + [_(m + 1) / k]_B)))
8458, 83sylan9eqr 1951 . . . . . . 7 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) /\ sum_k e. (M...m)(A + B) = (sum_k e. (M...m)A + sum_k e. (M...m)B)) -> (sum_k e. (M...m)(A + B) + [_(m + 1) / k]_(A + B)) = ((sum_k e. (M...m)A + [_(m + 1) / k]_A) + (sum_k e. (M...m)B + [_(m + 1) / k]_B)))
85 fsump1s 8273 . . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A + B) e. CC) -> sum_k e. (M...(m + 1))(A + B) = (sum_k e. (M...m)(A + B) + [_(m + 1) / k]_(A + B)))
8636ralimi 2168 . . . . . . . . 9 |- (A.k e. (M...(m + 1))(A e. CC /\ B e. CC) -> A.k e. (M...(m + 1))(A + B) e. CC)
8785, 86sylan2 500 . . . . . . . 8 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) -> sum_k e. (M...(m + 1))(A + B) = (sum_k e. (M...m)(A + B) + [_(m + 1) / k]_(A + B)))
8887adantr 425 . . . . . . 7 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) /\ sum_k e. (M...m)(A + B) = (sum_k e. (M...m)A + sum_k e. (M...m)B)) -> sum_k e. (M...(m + 1))(A + B) = (sum_k e. (M...m)(A + B) + [_(m + 1) / k]_(A + B)))
89 fsump1s 8273 . . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. (M...(m + 1))A = (sum_k e. (M...m)A + [_(m + 1) / k]_A))
9089, 72sylan2 500 . . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) -> sum_k e. (M...(m + 1))A = (sum_k e. (M...m)A + [_(m + 1) / k]_A))
91 fsump1s 8273 . . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))B e. CC) -> sum_k e. (M...(m + 1))B = (sum_k e. (M...m)B + [_(m + 1) / k]_B))
9291, 75sylan2 500 . . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) -> sum_k e. (M...(m + 1))B = (sum_k e. (M...m)B + [_(m + 1) / k]_B))
9390, 92opreq12d 4900 . . . . . . . 8 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) -> (sum_k e. (M...(m + 1))A + sum_k e. (M...(m + 1))B) = ((sum_k e. (M...m)A + [_(m + 1) / k]_A) + (sum_k e. (M...m)B + [_(m + 1) / k]_B)))
9493adantr 425 . . . . . . 7 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) /\ sum_k e. (M...m)(A + B) = (sum_k e. (M...m)A + sum_k e. (M...m)B)) -> (sum_k e. (M...(m + 1))A + sum_k e. (M...(m + 1))B) = ((sum_k e. (M...m)A + [_(m + 1) / k]_A) + (sum_k e. (M...m)B + [_(m + 1) / k]_B)))
9584, 88, 943eqtr4d 1937 . . . . . 6 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. CC /\ B e. CC)) /\ sum_k e. (M...m)(A + B) = (sum_k e. (M...m)A + sum_k e. (M...m)B)) -> sum_k e. (M...(m + 1))(A + B) = (sum_k e. (M...(m + 1))A + sum_k e. (M...(m + 1))B))
9695exp31 407 . . . . 5 |- (m e. (ZZ>=` M) -> (A.k e. (M...(m + 1))(A e. CC /\ B e. CC) -> (sum_k e. (M...m)(A + B) = (sum_k e. (M...m)A + sum_k e. (M...m)B) -> sum_k e. (M...(m + 1))(A + B) = (sum_k e. (M...(m + 1))A + sum_k e. (M...(m + 1))B))))
9796a2d 16 . . . 4 |- (m e. (ZZ>=` M) -> ((A.k e. (M...(m + 1))(A e. CC /\ B e. CC) -> sum_k e. (M...m)(A + B) = (sum_k e. (M...m)A + sum_k e. (M...m)B)) -> (A.k e. (M...(m + 1))(A e. CC /\ B e. CC) -> sum_k e. (M...(m + 1))(A + B) = (sum_k e. (M...(m + 1))A + sum_k e. (M...(m + 1))B))))
9857, 97syld 30 . . 3 |- (m e. (ZZ>=` M) -> ((A.k e. (M...m)(A e. CC /\ B e. CC) -> sum_k e. (M...m)(A + B) = (sum_k e. (M...m)A + sum_k e. (M...m)B)) -> (A.k e. (M...(m + 1))(A e. CC /\ B e. CC) -> sum_k e. (M...(m + 1))(A + B) = (sum_k e. (M...(m + 1))A + sum_k e. (M...(m + 1))B))))
998, 16, 24, 32, 49, 98uzind4 7619 . 2 |- (N e. (ZZ>=` M) -> (A.k e. (M...N)(A e. CC /\ B e. CC) -> sum_k e. (M...N)(A + B) = (sum_k e. (M...N)A + sum_k e. (M...N)B)))
10099imp 377 1 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. CC /\ B e. CC)) -> sum_k e. (M...N)(A + B) = (sum_k e. (M...N)A + sum_k e. (M...N)B))
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  [_csb 2540   C_ wss 2593  ` cfv 3998  (class class class)co 4884  CCcc 6384  1c1 6387   + caddc 6389  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637  sum_csu 8239
This theorem is referenced by:  fsumcom 8288  binomlem5 8330  arisumi 8487  fsum0diaglem2 8519  efaddlem6 8605  csbrni 15832  trirni 15833
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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