Metamath Proof Explorer


Theorem esumfsupre

Description: Formulating an extended sum over integers using the recursive sequence builder. This version is limited to real-valued functions. (Contributed by Thierry Arnoux, 19-Oct-2017)

Ref Expression
Hypothesis esumfsup.1 _ k F
Assertion esumfsupre F : 0 +∞ * k F k = sup ran seq 1 + F * <

Proof

Step Hyp Ref Expression
1 esumfsup.1 _ k F
2 icossicc 0 +∞ 0 +∞
3 fss F : 0 +∞ 0 +∞ 0 +∞ F : 0 +∞
4 2 3 mpan2 F : 0 +∞ F : 0 +∞
5 1 esumfsup F : 0 +∞ * k F k = sup ran seq 1 + 𝑒 F * <
6 4 5 syl F : 0 +∞ * k F k = sup ran seq 1 + 𝑒 F * <
7 1zzd F : 0 +∞ 1
8 elnnuz x x 1
9 ffvelrn F : 0 +∞ x F x 0 +∞
10 8 9 sylan2br F : 0 +∞ x 1 F x 0 +∞
11 ge0addcl x 0 +∞ y 0 +∞ x + y 0 +∞
12 11 adantl F : 0 +∞ x 0 +∞ y 0 +∞ x + y 0 +∞
13 rge0ssre 0 +∞
14 simprl F : 0 +∞ x 0 +∞ y 0 +∞ x 0 +∞
15 13 14 sselid F : 0 +∞ x 0 +∞ y 0 +∞ x
16 simprr F : 0 +∞ x 0 +∞ y 0 +∞ y 0 +∞
17 13 16 sselid F : 0 +∞ x 0 +∞ y 0 +∞ y
18 rexadd x y x + 𝑒 y = x + y
19 18 eqcomd x y x + y = x + 𝑒 y
20 15 17 19 syl2anc F : 0 +∞ x 0 +∞ y 0 +∞ x + y = x + 𝑒 y
21 7 10 12 20 seqfeq3 F : 0 +∞ seq 1 + F = seq 1 + 𝑒 F
22 21 rneqd F : 0 +∞ ran seq 1 + F = ran seq 1 + 𝑒 F
23 22 supeq1d F : 0 +∞ sup ran seq 1 + F * < = sup ran seq 1 + 𝑒 F * <
24 6 23 eqtr4d F : 0 +∞ * k F k = sup ran seq 1 + F * <