Metamath Proof Explorer


Theorem seqfeq3

Description: Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015) (Revised by Mario Carneiro, 25-Apr-2016)

Ref Expression
Hypotheses seqfeq3.m φ M
seqfeq3.f φ x M F x S
seqfeq3.cl φ x S y S x + ˙ y S
seqfeq3.id φ x S y S x + ˙ y = x Q y
Assertion seqfeq3 φ seq M + ˙ F = seq M Q F

Proof

Step Hyp Ref Expression
1 seqfeq3.m φ M
2 seqfeq3.f φ x M F x S
3 seqfeq3.cl φ x S y S x + ˙ y S
4 seqfeq3.id φ x S y S x + ˙ y = x Q y
5 seqfn M seq M + ˙ F Fn M
6 1 5 syl φ seq M + ˙ F Fn M
7 seqfn M seq M Q F Fn M
8 1 7 syl φ seq M Q F Fn M
9 simpr φ a M a M
10 simpll φ a M x M a φ
11 elfzuz x M a x M
12 11 adantl φ a M x M a x M
13 10 12 2 syl2anc φ a M x M a F x S
14 3 adantlr φ a M x S y S x + ˙ y S
15 4 adantlr φ a M x S y S x + ˙ y = x Q y
16 9 13 14 15 seqfeq4 φ a M seq M + ˙ F a = seq M Q F a
17 6 8 16 eqfnfvd φ seq M + ˙ F = seq M Q F