Step |
Hyp |
Ref |
Expression |
1 |
|
seqfeq3.m |
|- ( ph -> M e. ZZ ) |
2 |
|
seqfeq3.f |
|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S ) |
3 |
|
seqfeq3.cl |
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) |
4 |
|
seqfeq3.id |
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) ) |
5 |
|
seqfn |
|- ( M e. ZZ -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) ) |
6 |
1 5
|
syl |
|- ( ph -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) ) |
7 |
|
seqfn |
|- ( M e. ZZ -> seq M ( Q , F ) Fn ( ZZ>= ` M ) ) |
8 |
1 7
|
syl |
|- ( ph -> seq M ( Q , F ) Fn ( ZZ>= ` M ) ) |
9 |
|
simpr |
|- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> a e. ( ZZ>= ` M ) ) |
10 |
|
simpll |
|- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ x e. ( M ... a ) ) -> ph ) |
11 |
|
elfzuz |
|- ( x e. ( M ... a ) -> x e. ( ZZ>= ` M ) ) |
12 |
11
|
adantl |
|- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ x e. ( M ... a ) ) -> x e. ( ZZ>= ` M ) ) |
13 |
10 12 2
|
syl2anc |
|- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ x e. ( M ... a ) ) -> ( F ` x ) e. S ) |
14 |
3
|
adantlr |
|- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) |
15 |
4
|
adantlr |
|- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) ) |
16 |
9 13 14 15
|
seqfeq4 |
|- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` a ) = ( seq M ( Q , F ) ` a ) ) |
17 |
6 8 16
|
eqfnfvd |
|- ( ph -> seq M ( .+ , F ) = seq M ( Q , F ) ) |