Theorem cvgcmp3cetlem2 7125

Description: Lemma for cvgcmp3cet 7126.

Hypotheses

Ref	Expression
cvgcmp3cetlem2.1	|- A e. V
cvgcmp3cetlem2.2	|- B e. NN
cvgcmp3cetlem2.3	|- (ph <-> ((F:NN-->RR /\ A.x e. NN 0 <_ (F` x)) /\ ( + seq1 F) ~~> A))

Assertion

Ref	Expression
cvgcmp3cetlem2	|- ((ph /\ ((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.x e. NN (B < x -> (abs` (G` x)) <_ (C x. (F` x)))))) -> E.y( + seq1 G) ~~> y)

Distinct variable groups:   x,y,B   x,F,y   x,G,y   x,C,y   ph,y

Proof of Theorem cvgcmp3cetlem2

Step	Hyp	Ref	Expression
1		fveq1 3708	. . . . . . . . . . 11 |- (F = if(ph, F, (NN X. {0})) -> (F` z) = (if(ph, F, (NN X. {0}))` z))
2	1	opreq2d 3961	. . . . . . . . . 10 |- (F = if(ph, F, (NN X. {0})) -> (C x. (F` z)) = (C x. (if(ph, F, (NN X. {0}))` z)))
3	2	breq2d 2620	. . . . . . . . 9 |- (F = if(ph, F, (NN X. {0})) -> ((abs` (G` z)) <_ (C x. (F` z)) <-> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z))))
4	3	imbi2d 610	. . . . . . . 8 |- (F = if(ph, F, (NN X. {0})) -> ((B < z -> (abs` (G` z)) <_ (C x. (F` z))) <-> (B < z -> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z)))))
5	4	ralbidv 1655	. . . . . . 7 |- (F = if(ph, F, (NN X. {0})) -> (A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (F` z))) <-> A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z)))))
6	5	anbi2d 614	. . . . . 6 |- (F = if(ph, F, (NN X. {0})) -> ((G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (F` z)))) <-> (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z))))))
7	6	anbi2d 614	. . . . 5 |- (F = if(ph, F, (NN X. {0})) -> (((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (F` z))))) <-> ((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z)))))))
8	7	imbi1d 611	. . . 4 |- (F = if(ph, F, (NN X. {0})) -> ((((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (F` z))))) -> E.y( + seq1 G) ~~> y) <-> (((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z))))) -> E.y( + seq1 G) ~~> y)))
9		cvgcmp3cetlem2.1	. . . . . 6 |- A e. V
10		0re 5412	. . . . . . 7 |- 0 e. RR
11	10	elisseti 1809	. . . . . 6 |- 0 e. V
12	9, 11	ifex 2390	. . . . 5 |- if(ph, A, 0) e. V
13		cvgcmp3cetlem2.2	. . . . 5 |- B e. NN
14		feq1 3606	. . . . . . . . . . 11 |- (F = if(ph, F, (NN X. {0})) -> (F:NN-->RR <-> if(ph, F, (NN X. {0})):NN-->RR))
15	1	breq2d 2620	. . . . . . . . . . . 12 |- (F = if(ph, F, (NN X. {0})) -> (0 <_ (F` z) <-> 0 <_ (if(ph, F, (NN X. {0}))` z)))
16	15	ralbidv 1655	. . . . . . . . . . 11 |- (F = if(ph, F, (NN X. {0})) -> (A.z e. NN 0 <_ (F` z) <-> A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)))
17	14, 16	anbi12d 626	. . . . . . . . . 10 |- (F = if(ph, F, (NN X. {0})) -> ((F:NN-->RR /\ A.z e. NN 0 <_ (F` z)) <-> (if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z))))
18		opreq2 3954	. . . . . . . . . . 11 |- (F = if(ph, F, (NN X. {0})) -> ( + seq1 F) = ( + seq1 if(ph, F, (NN X. {0}))))
19	18	breq1d 2619	. . . . . . . . . 10 |- (F = if(ph, F, (NN X. {0})) -> (( + seq1 F) ~~> A <-> ( + seq1 if(ph, F, (NN X. {0}))) ~~> A))
20	17, 19	anbi12d 626	. . . . . . . . 9 |- (F = if(ph, F, (NN X. {0})) -> (((F:NN-->RR /\ A.z e. NN 0 <_ (F` z)) /\ ( + seq1 F) ~~> A) <-> ((if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)) /\ ( + seq1 if(ph, F, (NN X. {0}))) ~~> A)))
21		cvgcmp3cetlem2.3	. . . . . . . . . 10 |- (ph <-> ((F:NN-->RR /\ A.x e. NN 0 <_ (F` x)) /\ ( + seq1 F) ~~> A))
22		fveq2 3709	. . . . . . . . . . . . . 14 |- (x = z -> (F` x) = (F` z))
23	22	breq2d 2620	. . . . . . . . . . . . 13 |- (x = z -> (0 <_ (F` x) <-> 0 <_ (F` z)))
24	23	cbvralv 1791	. . . . . . . . . . . 12 |- (A.x e. NN 0 <_ (F` x) <-> A.z e. NN 0 <_ (F` z))
25	24	anbi2i 479	. . . . . . . . . . 11 |- ((F:NN-->RR /\ A.x e. NN 0 <_ (F` x)) <-> (F:NN-->RR /\ A.z e. NN 0 <_ (F` z)))
26	25	anbi1i 480	. . . . . . . . . 10 |- (((F:NN-->RR /\ A.x e. NN 0 <_ (F` x)) /\ ( + seq1 F) ~~> A) <-> ((F:NN-->RR /\ A.z e. NN 0 <_ (F` z)) /\ ( + seq1 F) ~~> A))
27	21, 26	bitr 173	. . . . . . . . 9 |- (ph <-> ((F:NN-->RR /\ A.z e. NN 0 <_ (F` z)) /\ ( + seq1 F) ~~> A))
28	20, 27	syl5bb 530	. . . . . . . 8 |- (F = if(ph, F, (NN X. {0})) -> (ph <-> ((if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)) /\ ( + seq1 if(ph, F, (NN X. {0}))) ~~> A)))
29		breq2 2613	. . . . . . . . 9 |- (A = if(ph, A, 0) -> (( + seq1 if(ph, F, (NN X. {0}))) ~~> A <-> ( + seq1 if(ph, F, (NN X. {0}))) ~~> if(ph, A, 0)))
30	29	anbi2d 614	. . . . . . . 8 |- (A = if(ph, A, 0) -> (((if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)) /\ ( + seq1 if(ph, F, (NN X. {0}))) ~~> A) <-> ((if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)) /\ ( + seq1 if(ph, F, (NN X. {0}))) ~~> if(ph, A, 0))))
31		feq1 3606	. . . . . . . . . 10 |- ((NN X. {0}) = if(ph, F, (NN X. {0})) -> ((NN X. {0}):NN-->RR <-> if(ph, F, (NN X. {0})):NN-->RR))
32		fveq1 3708	. . . . . . . . . . . 12 |- ((NN X. {0}) = if(ph, F, (NN X. {0})) -> ((NN X. {0})` z) = (if(ph, F, (NN X. {0}))` z))
33	32	breq2d 2620	. . . . . . . . . . 11 |- ((NN X. {0}) = if(ph, F, (NN X. {0})) -> (0 <_ ((NN X. {0})` z) <-> 0 <_ (if(ph, F, (NN X. {0}))` z)))
34	33	ralbidv 1655	. . . . . . . . . 10 |- ((NN X. {0}) = if(ph