Theorem caucvgsr 7724

Description: A Cauchy sequence of signed reals with a modulus of convergence converges to a signed real. This is basically Corollary 11.2.13 of [HoTT], p. (varies). The HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / n of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis).

This is similar to caucvgprpr 7634 but is for signed reals rather than positive reals.

Here is an outline of how we prove it:

1. Choose a lower bound for the sequence (see caucvgsrlembnd 7723).

2. Offset each element of the sequence so that each element of the resulting sequence is greater than one (greater than zero would not suffice, because the limit as well as the elements of the sequence need to be positive) (see caucvgsrlemofff 7719).

3. Since a signed real (element of R.) which is greater than zero can be mapped to a positive real (element of P.), perform that mapping on each element of the sequence and invoke caucvgprpr 7634 to get a limit (see caucvgsrlemgt1 7717).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7717).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7722). (Contributed by Jim Kingdon, 20-Jun-2021.)

Hypotheses

Ref	Expression
caucvgsr.f	|- ( ph -> F : N. --> R. )
caucvgsr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )

Assertion

Ref	Expression
caucvgsr	|- ( ph -> E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <R ( y +R x ) /\ y <R ( ( F ` k ) +R x ) ) ) ) )

Distinct variable groups:   j, F, k, l, u   n, F, k, l, u   x, F, y, j, k   ph, j, k, x   ph, n

Allowed substitution hints:   ph( y, u, l)

Proof of Theorem caucvgsr

Dummy variables f g h m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgsr.f	. 2 |- ( ph -> F : N. --> R. )
2		caucvgsr.cau	. 2 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
3		breq1 3970	. . . . . . . . . . . . 13 |- ( n = 1o -> ( n <N k <-> 1o <N k ) )
4		fveq2 5470	. . . . . . . . . . . . . . 15 |- ( n = 1o -> ( F ` n ) = ( F ` 1o ) )
5		opeq1 3743	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( n = 1o -> <. n , 1o >. = <. 1o , 1o >. )
6	5	eceq1d 6518	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( n = 1o -> [ <. n , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
7	6	fveq2d 5474	. . . . . . . . . . . . . . . . . . . . . 22 |- ( n = 1o -> ( *Q ` [ <. n , 1o >. ] ~Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q ) )
8	7	breq2d 3979	. . . . . . . . . . . . . . . . . . . . 21 |- ( n = 1o -> ( l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) )
9	8	abbidv 2275	. . . . . . . . . . . . . . . . . . . 20 |- ( n = 1o -> { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } )
10	7	breq1d 3977	. . . . . . . . . . . . . . . . . . . . 21 |- ( n = 1o -> ( ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u ) )
11	10	abbidv 2275	. . . . . . . . . . . . . . . . . . . 20 |- ( n = 1o -> { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } = { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } )
12	9, 11	opeq12d 3751	. . . . . . . . . . . . . . . . . . 19 |- ( n = 1o -> <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. = <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. )
13	12	oveq1d 5841	. . . . . . . . . . . . . . . . . 18 |- ( n = 1o -> ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) = ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) )
14	13	opeq1d 3749	. . . . . . . . . . . . . . . . 17 |- ( n = 1o -> <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. = <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. )
15	14	eceq1d 6518	. . . . . . . . . . . . . . . 16 |- ( n = 1o -> [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R = [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R )
16	15	oveq2d 5842	. . . . . . . . . . . . . . 15 |- ( n = 1o -> ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) = ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
17	4, 16	breq12d 3980	. . . . . . . . . . . . . 14 |- ( n = 1o -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) <-> ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
18	4, 15	oveq12d 5844	. . . . . . . . . . . . . . 15 |- ( n = 1o -> ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) = ( ( F ` 1o ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
19	18	breq2d 3979	. . . . . . . . . . . . . 14 |- ( n = 1o -> ( ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) <-> ( F ` k ) <R ( ( F ` 1o ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
20	17, 19	anbi12d 465	. . . . . . . . . . . . 13 |- ( n = 1o -> ( ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) <-> ( ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` 1o ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
21	3, 20	imbi12d 233	. . . . . . . . . . . 12 |- ( n = 1o -> ( ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) <-> ( 1o <N k -> ( ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` 1o ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) ) )
22	21	ralbidv 2457	. . . . . . . . . . 11 |- ( n = 1o -> ( A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) <-> A. k e. N. ( 1o <N k -> ( ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` 1o ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) ) )
23		1pi 7237	. . . . . . . . . . . 12 |- 1o e. N.
24	23	a1i 9	. . . . . . . . . . 11 |- ( ph -> 1o e. N. )
25	22, 2, 24	rspcdva 2821	. . . . . . . . . 10 |- ( ph -> A. k e. N. ( 1o <N k -> ( ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` 1o ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
26		simpl 108	. . . . . . . . . . . 12 |- ( ( ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` 1o ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) -> ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
27	26	imim2i 12	. . . . . . . . . . 11 |- ( ( 1o <N k -> ( ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` 1o ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) -> ( 1o <N k -> ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
28	27	ralimi 2520	. . . . . . . . . 10 |- ( A. k e. N. ( 1o <N k -> ( ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` 1o ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) -> A. k e. N. ( 1o <N k -> ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
29	25, 28	syl 14	. . . . . . . . 9 |- ( ph -> A. k e. N. ( 1o <N k -> ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
30		breq2 3971	. . . . . . . . . . 11 |- ( k = m -> ( 1o <N k <-> 1o <N m ) )
31		fveq2 5470	. . . . . . . . . . . . 13 |- ( k = m -> ( F ` k ) = ( F ` m ) )
32	31	oveq1d 5841	. . . . . . . . . . . 12 |- ( k = m -> ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) = ( ( F ` m ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
33	32	breq2d 3979	. . . . . . . . . . 11 |- ( k = m -> ( ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) <-> ( F ` 1o ) <R ( ( F ` m ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
34	30, 33	imbi12d 233	. . . . . . . . . 10 |- ( k = m -> ( ( 1o <N k -> ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) <-> ( 1o <N m -> ( F ` 1o ) <R ( ( F ` m ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
35	34	rspcv 2812	. . . . . . . . 9 |- ( m e. N. -> ( A. k e. N. ( 1o <N k -> ( F ` 1o ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) -> ( 1o <N m -> ( F ` 1o ) <R ( ( F ` m ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
36	29, 35	mpan9 279	. . . . . . . 8 |- ( ( ph /\ m e. N. ) -> ( 1o <N m -> ( F ` 1o ) <R ( ( F ` m ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
37		df-1nqqs 7273	. . . . . . . . . . . . . . . . . . . 20 |- 1Q = [ <. 1o , 1o >. ] ~Q
38	37	fveq2i 5473	. . . . . . . . . . . . . . . . . . 19 |- ( *Q ` 1Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q )
39		rec1nq 7317	. . . . . . . . . . . . . . . . . . 19 |- ( *Q ` 1Q ) = 1Q
40	38, 39	eqtr3i 2180	. . . . . . . . . . . . . . . . . 18 |- ( *Q ` [ <. 1o , 1o >. ] ~Q ) = 1Q
41	40	breq2i 3975	. . . . . . . . . . . . . . . . 17 |- ( l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) <-> l <Q 1Q )
42	41	abbii 2273	. . . . . . . . . . . . . . . 16 |- { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } = { l | l <Q 1Q }
43	40	breq1i 3974	. . . . . . . . . . . . . . . . 17 |- ( ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u <-> 1Q <Q u )
44	43	abbii 2273	. . . . . . . . . . . . . . . 16 |- { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } = { u | 1Q <Q u }
45	42, 44	opeq12i 3748	. . . . . . . . . . . . . . 15 |- <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
46		df-i1p 7389	. . . . . . . . . . . . . . 15 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
47	45, 46	eqtr4i 2181	. . . . . . . . . . . . . 14 |- <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. = 1P
48	47	oveq1i 5836	. . . . . . . . . . . . 13 |- ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) = ( 1P +P. 1P )
49	48	opeq1i 3746	. . . . . . . . . . . 12 |- <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. = <. ( 1P +P. 1P ) , 1P >.
50		eceq1 6517	. . . . . . . . . . . 12 |- ( <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. = <. ( 1P +P. 1P ) , 1P >. -> [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R )
51	49, 50	ax-mp 5	. . . . . . . . . . 11 |- [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
52		df-1r 7654	. . . . . . . . . . 11 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
53	51, 52	eqtr4i 2181	. . . . . . . . . 10 |- [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R = 1R
54	53	oveq2i 5837	. . . . . . . . 9 |- ( ( F ` m ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) = ( ( F ` m ) +R 1R )
55	54	breq2i 3975	. . . . . . . 8 |- ( ( F ` 1o ) <R ( ( F ` m ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) <-> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
56	36, 55	syl6ib 160	. . . . . . 7 |- ( ( ph /\ m e. N. ) -> ( 1o <N m -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) ) )
57	56	imp 123	. . . . . 6 |- ( ( ( ph /\ m e. N. ) /\ 1o <N m ) -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
58	1	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ m e. N. ) -> F : N. --> R. )
59	23	a1i 9	. . . . . . . . . 10 |- ( ( ph /\ m e. N. ) -> 1o e. N. )
60	58, 59	ffvelrnd 5605	. . . . . . . . 9 |- ( ( ph /\ m e. N. ) -> ( F ` 1o ) e. R. )
61		ltadd1sr 7698	. . . . . . . . 9 |- ( ( F ` 1o ) e. R. -> ( F ` 1o ) <R ( ( F ` 1o ) +R 1R ) )
62	60, 61	syl 14	. . . . . . . 8 |- ( ( ph /\ m e. N. ) -> ( F ` 1o ) <R ( ( F ` 1o ) +R 1R ) )
63	62	adantr 274	. . . . . . 7 |- ( ( ( ph /\ m e. N. ) /\ 1o = m ) -> ( F ` 1o ) <R ( ( F ` 1o ) +R 1R ) )
64		fveq2 5470	. . . . . . . . 9 |- ( 1o = m -> ( F ` 1o ) = ( F ` m ) )
65	64	oveq1d 5841	. . . . . . . 8 |- ( 1o = m -> ( ( F ` 1o ) +R 1R ) = ( ( F ` m ) +R 1R ) )
66	65	adantl 275	. . . . . . 7 |- ( ( ( ph /\ m e. N. ) /\ 1o = m ) -> ( ( F ` 1o ) +R 1R ) = ( ( F ` m ) +R 1R ) )
67	63, 66	breqtrd 3992	. . . . . 6 |- ( ( ( ph /\ m e. N. ) /\ 1o = m ) -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
68		nlt1pig 7263	. . . . . . . . 9 |- ( m e. N. -> -. m <N 1o )
69	68	adantl 275	. . . . . . . 8 |- ( ( ph /\ m e. N. ) -> -. m <N 1o )
70	69	pm2.21d 609	. . . . . . 7 |- ( ( ph /\ m e. N. ) -> ( m <N 1o -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) ) )
71	70	imp 123	. . . . . 6 |- ( ( ( ph /\ m e. N. ) /\ m <N 1o ) -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
72		pitri3or 7244	. . . . . . . 8 |- ( ( 1o e. N. /\ m e. N. ) -> ( 1o <N m \/ 1o = m \/ m <N 1o ) )
73	23, 72	mpan 421	. . . . . . 7 |- ( m e. N. -> ( 1o <N m \/ 1o = m \/ m <N 1o ) )
74	73	adantl 275	. . . . . 6 |- ( ( ph /\ m e. N. ) -> ( 1o <N m \/ 1o = m \/ m <N 1o ) )
75	57, 67, 71, 74	mpjao3dan 1289	. . . . 5 |- ( ( ph /\ m e. N. ) -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
76		ltasrg 7692	. . . . . . 7 |- ( ( f e. R. /\ g e. R. /\ h e. R. ) -> ( f <R g <-> ( h +R f ) <R ( h +R g ) ) )
77	76	adantl 275	. . . . . 6 |- ( ( ( ph /\ m e. N. ) /\ ( f e. R. /\ g e. R. /\ h e. R. ) ) -> ( f <R g <-> ( h +R f ) <R ( h +R g ) ) )
78	1	ffvelrnda 5604	. . . . . . 7 |- ( ( ph /\ m e. N. ) -> ( F ` m ) e. R. )
79		1sr 7673	. . . . . . 7 |- 1R e. R.
80		addclsr 7675	. . . . . . 7 |- ( ( ( F ` m ) e. R. /\ 1R e. R. ) -> ( ( F ` m ) +R 1R ) e. R. )
81	78, 79, 80	sylancl 410	. . . . . 6 |- ( ( ph /\ m e. N. ) -> ( ( F ` m ) +R 1R ) e. R. )
82		m1r 7674	. . . . . . 7 |- -1R e. R.
83	82	a1i 9	. . . . . 6 |- ( ( ph /\ m e. N. ) -> -1R e. R. )
84		addcomsrg 7677	. . . . . . 7 |- ( ( f e. R. /\ g e. R. ) -> ( f +R g ) = ( g +R f ) )
85	84	adantl 275	. . . . . 6 |- ( ( ( ph /\ m e. N. ) /\ ( f e. R. /\ g e. R. ) ) -> ( f +R g ) = ( g +R f ) )
86	77, 60, 81, 83, 85	caovord2d 5992	. . . . 5 |- ( ( ph /\ m e. N. ) -> ( ( F ` 1o ) <R ( ( F ` m ) +R 1R ) <-> ( ( F ` 1o ) +R -1R ) <R ( ( ( F ` m ) +R 1R ) +R -1R ) ) )
87	75, 86	mpbid 146	. . . 4 |- ( ( ph /\ m e. N. ) -> ( ( F ` 1o ) +R -1R ) <R ( ( ( F ` m ) +R 1R ) +R -1R ) )
88	79	a1i 9	. . . . . 6 |- ( ( ph /\ m e. N. ) -> 1R e. R. )
89		addasssrg 7678	. . . . . 6 |- ( ( ( F ` m ) e. R. /\ 1R e. R. /\ -1R e. R. ) -> ( ( ( F ` m ) +R 1R ) +R -1R ) = ( ( F ` m ) +R ( 1R +R -1R ) ) )
90	78, 88, 83, 89	syl3anc 1220	. . . . 5 |- ( ( ph /\ m e. N. ) -> ( ( ( F ` m ) +R 1R ) +R -1R ) = ( ( F ` m ) +R ( 1R +R -1R ) ) )
91		addcomsrg 7677	. . . . . . . . 9 |- ( ( 1R e. R. /\ -1R e. R. ) -> ( 1R +R -1R ) = ( -1R +R 1R ) )
92	79, 82, 91	mp2an 423	. . . . . . . 8 |- ( 1R +R -1R ) = ( -1R +R 1R )
93		m1p1sr 7682	. . . . . . . 8 |- ( -1R +R 1R ) = 0R
94	92, 93	eqtri 2178	. . . . . . 7 |- ( 1R +R -1R ) = 0R
95	94	oveq2i 5837	. . . . . 6 |- ( ( F ` m ) +R ( 1R +R -1R ) ) = ( ( F ` m ) +R 0R )
96		0idsr 7689	. . . . . . 7 |- ( ( F ` m ) e. R. -> ( ( F ` m ) +R 0R ) = ( F ` m ) )
97	78, 96	syl 14	. . . . . 6 |- ( ( ph /\ m e. N. ) -> ( ( F ` m ) +R 0R ) = ( F ` m ) )
98	95, 97	syl5eq 2202	. . . . 5 |- ( ( ph /\ m e. N. ) -> ( ( F ` m ) +R ( 1R +R -1R ) ) = ( F ` m ) )
99	90, 98	eqtrd 2190	. . . 4 |- ( ( ph /\ m e. N. ) -> ( ( ( F ` m ) +R 1R ) +R -1R ) = ( F ` m ) )
100	87, 99	breqtrd 3992	. . 3 |- ( ( ph /\ m e. N. ) -> ( ( F ` 1o ) +R -1R ) <R ( F ` m ) )
101	100	ralrimiva 2530	. 2 |- ( ph -> A. m e. N. ( ( F ` 1o ) +R -1R ) <R ( F ` m ) )
102	1, 2, 101	caucvgsrlembnd 7723	1 |- ( ph -> E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <R ( y +R x ) /\ y <R ( ( F ` k ) +R x ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 962   /\ w3a 963   = wceq 1335   e. wcel 2128   {cab 2143   A.wral 2435   E.wrex 2436   <.cop 3564   class class class wbr 3967   -->wf 5168   ` cfv 5172  (class class class)co 5826   1oc1o 6358   [cec 6480   N.cnpi 7194   <N clti 7197   ~Q ceq 7201   1Qc1q 7203   *Qcrq 7206   <Q cltq 7207   1Pc1p 7214   +P. cpp 7215   ~R cer 7218   R.cnr 7219   0Rc0r 7220   1Rc1r 7221   -1Rcm1r 7222   +R cplr 7223   <R cltr 7225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-eprel 4251  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-er 6482  df-ec 6484  df-qs 6488  df-ni 7226  df-pli 7227  df-mi 7228  df-lti 7229  df-plpq 7266  df-mpq 7267  df-enq 7269  df-nqqs 7270  df-plqqs 7271  df-mqqs 7272  df-1nqqs 7273  df-rq 7274  df-ltnqqs 7275  df-enq0 7346  df-nq0 7347  df-0nq0 7348  df-plq0 7349  df-mq0 7350  df-inp 7388  df-i1p 7389  df-iplp 7390  df-imp 7391  df-iltp 7392  df-enr 7648  df-nr 7649  df-plr 7650  df-mr 7651  df-ltr 7652  df-0r 7653  df-1r 7654  df-m1r 7655

This theorem is referenced by:  axcaucvglemres  7821