Theorem caucvgsr 7914

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 7824 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 7913).

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 7909).

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 7824 to get a limit (see caucvgsrlemgt1 7907).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7907).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7912). (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 4046	. . . . . . . . . . . . 13 |- ( n = 1o -> ( n <N k <-> 1o <N k ) )
4		fveq2 5575	. . . . . . . . . . . . . . 15 |- ( n = 1o -> ( F ` n ) = ( F ` 1o ) )
5		opeq1 3818	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( n = 1o -> <. n , 1o >. = <. 1o , 1o >. )
6	5	eceq1d 6655	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( n = 1o -> [ <. n , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
7	6	fveq2d 5579	. . . . . . . . . . . . . . . . . . . . . 22 |- ( n = 1o -> ( *Q ` [ <. n , 1o >. ] ~Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q ) )
8	7	breq2d 4055	. . . . . . . . . . . . . . . . . . . . 21 |- ( n = 1o -> ( l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) )
9	8	abbidv 2322	. . . . . . . . . . . . . . . . . . . 20 |- ( n = 1o -> { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } )
10	7	breq1d 4053	. . . . . . . . . . . . . . . . . . . . 21 |- ( n = 1o -> ( ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u ) )
11	10	abbidv 2322	. . . . . . . . . . . . . . . . . . . 20 |- ( n = 1o -> { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } = { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } )
12	9, 11	opeq12d 3826	. . . . . . . . . . . . . . . . . . 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 5958	. . . . . . . . . . . . . . . . . 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 3824	. . . . . . . . . . . . . . . . 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 6655	. . . . . . . . . . . . . . . 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 5959	. . . . . . . . . . . . . . 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 4056	. . . . . . . . . . . . . 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 5961	. . . . . . . . . . . . . . 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 4055	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 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 2505	. . . . . . . . . . 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 7427	. . . . . . . . . . . 12 |- 1o e. N.
24	23	a1i 9	. . . . . . . . . . 11 |- ( ph -> 1o e. N. )
25	22, 2, 24	rspcdva 2881	. . . . . . . . . 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 109	. . . . . . . . . . . 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 2568	. . . . . . . . . 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 4047	. . . . . . . . . . 11 |- ( k = m -> ( 1o <N k <-> 1o <N m ) )
31		fveq2 5575	. . . . . . . . . . . . 13 |- ( k = m -> ( F ` k ) = ( F ` m ) )
32	31	oveq1d 5958	. . . . . . . . . . . 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 4055	. . . . . . . . . . 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 234	. . . . . . . . . 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 2872	. . . . . . . . 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 281	. . . . . . . 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 7463	. . . . . . . . . . . . . . . . . . . 20 |- 1Q = [ <. 1o , 1o >. ] ~Q
38	37	fveq2i 5578	. . . . . . . . . . . . . . . . . . 19 |- ( *Q ` 1Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q )
39		rec1nq 7507	. . . . . . . . . . . . . . . . . . 19 |- ( *Q ` 1Q ) = 1Q
40	38, 39	eqtr3i 2227	. . . . . . . . . . . . . . . . . 18 |- ( *Q ` [ <. 1o , 1o >. ] ~Q ) = 1Q
41	40	breq2i 4051	. . . . . . . . . . . . . . . . 17 |- ( l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) <-> l <Q 1Q )
42	41	abbii 2320	. . . . . . . . . . . . . . . 16 |- { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } = { l | l <Q 1Q }
43	40	breq1i 4050	. . . . . . . . . . . . . . . . 17 |- ( ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u <-> 1Q <Q u )
44	43	abbii 2320	. . . . . . . . . . . . . . . 16 |- { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } = { u | 1Q <Q u }
45	42, 44	opeq12i 3823	. . . . . . . . . . . . . . 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 7579	. . . . . . . . . . . . . . 15 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
47	45, 46	eqtr4i 2228	. . . . . . . . . . . . . 14 |- <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. = 1P
48	47	oveq1i 5953	. . . . . . . . . . . . 13 |- ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) = ( 1P +P. 1P )
49	48	opeq1i 3821	. . . . . . . . . . . 12 |- <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. = <. ( 1P +P. 1P ) , 1P >.
50		eceq1 6654	. . . . . . . . . . . 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 7844	. . . . . . . . . . 11 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
53	51, 52	eqtr4i 2228	. . . . . . . . . 10 |- [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R = 1R
54	53	oveq2i 5954	. . . . . . . . 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 4051	. . . . . . . 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	imbitrdi 161	. . . . . . 7 |- ( ( ph /\ m e. N. ) -> ( 1o <N m -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) ) )
57	56	imp 124	. . . . . 6 |- ( ( ( ph /\ m e. N. ) /\ 1o <N m ) -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
58	1	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ m e. N. ) -> F : N. --> R. )
59	23	a1i 9	. . . . . . . . . 10 |- ( ( ph /\ m e. N. ) -> 1o e. N. )
60	58, 59	ffvelcdmd 5715	. . . . . . . . 9 |- ( ( ph /\ m e. N. ) -> ( F ` 1o ) e. R. )
61		ltadd1sr 7888	. . . . . . . . 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 276	. . . . . . 7 |- ( ( ( ph /\ m e. N. ) /\ 1o = m ) -> ( F ` 1o ) <R ( ( F ` 1o ) +R 1R ) )
64		fveq2 5575	. . . . . . . . 9 |- ( 1o = m -> ( F ` 1o ) = ( F ` m ) )
65	64	oveq1d 5958	. . . . . . . 8 |- ( 1o = m -> ( ( F ` 1o ) +R 1R ) = ( ( F ` m ) +R 1R ) )
66	65	adantl 277	. . . . . . 7 |- ( ( ( ph /\ m e. N. ) /\ 1o = m ) -> ( ( F ` 1o ) +R 1R ) = ( ( F ` m ) +R 1R ) )
67	63, 66	breqtrd 4069	. . . . . 6 |- ( ( ( ph /\ m e. N. ) /\ 1o = m ) -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
68		nlt1pig 7453	. . . . . . . . 9 |- ( m e. N. -> -. m <N 1o )
69	68	adantl 277	. . . . . . . 8 |- ( ( ph /\ m e. N. ) -> -. m <N 1o )
70	69	pm2.21d 620	. . . . . . 7 |- ( ( ph /\ m e. N. ) -> ( m <N 1o -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) ) )
71	70	imp 124	. . . . . 6 |- ( ( ( ph /\ m e. N. ) /\ m <N 1o ) -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
72		pitri3or 7434	. . . . . . . 8 |- ( ( 1o e. N. /\ m e. N. ) -> ( 1o <N m \/ 1o = m \/ m <N 1o ) )
73	23, 72	mpan 424	. . . . . . 7 |- ( m e. N. -> ( 1o <N m \/ 1o = m \/ m <N 1o ) )
74	73	adantl 277	. . . . . 6 |- ( ( ph /\ m e. N. ) -> ( 1o <N m \/ 1o = m \/ m <N 1o ) )
75	57, 67, 71, 74	mpjao3dan 1319	. . . . 5 |- ( ( ph /\ m e. N. ) -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
76		ltasrg 7882	. . . . . . 7 |- ( ( f e. R. /\ g e. R. /\ h e. R. ) -> ( f <R g <-> ( h +R f ) <R ( h +R g ) ) )
77	76	adantl 277	. . . . . 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	ffvelcdmda 5714	. . . . . . 7 |- ( ( ph /\ m e. N. ) -> ( F ` m ) e. R. )
79		1sr 7863	. . . . . . 7 |- 1R e. R.
80		addclsr 7865	. . . . . . 7 |- ( ( ( F ` m ) e. R. /\ 1R e. R. ) -> ( ( F ` m ) +R 1R ) e. R. )
81	78, 79, 80	sylancl 413	. . . . . 6 |- ( ( ph /\ m e. N. ) -> ( ( F ` m ) +R 1R ) e. R. )
82		m1r 7864	. . . . . . 7 |- -1R e. R.
83	82	a1i 9	. . . . . 6 |- ( ( ph /\ m e. N. ) -> -1R e. R. )
84		addcomsrg 7867	. . . . . . 7 |- ( ( f e. R. /\ g e. R. ) -> ( f +R g ) = ( g +R f ) )
85	84	adantl 277	. . . . . 6 |- ( ( ( ph /\ m e. N. ) /\ ( f e. R. /\ g e. R. ) ) -> ( f +R g ) = ( g +R f ) )
86	77, 60, 81, 83, 85	caovord2d 6115	. . . . 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 147	. . . 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 7868	. . . . . 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 1249	. . . . 5 |- ( ( ph /\ m e. N. ) -> ( ( ( F ` m ) +R 1R ) +R -1R ) = ( ( F ` m ) +R ( 1R +R -1R ) ) )
91		addcomsrg 7867	. . . . . . . . 9 |- ( ( 1R e. R. /\ -1R e. R. ) -> ( 1R +R -1R ) = ( -1R +R 1R ) )
92	79, 82, 91	mp2an 426	. . . . . . . 8 |- ( 1R +R -1R ) = ( -1R +R 1R )
93		m1p1sr 7872	. . . . . . . 8 |- ( -1R +R 1R ) = 0R
94	92, 93	eqtri 2225	. . . . . . 7 |- ( 1R +R -1R ) = 0R
95	94	oveq2i 5954	. . . . . 6 |- ( ( F ` m ) +R ( 1R +R -1R ) ) = ( ( F ` m ) +R 0R )
96		0idsr 7879	. . . . . . 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	eqtrid 2249	. . . . 5 |- ( ( ph /\ m e. N. ) -> ( ( F ` m ) +R ( 1R +R -1R ) ) = ( F ` m ) )
99	90, 98	eqtrd 2237	. . . 4 |- ( ( ph /\ m e. N. ) -> ( ( ( F ` m ) +R 1R ) +R -1R ) = ( F ` m ) )
100	87, 99	breqtrd 4069	. . 3 |- ( ( ph /\ m e. N. ) -> ( ( F ` 1o ) +R -1R ) <R ( F ` m ) )
101	100	ralrimiva 2578	. 2 |- ( ph -> A. m e. N. ( ( F ` 1o ) +R -1R ) <R ( F ` m ) )
102	1, 2, 101	caucvgsrlembnd 7913	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 104   <-> wb 105   \/ w3o 979   /\ w3a 980   = wceq 1372   e. wcel 2175   {cab 2190   A.wral 2483   E.wrex 2484   <.cop 3635   class class class wbr 4043   -->wf 5266   ` cfv 5270  (class class class)co 5943   1oc1o 6494   [cec 6617   N.cnpi 7384   <N clti 7387   ~Q ceq 7391   1Qc1q 7393   *Qcrq 7396   <Q cltq 7397   1Pc1p 7404   +P. cpp 7405   ~R cer 7408   R.cnr 7409   0Rc0r 7410   1Rc1r 7411   -1Rcm1r 7412   +R cplr 7413   <R cltr 7415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4335  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-1o 6501  df-2o 6502  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-pli 7417  df-mi 7418  df-lti 7419  df-plpq 7456  df-mpq 7457  df-enq 7459  df-nqqs 7460  df-plqqs 7461  df-mqqs 7462  df-1nqqs 7463  df-rq 7464  df-ltnqqs 7465  df-enq0 7536  df-nq0 7537  df-0nq0 7538  df-plq0 7539  df-mq0 7540  df-inp 7578  df-i1p 7579  df-iplp 7580  df-imp 7581  df-iltp 7582  df-enr 7838  df-nr 7839  df-plr 7840  df-mr 7841  df-ltr 7842  df-0r 7843  df-1r 7844  df-m1r 7845

This theorem is referenced by:  axcaucvglemres  8011