Theorem caucvgsr 7603

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 7513 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 7602).

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 7598).

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 7513 to get a limit (see caucvgsrlemgt1 7596).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7596).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7601). (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 3927	. . . . . . . . . . . . 13 |- ( n = 1o -> ( n <N k <-> 1o <N k ) )
4		fveq2 5414	. . . . . . . . . . . . . . 15 |- ( n = 1o -> ( F ` n ) = ( F ` 1o ) )
5		opeq1 3700	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( n = 1o -> <. n , 1o >. = <. 1o , 1o >. )
6	5	eceq1d 6458	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( n = 1o -> [ <. n , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
7	6	fveq2d 5418	. . . . . . . . . . . . . . . . . . . . . 22 |- ( n = 1o -> ( *Q ` [ <. n , 1o >. ] ~Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q ) )
8	7	breq2d 3936	. . . . . . . . . . . . . . . . . . . . 21 |- ( n = 1o -> ( l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) )
9	8	abbidv 2255	. . . . . . . . . . . . . . . . . . . 20 |- ( n = 1o -> { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } )
10	7	breq1d 3934	. . . . . . . . . . . . . . . . . . . . 21 |- ( n = 1o -> ( ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u ) )
11	10	abbidv 2255	. . . . . . . . . . . . . . . . . . . 20 |- ( n = 1o -> { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } = { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } )
12	9, 11	opeq12d 3708	. . . . . . . . . . . . . . . . . . 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 5782	. . . . . . . . . . . . . . . . . 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 3706	. . . . . . . . . . . . . . . . 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 6458	. . . . . . . . . . . . . . . 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 5783	. . . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . 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 5785	. . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . 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 464	. . . . . . . . . . . . 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 2435	. . . . . . . . . . 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 7116	. . . . . . . . . . . 12 |- 1o e. N.
24	23	a1i 9	. . . . . . . . . . 11 |- ( ph -> 1o e. N. )
25	22, 2, 24	rspcdva 2789	. . . . . . . . . 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 2493	. . . . . . . . . 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 3928	. . . . . . . . . . 11 |- ( k = m -> ( 1o <N k <-> 1o <N m ) )
31		fveq2 5414	. . . . . . . . . . . . 13 |- ( k = m -> ( F ` k ) = ( F ` m ) )
32	31	oveq1d 5782	. . . . . . . . . . . 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 3936	. . . . . . . . . . 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 2780	. . . . . . . . 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 7152	. . . . . . . . . . . . . . . . . . . 20 |- 1Q = [ <. 1o , 1o >. ] ~Q
38	37	fveq2i 5417	. . . . . . . . . . . . . . . . . . 19 |- ( *Q ` 1Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q )
39		rec1nq 7196	. . . . . . . . . . . . . . . . . . 19 |- ( *Q ` 1Q ) = 1Q
40	38, 39	eqtr3i 2160	. . . . . . . . . . . . . . . . . 18 |- ( *Q ` [ <. 1o , 1o >. ] ~Q ) = 1Q
41	40	breq2i 3932	. . . . . . . . . . . . . . . . 17 |- ( l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) <-> l <Q 1Q )
42	41	abbii 2253	. . . . . . . . . . . . . . . 16 |- { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } = { l | l <Q 1Q }
43	40	breq1i 3931	. . . . . . . . . . . . . . . . 17 |- ( ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u <-> 1Q <Q u )
44	43	abbii 2253	. . . . . . . . . . . . . . . 16 |- { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } = { u | 1Q <Q u }
45	42, 44	opeq12i 3705	. . . . . . . . . . . . . . 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 7268	. . . . . . . . . . . . . . 15 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
47	45, 46	eqtr4i 2161	. . . . . . . . . . . . . 14 |- <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. = 1P
48	47	oveq1i 5777	. . . . . . . . . . . . 13 |- ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) = ( 1P +P. 1P )
49	48	opeq1i 3703	. . . . . . . . . . . 12 |- <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. = <. ( 1P +P. 1P ) , 1P >.
50		eceq1 6457	. . . . . . . . . . . 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 7533	. . . . . . . . . . 11 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
53	51, 52	eqtr4i 2161	. . . . . . . . . 10 |- [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R = 1R
54	53	oveq2i 5778	. . . . . . . . 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 3932	. . . . . . . 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 5549	. . . . . . . . 9 |- ( ( ph /\ m e. N. ) -> ( F ` 1o ) e. R. )
61		ltadd1sr 7577	. . . . . . . . 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 5414	. . . . . . . . 9 |- ( 1o = m -> ( F ` 1o ) = ( F ` m ) )
65	64	oveq1d 5782	. . . . . . . 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 3949	. . . . . 6 |- ( ( ( ph /\ m e. N. ) /\ 1o = m ) -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
68		nlt1pig 7142	. . . . . . . . 9 |- ( m e. N. -> -. m <N 1o )
69	68	adantl 275	. . . . . . . 8 |- ( ( ph /\ m e. N. ) -> -. m <N 1o )
70	69	pm2.21d 608	. . . . . . 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 7123	. . . . . . . 8 |- ( ( 1o e. N. /\ m e. N. ) -> ( 1o <N m \/ 1o = m \/ m <N 1o ) )
73	23, 72	mpan 420	. . . . . . 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 1285	. . . . 5 |- ( ( ph /\ m e. N. ) -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
76		ltasrg 7571	. . . . . . 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 5548	. . . . . . 7 |- ( ( ph /\ m e. N. ) -> ( F ` m ) e. R. )
79		1sr 7552	. . . . . . 7 |- 1R e. R.
80		addclsr 7554	. . . . . . 7 |- ( ( ( F ` m ) e. R. /\ 1R e. R. ) -> ( ( F ` m ) +R 1R ) e. R. )
81	78, 79, 80	sylancl 409	. . . . . 6 |- ( ( ph /\ m e. N. ) -> ( ( F ` m ) +R 1R ) e. R. )
82		m1r 7553	. . . . . . 7 |- -1R e. R.
83	82	a1i 9	. . . . . 6 |- ( ( ph /\ m e. N. ) -> -1R e. R. )
84		addcomsrg 7556	. . . . . . 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 5933	. . . . 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 7557	. . . . . 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 1216	. . . . 5 |- ( ( ph /\ m e. N. ) -> ( ( ( F ` m ) +R 1R ) +R -1R ) = ( ( F ` m ) +R ( 1R +R -1R ) ) )
91		addcomsrg 7556	. . . . . . . . 9 |- ( ( 1R e. R. /\ -1R e. R. ) -> ( 1R +R -1R ) = ( -1R +R 1R ) )
92	79, 82, 91	mp2an 422	. . . . . . . 8 |- ( 1R +R -1R ) = ( -1R +R 1R )
93		m1p1sr 7561	. . . . . . . 8 |- ( -1R +R 1R ) = 0R
94	92, 93	eqtri 2158	. . . . . . 7 |- ( 1R +R -1R ) = 0R
95	94	oveq2i 5778	. . . . . 6 |- ( ( F ` m ) +R ( 1R +R -1R ) ) = ( ( F ` m ) +R 0R )
96		0idsr 7568	. . . . . . 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 2182	. . . . 5 |- ( ( ph /\ m e. N. ) -> ( ( F ` m ) +R ( 1R +R -1R ) ) = ( F ` m ) )
99	90, 98	eqtrd 2170	. . . 4 |- ( ( ph /\ m e. N. ) -> ( ( ( F ` m ) +R 1R ) +R -1R ) = ( F ` m ) )
100	87, 99	breqtrd 3949	. . 3 |- ( ( ph /\ m e. N. ) -> ( ( F ` 1o ) +R -1R ) <R ( F ` m ) )
101	100	ralrimiva 2503	. 2 |- ( ph -> A. m e. N. ( ( F ` 1o ) +R -1R ) <R ( F ` m ) )
102	1, 2, 101	caucvgsrlembnd 7602	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 961   /\ w3a 962   = wceq 1331   e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   <.cop 3525   class class class wbr 3924   -->wf 5114   ` cfv 5118  (class class class)co 5767   1oc1o 6299   [cec 6420   N.cnpi 7073   <N clti 7076   ~Q ceq 7080   1Qc1q 7082   *Qcrq 7085   <Q cltq 7086   1Pc1p 7093   +P. cpp 7094   ~R cer 7097   R.cnr 7098   0Rc0r 7099   1Rc1r 7100   -1Rcm1r 7101   +R cplr 7102   <R cltr 7104

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-i1p 7268  df-iplp 7269  df-imp 7270  df-iltp 7271  df-enr 7527  df-nr 7528  df-plr 7529  df-mr 7530  df-ltr 7531  df-0r 7532  df-1r 7533  df-m1r 7534

This theorem is referenced by:  axcaucvglemres  7700