Theorem caucvgsr 8012

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 7922 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 8011).

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 8007).

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 7922 to get a limit (see caucvgsrlemgt1 8005).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 8005).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 8010). (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 4089	. . . . . . . . . . . . 13 |- ( n = 1o -> ( n <N k <-> 1o <N k ) )
4		fveq2 5635	. . . . . . . . . . . . . . 15 |- ( n = 1o -> ( F ` n ) = ( F ` 1o ) )
5		opeq1 3860	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( n = 1o -> <. n , 1o >. = <. 1o , 1o >. )
6	5	eceq1d 6733	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( n = 1o -> [ <. n , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
7	6	fveq2d 5639	. . . . . . . . . . . . . . . . . . . . . 22 |- ( n = 1o -> ( *Q ` [ <. n , 1o >. ] ~Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q ) )
8	7	breq2d 4098	. . . . . . . . . . . . . . . . . . . . 21 |- ( n = 1o -> ( l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) )
9	8	abbidv 2347	. . . . . . . . . . . . . . . . . . . 20 |- ( n = 1o -> { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } )
10	7	breq1d 4096	. . . . . . . . . . . . . . . . . . . . 21 |- ( n = 1o -> ( ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u ) )
11	10	abbidv 2347	. . . . . . . . . . . . . . . . . . . 20 |- ( n = 1o -> { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } = { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } )
12	9, 11	opeq12d 3868	. . . . . . . . . . . . . . . . . . 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 6028	. . . . . . . . . . . . . . . . . 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 3866	. . . . . . . . . . . . . . . . 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 6733	. . . . . . . . . . . . . . . 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 6029	. . . . . . . . . . . . . . 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 4099	. . . . . . . . . . . . . 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 6031	. . . . . . . . . . . . . . 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 4098	. . . . . . . . . . . . . 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 2530	. . . . . . . . . . 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 7525	. . . . . . . . . . . 12 |- 1o e. N.
24	23	a1i 9	. . . . . . . . . . 11 |- ( ph -> 1o e. N. )
25	22, 2, 24	rspcdva 2913	. . . . . . . . . 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 2593	. . . . . . . . . 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 4090	. . . . . . . . . . 11 |- ( k = m -> ( 1o <N k <-> 1o <N m ) )
31		fveq2 5635	. . . . . . . . . . . . 13 |- ( k = m -> ( F ` k ) = ( F ` m ) )
32	31	oveq1d 6028	. . . . . . . . . . . 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 4098	. . . . . . . . . . 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 2904	. . . . . . . . 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 7561	. . . . . . . . . . . . . . . . . . . 20 |- 1Q = [ <. 1o , 1o >. ] ~Q
38	37	fveq2i 5638	. . . . . . . . . . . . . . . . . . 19 |- ( *Q ` 1Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q )
39		rec1nq 7605	. . . . . . . . . . . . . . . . . . 19 |- ( *Q ` 1Q ) = 1Q
40	38, 39	eqtr3i 2252	. . . . . . . . . . . . . . . . . 18 |- ( *Q ` [ <. 1o , 1o >. ] ~Q ) = 1Q
41	40	breq2i 4094	. . . . . . . . . . . . . . . . 17 |- ( l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) <-> l <Q 1Q )
42	41	abbii 2345	. . . . . . . . . . . . . . . 16 |- { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } = { l | l <Q 1Q }
43	40	breq1i 4093	. . . . . . . . . . . . . . . . 17 |- ( ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u <-> 1Q <Q u )
44	43	abbii 2345	. . . . . . . . . . . . . . . 16 |- { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } = { u | 1Q <Q u }
45	42, 44	opeq12i 3865	. . . . . . . . . . . . . . 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 7677	. . . . . . . . . . . . . . 15 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
47	45, 46	eqtr4i 2253	. . . . . . . . . . . . . 14 |- <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. = 1P
48	47	oveq1i 6023	. . . . . . . . . . . . 13 |- ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) = ( 1P +P. 1P )
49	48	opeq1i 3863	. . . . . . . . . . . 12 |- <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. = <. ( 1P +P. 1P ) , 1P >.
50		eceq1 6732	. . . . . . . . . . . 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 7942	. . . . . . . . . . 11 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
53	51, 52	eqtr4i 2253	. . . . . . . . . 10 |- [ <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R = 1R
54	53	oveq2i 6024	. . . . . . . . 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 4094	. . . . . . . 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 5779	. . . . . . . . 9 |- ( ( ph /\ m e. N. ) -> ( F ` 1o ) e. R. )
61		ltadd1sr 7986	. . . . . . . . 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 5635	. . . . . . . . 9 |- ( 1o = m -> ( F ` 1o ) = ( F ` m ) )
65	64	oveq1d 6028	. . . . . . . 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 4112	. . . . . 6 |- ( ( ( ph /\ m e. N. ) /\ 1o = m ) -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
68		nlt1pig 7551	. . . . . . . . 9 |- ( m e. N. -> -. m <N 1o )
69	68	adantl 277	. . . . . . . 8 |- ( ( ph /\ m e. N. ) -> -. m <N 1o )
70	69	pm2.21d 622	. . . . . . 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 7532	. . . . . . . 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 1341	. . . . 5 |- ( ( ph /\ m e. N. ) -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
76		ltasrg 7980	. . . . . . 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 5778	. . . . . . 7 |- ( ( ph /\ m e. N. ) -> ( F ` m ) e. R. )
79		1sr 7961	. . . . . . 7 |- 1R e. R.
80		addclsr 7963	. . . . . . 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 7962	. . . . . . 7 |- -1R e. R.
83	82	a1i 9	. . . . . 6 |- ( ( ph /\ m e. N. ) -> -1R e. R. )
84		addcomsrg 7965	. . . . . . 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 6187	. . . . 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 7966	. . . . . 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 1271	. . . . 5 |- ( ( ph /\ m e. N. ) -> ( ( ( F ` m ) +R 1R ) +R -1R ) = ( ( F ` m ) +R ( 1R +R -1R ) ) )
91		addcomsrg 7965	. . . . . . . . 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 7970	. . . . . . . 8 |- ( -1R +R 1R ) = 0R
94	92, 93	eqtri 2250	. . . . . . 7 |- ( 1R +R -1R ) = 0R
95	94	oveq2i 6024	. . . . . 6 |- ( ( F ` m ) +R ( 1R +R -1R ) ) = ( ( F ` m ) +R 0R )
96		0idsr 7977	. . . . . . 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 2274	. . . . 5 |- ( ( ph /\ m e. N. ) -> ( ( F ` m ) +R ( 1R +R -1R ) ) = ( F ` m ) )
99	90, 98	eqtrd 2262	. . . 4 |- ( ( ph /\ m e. N. ) -> ( ( ( F ` m ) +R 1R ) +R -1R ) = ( F ` m ) )
100	87, 99	breqtrd 4112	. . 3 |- ( ( ph /\ m e. N. ) -> ( ( F ` 1o ) +R -1R ) <R ( F ` m ) )
101	100	ralrimiva 2603	. 2 |- ( ph -> A. m e. N. ( ( F ` 1o ) +R -1R ) <R ( F ` m ) )
102	1, 2, 101	caucvgsrlembnd 8011	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 1001   /\ w3a 1002   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   <.cop 3670   class class class wbr 4086   -->wf 5320   ` cfv 5324  (class class class)co 6013   1oc1o 6570   [cec 6695   N.cnpi 7482   <N clti 7485   ~Q ceq 7489   1Qc1q 7491   *Qcrq 7494   <Q cltq 7495   1Pc1p 7502   +P. cpp 7503   ~R cer 7506   R.cnr 7507   0Rc0r 7508   1Rc1r 7509   -1Rcm1r 7510   +R cplr 7511   <R cltr 7513

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-i1p 7677  df-iplp 7678  df-imp 7679  df-iltp 7680  df-enr 7936  df-nr 7937  df-plr 7938  df-mr 7939  df-ltr 7940  df-0r 7941  df-1r 7942  df-m1r 7943

This theorem is referenced by:  axcaucvglemres  8109