Theorem caucvgsr 8000

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 7910 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 7999).

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 7995).

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 7910 to get a limit (see caucvgsrlemgt1 7993).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7993).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7998). (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 4086	. . . . . . . . . . . . 13 |- ( n = 1o -> ( n <N k <-> 1o <N k ) )
4		fveq2 5629	. . . . . . . . . . . . . . 15 |- ( n = 1o -> ( F ` n ) = ( F ` 1o ) )
5		opeq1 3857	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( n = 1o -> <. n , 1o >. = <. 1o , 1o >. )
6	5	eceq1d 6724	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( n = 1o -> [ <. n , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
7	6	fveq2d 5633	. . . . . . . . . . . . . . . . . . . . . 22 |- ( n = 1o -> ( *Q ` [ <. n , 1o >. ] ~Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q ) )
8	7	breq2d 4095	. . . . . . . . . . . . . . . . . . . . 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 4093	. . . . . . . . . . . . . . . . . . . . 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 3865	. . . . . . . . . . . . . . . . . . 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 6022	. . . . . . . . . . . . . . . . . 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 3863	. . . . . . . . . . . . . . . . 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 6724	. . . . . . . . . . . . . . . 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 6023	. . . . . . . . . . . . . . 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 4096	. . . . . . . . . . . . . 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 6025	. . . . . . . . . . . . . . 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 4095	. . . . . . . . . . . . . 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 7513	. . . . . . . . . . . 12 |- 1o e. N.
24	23	a1i 9	. . . . . . . . . . 11 |- ( ph -> 1o e. N. )
25	22, 2, 24	rspcdva 2912	. . . . . . . . . 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 4087	. . . . . . . . . . 11 |- ( k = m -> ( 1o <N k <-> 1o <N m ) )
31		fveq2 5629	. . . . . . . . . . . . 13 |- ( k = m -> ( F ` k ) = ( F ` m ) )
32	31	oveq1d 6022	. . . . . . . . . . . 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 4095	. . . . . . . . . . 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 2903	. . . . . . . . 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 7549	. . . . . . . . . . . . . . . . . . . 20 |- 1Q = [ <. 1o , 1o >. ] ~Q
38	37	fveq2i 5632	. . . . . . . . . . . . . . . . . . 19 |- ( *Q ` 1Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q )
39		rec1nq 7593	. . . . . . . . . . . . . . . . . . 19 |- ( *Q ` 1Q ) = 1Q
40	38, 39	eqtr3i 2252	. . . . . . . . . . . . . . . . . 18 |- ( *Q ` [ <. 1o , 1o >. ] ~Q ) = 1Q
41	40	breq2i 4091	. . . . . . . . . . . . . . . . 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 4090	. . . . . . . . . . . . . . . . 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 3862	. . . . . . . . . . . . . . 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 7665	. . . . . . . . . . . . . . 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 6017	. . . . . . . . . . . . 13 |- ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) = ( 1P +P. 1P )
49	48	opeq1i 3860	. . . . . . . . . . . 12 |- <. ( <. { l | l <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. = <. ( 1P +P. 1P ) , 1P >.
50		eceq1 6723	. . . . . . . . . . . 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 7930	. . . . . . . . . . 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 6018	. . . . . . . . 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 4091	. . . . . . . 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 5773	. . . . . . . . 9 |- ( ( ph /\ m e. N. ) -> ( F ` 1o ) e. R. )
61		ltadd1sr 7974	. . . . . . . . 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 5629	. . . . . . . . 9 |- ( 1o = m -> ( F ` 1o ) = ( F ` m ) )
65	64	oveq1d 6022	. . . . . . . 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 4109	. . . . . 6 |- ( ( ( ph /\ m e. N. ) /\ 1o = m ) -> ( F ` 1o ) <R ( ( F ` m ) +R 1R ) )
68		nlt1pig 7539	. . . . . . . . 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 7520	. . . . . . . 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 7968	. . . . . . 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 5772	. . . . . . 7 |- ( ( ph /\ m e. N. ) -> ( F ` m ) e. R. )
79		1sr 7949	. . . . . . 7 |- 1R e. R.
80		addclsr 7951	. . . . . . 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 7950	. . . . . . 7 |- -1R e. R.
83	82	a1i 9	. . . . . 6 |- ( ( ph /\ m e. N. ) -> -1R e. R. )
84		addcomsrg 7953	. . . . . . 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 6181	. . . . 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 7954	. . . . . 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 7953	. . . . . . . . 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 7958	. . . . . . . 8 |- ( -1R +R 1R ) = 0R
94	92, 93	eqtri 2250	. . . . . . 7 |- ( 1R +R -1R ) = 0R
95	94	oveq2i 6018	. . . . . 6 |- ( ( F ` m ) +R ( 1R +R -1R ) ) = ( ( F ` m ) +R 0R )
96		0idsr 7965	. . . . . . 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 4109	. . 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 7999	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 3669   class class class wbr 4083   -->wf 5314   ` cfv 5318  (class class class)co 6007   1oc1o 6561   [cec 6686   N.cnpi 7470   <N clti 7473   ~Q ceq 7477   1Qc1q 7479   *Qcrq 7482   <Q cltq 7483   1Pc1p 7490   +P. cpp 7491   ~R cer 7494   R.cnr 7495   0Rc0r 7496   1Rc1r 7497   -1Rcm1r 7498   +R cplr 7499   <R cltr 7501

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-2o 6569  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7502  df-pli 7503  df-mi 7504  df-lti 7505  df-plpq 7542  df-mpq 7543  df-enq 7545  df-nqqs 7546  df-plqqs 7547  df-mqqs 7548  df-1nqqs 7549  df-rq 7550  df-ltnqqs 7551  df-enq0 7622  df-nq0 7623  df-0nq0 7624  df-plq0 7625  df-mq0 7626  df-inp 7664  df-i1p 7665  df-iplp 7666  df-imp 7667  df-iltp 7668  df-enr 7924  df-nr 7925  df-plr 7926  df-mr 7927  df-ltr 7928  df-0r 7929  df-1r 7930  df-m1r 7931

This theorem is referenced by:  axcaucvglemres  8097