Theorem ivthreinc 15161

Description: Restating the intermediate value theorem. Given a hypothesis stating the intermediate value theorem (in a strong form which is not provable given our axioms alone), provide a conclusion similar to the theorem as stated in the Metamath Proof Explorer (which is also similar to how we state the theorem for a strictly monotonic function at ivthinc 15159). Being able to have a hypothesis stating the intermediate value theorem will be helpful when it comes time to show that it implies a constructive taboo. This version of the theorem requires that the function F is continuous on the entire real line, not just ( A [,] B ) which may be an unnecessary condition but which is sufficient for the way we want to use it. (Contributed by Jim Kingdon, 7-Jul-2025.)

Hypotheses

Ref	Expression
ivthreinc.1	|- ( ph -> A e. RR )
ivthreinc.2	|- ( ph -> B e. RR )
ivthreinc.3	|- ( ph -> U e. RR )
ivthreinc.4	|- ( ph -> A < B )
ivthreinc.7	|- ( ph -> F e. ( RR -cn-> RR ) )
ivthreinc.9	|- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
ivthreinc.i	|- ( ph -> A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) )

Assertion

Ref	Expression
ivthreinc	|- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )

Distinct variable groups:   A, a, b, x   A, c, x   B, b, x   B, c   F, a, b, f, x   F, c   U, a, b, f, x   U, c   ph, x

Allowed substitution hints:   ph( f, a, b, c)   A( f)   B( f, a)

Proof of Theorem ivthreinc

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ivthreinc.4	. . . 4 |- ( ph -> A < B )
2		eqid 2206	. . . . . 6 |- ( r e. RR |-> ( ( F ` r ) - U ) ) = ( r e. RR |-> ( ( F ` r ) - U ) )
3		fveq2 5583	. . . . . . 7 |- ( r = A -> ( F ` r ) = ( F ` A ) )
4	3	oveq1d 5966	. . . . . 6 |- ( r = A -> ( ( F ` r ) - U ) = ( ( F ` A ) - U ) )
5		ivthreinc.1	. . . . . 6 |- ( ph -> A e. RR )
6		ivthreinc.7	. . . . . . . . 9 |- ( ph -> F e. ( RR -cn-> RR ) )
7		cncff 15093	. . . . . . . . 9 |- ( F e. ( RR -cn-> RR ) -> F : RR --> RR )
8	6, 7	syl 14	. . . . . . . 8 |- ( ph -> F : RR --> RR )
9	8, 5	ffvelcdmd 5723	. . . . . . 7 |- ( ph -> ( F ` A ) e. RR )
10		ivthreinc.3	. . . . . . 7 |- ( ph -> U e. RR )
11	9, 10	resubcld 8460	. . . . . 6 |- ( ph -> ( ( F ` A ) - U ) e. RR )
12	2, 4, 5, 11	fvmptd3 5680	. . . . 5 |- ( ph -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) = ( ( F ` A ) - U ) )
13		ivthreinc.9	. . . . . . 7 |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
14	13	simpld 112	. . . . . 6 |- ( ph -> ( F ` A ) < U )
15	9, 10	sublt0d 8650	. . . . . 6 |- ( ph -> ( ( ( F ` A ) - U ) < 0 <-> ( F ` A ) < U ) )
16	14, 15	mpbird 167	. . . . 5 |- ( ph -> ( ( F ` A ) - U ) < 0 )
17	12, 16	eqbrtrd 4069	. . . 4 |- ( ph -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 )
18	13	simprd 114	. . . . . 6 |- ( ph -> U < ( F ` B ) )
19		ivthreinc.2	. . . . . . . 8 |- ( ph -> B e. RR )
20	8, 19	ffvelcdmd 5723	. . . . . . 7 |- ( ph -> ( F ` B ) e. RR )
21	10, 20	posdifd 8612	. . . . . 6 |- ( ph -> ( U < ( F ` B ) <-> 0 < ( ( F ` B ) - U ) ) )
22	18, 21	mpbid 147	. . . . 5 |- ( ph -> 0 < ( ( F ` B ) - U ) )
23		fveq2 5583	. . . . . . 7 |- ( r = B -> ( F ` r ) = ( F ` B ) )
24	23	oveq1d 5966	. . . . . 6 |- ( r = B -> ( ( F ` r ) - U ) = ( ( F ` B ) - U ) )
25	20, 10	resubcld 8460	. . . . . 6 |- ( ph -> ( ( F ` B ) - U ) e. RR )
26	2, 24, 19, 25	fvmptd3 5680	. . . . 5 |- ( ph -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) = ( ( F ` B ) - U ) )
27	22, 26	breqtrrd 4075	. . . 4 |- ( ph -> 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) )
28	1, 17, 27	3jca 1180	. . 3 |- ( ph -> ( A < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) ) )
29		breq2 4051	. . . . . 6 |- ( b = B -> ( A < b <-> A < B ) )
30		fveq2 5583	. . . . . . 7 |- ( b = B -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) )
31	30	breq2d 4059	. . . . . 6 |- ( b = B -> ( 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) <-> 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) ) )
32	29, 31	3anbi13d 1327	. . . . 5 |- ( b = B -> ( ( A < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) <-> ( A < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) ) ) )
33		breq2 4051	. . . . . . 7 |- ( b = B -> ( x < b <-> x < B ) )
34	33	3anbi2d 1330	. . . . . 6 |- ( b = B -> ( ( A < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) <-> ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) )
35	34	rexbidv 2508	. . . . 5 |- ( b = B -> ( E. x e. RR ( A < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) <-> E. x e. RR ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) )
36	32, 35	imbi12d 234	. . . 4 |- ( b = B -> ( ( ( A < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) -> E. x e. RR ( A < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) <-> ( ( A < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) ) -> E. x e. RR ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) )
37		breq1 4050	. . . . . . . 8 |- ( a = A -> ( a < b <-> A < b ) )
38		fveq2 5583	. . . . . . . . 9 |- ( a = A -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) )
39	38	breq1d 4057	. . . . . . . 8 |- ( a = A -> ( ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 <-> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 ) )
40	37, 39	3anbi12d 1326	. . . . . . 7 |- ( a = A -> ( ( a < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) <-> ( A < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) ) )
41		breq1 4050	. . . . . . . . 9 |- ( a = A -> ( a < x <-> A < x ) )
42	41	3anbi1d 1329	. . . . . . . 8 |- ( a = A -> ( ( a < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) <-> ( A < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) )
43	42	rexbidv 2508	. . . . . . 7 |- ( a = A -> ( E. x e. RR ( a < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) <-> E. x e. RR ( A < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) )
44	40, 43	imbi12d 234	. . . . . 6 |- ( a = A -> ( ( ( a < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) <-> ( ( A < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) -> E. x e. RR ( A < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) )
45	44	ralbidv 2507	. . . . 5 |- ( a = A -> ( A. b e. RR ( ( a < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) <-> A. b e. RR ( ( A < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) -> E. x e. RR ( A < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) )
46	8	ffvelcdmda 5722	. . . . . . . . 9 |- ( ( ph /\ r e. RR ) -> ( F ` r ) e. RR )
47	10	adantr 276	. . . . . . . . 9 |- ( ( ph /\ r e. RR ) -> U e. RR )
48	46, 47	resubcld 8460	. . . . . . . 8 |- ( ( ph /\ r e. RR ) -> ( ( F ` r ) - U ) e. RR )
49	48	fmpttd 5742	. . . . . . 7 |- ( ph -> ( r e. RR |-> ( ( F ` r ) - U ) ) : RR --> RR )
50		ax-resscn 8024	. . . . . . . . 9 |- RR C_ CC
51	50	a1i 9	. . . . . . . 8 |- ( ph -> RR C_ CC )
52	8	feqmptd 5639	. . . . . . . . . 10 |- ( ph -> F = ( r e. RR |-> ( F ` r ) ) )
53		ssid 3214	. . . . . . . . . . . 12 |- CC C_ CC
54		cncfss 15099	. . . . . . . . . . . 12 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( RR -cn-> RR ) C_ ( RR -cn-> CC ) )
55	50, 53, 54	mp2an 426	. . . . . . . . . . 11 |- ( RR -cn-> RR ) C_ ( RR -cn-> CC )
56	55, 6	sselid 3192	. . . . . . . . . 10 |- ( ph -> F e. ( RR -cn-> CC ) )
57	52, 56	eqeltrrd 2284	. . . . . . . . 9 |- ( ph -> ( r e. RR |-> ( F ` r ) ) e. ( RR -cn-> CC ) )
58	10	recnd 8108	. . . . . . . . . 10 |- ( ph -> U e. CC )
59	53	a1i 9	. . . . . . . . . 10 |- ( ph -> CC C_ CC )
60		cncfmptc 15112	. . . . . . . . . 10 |- ( ( U e. CC /\ RR C_ CC /\ CC C_ CC ) -> ( r e. RR |-> U ) e. ( RR -cn-> CC ) )
61	58, 51, 59, 60	syl3anc 1250	. . . . . . . . 9 |- ( ph -> ( r e. RR |-> U ) e. ( RR -cn-> CC ) )
62	57, 61	subcncf 15129	. . . . . . . 8 |- ( ph -> ( r e. RR |-> ( ( F ` r ) - U ) ) e. ( RR -cn-> CC ) )
63		cncfcdm 15098	. . . . . . . 8 |- ( ( RR C_ CC /\ ( r e. RR |-> ( ( F ` r ) - U ) ) e. ( RR -cn-> CC ) ) -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) e. ( RR -cn-> RR ) <-> ( r e. RR |-> ( ( F ` r ) - U ) ) : RR --> RR ) )
64	51, 62, 63	syl2anc 411	. . . . . . 7 |- ( ph -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) e. ( RR -cn-> RR ) <-> ( r e. RR |-> ( ( F ` r ) - U ) ) : RR --> RR ) )
65	49, 64	mpbird 167	. . . . . 6 |- ( ph -> ( r e. RR |-> ( ( F ` r ) - U ) ) e. ( RR -cn-> RR ) )
66		ivthreinc.i	. . . . . . 7 |- ( ph -> A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) )
67		reex 8066	. . . . . . . . 9 |- RR e. _V
68	67	mptex 5817	. . . . . . . 8 |- ( r e. RR |-> ( ( F ` r ) - U ) ) e. _V
69		eleq1 2269	. . . . . . . . 9 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( f e. ( RR -cn-> RR ) <-> ( r e. RR |-> ( ( F ` r ) - U ) ) e. ( RR -cn-> RR ) ) )
70		fveq1 5582	. . . . . . . . . . . . . 14 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( f ` a ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) )
71	70	breq1d 4057	. . . . . . . . . . . . 13 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( ( f ` a ) < 0 <-> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 ) )
72		fveq1 5582	. . . . . . . . . . . . . 14 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( f ` b ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) )
73	72	breq2d 4059	. . . . . . . . . . . . 13 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( 0 < ( f ` b ) <-> 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) )
74	71, 73	3anbi23d 1328	. . . . . . . . . . . 12 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) <-> ( a < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) ) )
75		fveq1 5582	. . . . . . . . . . . . . . 15 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( f ` x ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) )
76	75	eqeq1d 2215	. . . . . . . . . . . . . 14 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( ( f ` x ) = 0 <-> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) )
77	76	3anbi3d 1331	. . . . . . . . . . . . 13 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( ( a < x /\ x < b /\ ( f ` x ) = 0 ) <-> ( a < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) )
78	77	rexbidv 2508	. . . . . . . . . . . 12 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) <-> E. x e. RR ( a < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) )
79	74, 78	imbi12d 234	. . . . . . . . . . 11 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) <-> ( ( a < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) )
80	79	ralbidv 2507	. . . . . . . . . 10 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) <-> A. b e. RR ( ( a < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) )
81	80	ralbidv 2507	. . . . . . . . 9 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) <-> A. a e. RR A. b e. RR ( ( a < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) )
82	69, 81	imbi12d 234	. . . . . . . 8 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) <-> ( ( r e. RR |-> ( ( F ` r ) - U ) ) e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) ) )
83	68, 82	spcv 2868	. . . . . . 7 |- ( A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) )
84	66, 83	syl 14	. . . . . 6 |- ( ph -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) )
85	65, 84	mpd 13	. . . . 5 |- ( ph -> A. a e. RR A. b e. RR ( ( a < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) )
86	45, 85, 5	rspcdva 2883	. . . 4 |- ( ph -> A. b e. RR ( ( A < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) -> E. x e. RR ( A < x /\ x < b /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) )
87	36, 86, 19	rspcdva 2883	. . 3 |- ( ph -> ( ( A < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) ) -> E. x e. RR ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) )
88	28, 87	mpd 13	. 2 |- ( ph -> E. x e. RR ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) )
89	5	adantr 276	. . . . . 6 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> A e. RR )
90	89	rexrd 8129	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> A e. RR* )
91	19	adantr 276	. . . . . 6 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> B e. RR )
92	91	rexrd 8129	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> B e. RR* )
93		simprl 529	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> x e. RR )
94	90, 92, 93	3jca 1180	. . . 4 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> ( A e. RR* /\ B e. RR* /\ x e. RR ) )
95		simprr1 1048	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> A < x )
96		simprr2 1049	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> x < B )
97	95, 96	jca 306	. . . 4 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> ( A < x /\ x < B ) )
98		elioo4g 10063	. . . 4 |- ( x e. ( A (,) B ) <-> ( ( A e. RR* /\ B e. RR* /\ x e. RR ) /\ ( A < x /\ x < B ) ) )
99	94, 97, 98	sylanbrc 417	. . 3 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> x e. ( A (,) B ) )
100	8	adantr 276	. . . . . 6 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> F : RR --> RR )
101	100, 93	ffvelcdmd 5723	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> ( F ` x ) e. RR )
102	101	recnd 8108	. . . 4 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> ( F ` x ) e. CC )
103	58	adantr 276	. . . 4 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> U e. CC )
104		fveq2 5583	. . . . . . 7 |- ( r = x -> ( F ` r ) = ( F ` x ) )
105	104	oveq1d 5966	. . . . . 6 |- ( r = x -> ( ( F ` r ) - U ) = ( ( F ` x ) - U ) )
106	10	adantr 276	. . . . . . 7 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> U e. RR )
107	101, 106	resubcld 8460	. . . . . 6 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> ( ( F ` x ) - U ) e. RR )
108	2, 105, 93, 107	fvmptd3 5680	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = ( ( F ` x ) - U ) )
109		simprr3 1050	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 )
110	108, 109	eqtr3d 2241	. . . 4 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> ( ( F ` x ) - U ) = 0 )
111	102, 103, 110	subeq0d 8398	. . 3 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> ( F ` x ) = U )
112		fveqeq2 5592	. . . 4 |- ( c = x -> ( ( F ` c ) = U <-> ( F ` x ) = U ) )
113	112	rspcev 2878	. . 3 |- ( ( x e. ( A (,) B ) /\ ( F ` x ) = U ) -> E. c e. ( A (,) B ) ( F ` c ) = U )
114	99, 111, 113	syl2anc 411	. 2 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> E. c e. ( A (,) B ) ( F ` c ) = U )
115	88, 114	rexlimddv 2629	1 |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   A.wal 1371   = wceq 1373   e. wcel 2177   A.wral 2485   E.wrex 2486   C_ wss 3167   class class class wbr 4047   |-> cmpt 4109   -->wf 5272   ` cfv 5276  (class class class)co 5951   CCcc 7930   RRcr 7931   0cc0 7932   RR*cxr 8113   < clt 8114   - cmin 8250   (,)cioo 10017   -cn->ccncf 15086

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-frec 6484  df-map 6744  df-sup 7093  df-inf 7094  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-xneg 9901  df-xadd 9902  df-ioo 10021  df-seqfrec 10600  df-exp 10691  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-rest 13117  df-topgen 13136  df-psmet 14349  df-xmet 14350  df-met 14351  df-bl 14352  df-mopn 14353  df-top 14514  df-topon 14527  df-bases 14559  df-cn 14704  df-cnp 14705  df-tx 14769  df-cncf 15087

This theorem is referenced by:  ivthdichlem  15167