Theorem ivthreinc 14824

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 14822). 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 2193	. . . . . 6 |- ( r e. RR |-> ( ( F ` r ) - U ) ) = ( r e. RR |-> ( ( F ` r ) - U ) )
3		fveq2 5555	. . . . . . 7 |- ( r = A -> ( F ` r ) = ( F ` A ) )
4	3	oveq1d 5934	. . . . . 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 14756	. . . . . . . . 9 |- ( F e. ( RR -cn-> RR ) -> F : RR --> RR )
8	6, 7	syl 14	. . . . . . . 8 |- ( ph -> F : RR --> RR )
9	8, 5	ffvelcdmd 5695	. . . . . . 7 |- ( ph -> ( F ` A ) e. RR )
10		ivthreinc.3	. . . . . . 7 |- ( ph -> U e. RR )
11	9, 10	resubcld 8402	. . . . . 6 |- ( ph -> ( ( F ` A ) - U ) e. RR )
12	2, 4, 5, 11	fvmptd3 5652	. . . . 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 8591	. . . . . 6 |- ( ph -> ( ( ( F ` A ) - U ) < 0 <-> ( F ` A ) < U ) )
16	14, 15	mpbird 167	. . . . 5 |- ( ph -> ( ( F ` A ) - U ) < 0 )
17	12, 16	eqbrtrd 4052	. . . 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 5695	. . . . . . 7 |- ( ph -> ( F ` B ) e. RR )
21	10, 20	posdifd 8553	. . . . . 6 |- ( ph -> ( U < ( F ` B ) <-> 0 < ( ( F ` B ) - U ) ) )
22	18, 21	mpbid 147	. . . . 5 |- ( ph -> 0 < ( ( F ` B ) - U ) )
23		fveq2 5555	. . . . . . 7 |- ( r = B -> ( F ` r ) = ( F ` B ) )
24	23	oveq1d 5934	. . . . . 6 |- ( r = B -> ( ( F ` r ) - U ) = ( ( F ` B ) - U ) )
25	20, 10	resubcld 8402	. . . . . 6 |- ( ph -> ( ( F ` B ) - U ) e. RR )
26	2, 24, 19, 25	fvmptd3 5652	. . . . 5 |- ( ph -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) = ( ( F ` B ) - U ) )
27	22, 26	breqtrrd 4058	. . . 4 |- ( ph -> 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) )
28	1, 17, 27	3jca 1179	. . 3 |- ( ph -> ( A < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) ) )
29		breq2 4034	. . . . . 6 |- ( b = B -> ( A < b <-> A < B ) )
30		fveq2 5555	. . . . . . 7 |- ( b = B -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) )
31	30	breq2d 4042	. . . . . 6 |- ( b = B -> ( 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) <-> 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) ) )
32	29, 31	3anbi13d 1325	. . . . 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 4034	. . . . . . 7 |- ( b = B -> ( x < b <-> x < B ) )
34	33	3anbi2d 1328	. . . . . 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 2495	. . . . 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 4033	. . . . . . . 8 |- ( a = A -> ( a < b <-> A < b ) )
38		fveq2 5555	. . . . . . . . 9 |- ( a = A -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) )
39	38	breq1d 4040	. . . . . . . 8 |- ( a = A -> ( ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 <-> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 ) )
40	37, 39	3anbi12d 1324	. . . . . . 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 4033	. . . . . . . . 9 |- ( a = A -> ( a < x <-> A < x ) )
42	41	3anbi1d 1327	. . . . . . . 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 2495	. . . . . . 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 2494	. . . . 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 5694	. . . . . . . . 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 8402	. . . . . . . 8 |- ( ( ph /\ r e. RR ) -> ( ( F ` r ) - U ) e. RR )
49	48	fmpttd 5714	. . . . . . 7 |- ( ph -> ( r e. RR |-> ( ( F ` r ) - U ) ) : RR --> RR )
50		ax-resscn 7966	. . . . . . . . 9 |- RR C_ CC
51	50	a1i 9	. . . . . . . 8 |- ( ph -> RR C_ CC )
52	8	feqmptd 5611	. . . . . . . . . 10 |- ( ph -> F = ( r e. RR |-> ( F ` r ) ) )
53		ssid 3200	. . . . . . . . . . . 12 |- CC C_ CC
54		cncfss 14762	. . . . . . . . . . . 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 3178	. . . . . . . . . 10 |- ( ph -> F e. ( RR -cn-> CC ) )
57	52, 56	eqeltrrd 2271	. . . . . . . . 9 |- ( ph -> ( r e. RR |-> ( F ` r ) ) e. ( RR -cn-> CC ) )
58	10	recnd 8050	. . . . . . . . . 10 |- ( ph -> U e. CC )
59	53	a1i 9	. . . . . . . . . 10 |- ( ph -> CC C_ CC )
60		cncfmptc 14775	. . . . . . . . . 10 |- ( ( U e. CC /\ RR C_ CC /\ CC C_ CC ) -> ( r e. RR |-> U ) e. ( RR -cn-> CC ) )
61	58, 51, 59, 60	syl3anc 1249	. . . . . . . . 9 |- ( ph -> ( r e. RR |-> U ) e. ( RR -cn-> CC ) )
62	57, 61	subcncf 14792	. . . . . . . 8 |- ( ph -> ( r e. RR |-> ( ( F ` r ) - U ) ) e. ( RR -cn-> CC ) )
63		cncfcdm 14761	. . . . . . . 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 8008	. . . . . . . . 9 |- RR e. _V
68	67	mptex 5785	. . . . . . . 8 |- ( r e. RR |-> ( ( F ` r ) - U ) ) e. _V
69		eleq1 2256	. . . . . . . . 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 5554	. . . . . . . . . . . . . 14 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( f ` a ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) )
71	70	breq1d 4040	. . . . . . . . . . . . 13 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( ( f ` a ) < 0 <-> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 ) )
72		fveq1 5554	. . . . . . . . . . . . . 14 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( f ` b ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) )
73	72	breq2d 4042	. . . . . . . . . . . . 13 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( 0 < ( f ` b ) <-> 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) )
74	71, 73	3anbi23d 1326	. . . . . . . . . . . 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 5554	. . . . . . . . . . . . . . 15 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( f ` x ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) )
76	75	eqeq1d 2202	. . . . . . . . . . . . . 14 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( ( f ` x ) = 0 <-> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) )
77	76	3anbi3d 1329	. . . . . . . . . . . . 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 2495	. . . . . . . . . . . 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 2494	. . . . . . . . . 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 2494	. . . . . . . . 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 2855	. . . . . . 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 2870	. . . 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 2870	. . 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 8071	. . . . 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 8071	. . . . 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 1179	. . . 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 1047	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> A < x )
96		simprr2 1048	. . . . 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 10003	. . . 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 5695	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> ( F ` x ) e. RR )
102	101	recnd 8050	. . . 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 5555	. . . . . . 7 |- ( r = x -> ( F ` r ) = ( F ` x ) )
105	104	oveq1d 5934	. . . . . 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 8402	. . . . . 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 5652	. . . . 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 1049	. . . . 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 2228	. . . 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 8340	. . 3 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> ( F ` x ) = U )
112		fveqeq2 5564	. . . 4 |- ( c = x -> ( ( F ` c ) = U <-> ( F ` x ) = U ) )
113	112	rspcev 2865	. . 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 2616	1 |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   A.wal 1362   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   C_ wss 3154   class class class wbr 4030   |-> cmpt 4091   -->wf 5251   ` cfv 5255  (class class class)co 5919   CCcc 7872   RRcr 7873   0cc0 7874   RR*cxr 8055   < clt 8056   - cmin 8192   (,)cioo 9957   -cn->ccncf 14749

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-map 6706  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-xneg 9841  df-xadd 9842  df-ioo 9961  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-rest 12855  df-topgen 12874  df-psmet 14042  df-xmet 14043  df-met 14044  df-bl 14045  df-mopn 14046  df-top 14177  df-topon 14190  df-bases 14222  df-cn 14367  df-cnp 14368  df-tx 14432  df-cncf 14750

This theorem is referenced by:  ivthdichlem  14830