Theorem ivthreinc 15871

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 15869). 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 2238	. . . . . 6 |- ( r e. RR |-> ( ( F ` r ) - U ) ) = ( r e. RR |-> ( ( F ` r ) - U ) )
3		fveq2 5698	. . . . . . 7 |- ( r = A -> ( F ` r ) = ( F ` A ) )
4	3	oveq1d 6103	. . . . . 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 15803	. . . . . . . . 9 |- ( F e. ( RR -cn-> RR ) -> F : RR --> RR )
8	6, 7	syl 14	. . . . . . . 8 |- ( ph -> F : RR --> RR )
9	8, 5	ffvelcdmd 5847	. . . . . . 7 |- ( ph -> ( F ` A ) e. RR )
10		ivthreinc.3	. . . . . . 7 |- ( ph -> U e. RR )
11	9, 10	resubcld 8715	. . . . . 6 |- ( ph -> ( ( F ` A ) - U ) e. RR )
12	2, 4, 5, 11	fvmptd3 5802	. . . . 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 8906	. . . . . 6 |- ( ph -> ( ( ( F ` A ) - U ) < 0 <-> ( F ` A ) < U ) )
16	14, 15	mpbird 167	. . . . 5 |- ( ph -> ( ( F ` A ) - U ) < 0 )
17	12, 16	eqbrtrd 4153	. . . 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 5847	. . . . . . 7 |- ( ph -> ( F ` B ) e. RR )
21	10, 20	posdifd 8867	. . . . . 6 |- ( ph -> ( U < ( F ` B ) <-> 0 < ( ( F ` B ) - U ) ) )
22	18, 21	mpbid 147	. . . . 5 |- ( ph -> 0 < ( ( F ` B ) - U ) )
23		fveq2 5698	. . . . . . 7 |- ( r = B -> ( F ` r ) = ( F ` B ) )
24	23	oveq1d 6103	. . . . . 6 |- ( r = B -> ( ( F ` r ) - U ) = ( ( F ` B ) - U ) )
25	20, 10	resubcld 8715	. . . . . 6 |- ( ph -> ( ( F ` B ) - U ) e. RR )
26	2, 24, 19, 25	fvmptd3 5802	. . . . 5 |- ( ph -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) = ( ( F ` B ) - U ) )
27	22, 26	breqtrrd 4159	. . . 4 |- ( ph -> 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) )
28	1, 17, 27	3jca 1208	. . 3 |- ( ph -> ( A < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 /\ 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) ) )
29		breq2 4135	. . . . . 6 |- ( b = B -> ( A < b <-> A < B ) )
30		fveq2 5698	. . . . . . 7 |- ( b = B -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) )
31	30	breq2d 4143	. . . . . 6 |- ( b = B -> ( 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) <-> 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` B ) ) )
32	29, 31	3anbi13d 1355	. . . . 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 4135	. . . . . . 7 |- ( b = B -> ( x < b <-> x < B ) )
34	33	3anbi2d 1358	. . . . . 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 2551	. . . . 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 4134	. . . . . . . 8 |- ( a = A -> ( a < b <-> A < b ) )
38		fveq2 5698	. . . . . . . . 9 |- ( a = A -> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) )
39	38	breq1d 4141	. . . . . . . 8 |- ( a = A -> ( ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 <-> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` A ) < 0 ) )
40	37, 39	3anbi12d 1354	. . . . . . 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 4134	. . . . . . . . 9 |- ( a = A -> ( a < x <-> A < x ) )
42	41	3anbi1d 1357	. . . . . . . 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 2551	. . . . . . 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 2550	. . . . 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 5846	. . . . . . . . 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 8715	. . . . . . . 8 |- ( ( ph /\ r e. RR ) -> ( ( F ` r ) - U ) e. RR )
49	48	fmpttd 5866	. . . . . . 7 |- ( ph -> ( r e. RR |-> ( ( F ` r ) - U ) ) : RR --> RR )
50		ax-resscn 8277	. . . . . . . . 9 |- RR C_ CC
51	50	a1i 9	. . . . . . . 8 |- ( ph -> RR C_ CC )
52	8	feqmptd 5759	. . . . . . . . . 10 |- ( ph -> F = ( r e. RR |-> ( F ` r ) ) )
53		ssid 3268	. . . . . . . . . . . 12 |- CC C_ CC
54		cncfss 15809	. . . . . . . . . . . 12 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( RR -cn-> RR ) C_ ( RR -cn-> CC ) )
55	50, 53, 54	mp2an 430	. . . . . . . . . . 11 |- ( RR -cn-> RR ) C_ ( RR -cn-> CC )
56	55, 6	sselid 3246	. . . . . . . . . 10 |- ( ph -> F e. ( RR -cn-> CC ) )
57	52, 56	eqeltrrd 2316	. . . . . . . . 9 |- ( ph -> ( r e. RR |-> ( F ` r ) ) e. ( RR -cn-> CC ) )
58	10	recnd 8360	. . . . . . . . . 10 |- ( ph -> U e. CC )
59	53	a1i 9	. . . . . . . . . 10 |- ( ph -> CC C_ CC )
60		cncfmptc 15822	. . . . . . . . . 10 |- ( ( U e. CC /\ RR C_ CC /\ CC C_ CC ) -> ( r e. RR |-> U ) e. ( RR -cn-> CC ) )
61	58, 51, 59, 60	syl3anc 1278	. . . . . . . . 9 |- ( ph -> ( r e. RR |-> U ) e. ( RR -cn-> CC ) )
62	57, 61	subcncf 15839	. . . . . . . 8 |- ( ph -> ( r e. RR |-> ( ( F ` r ) - U ) ) e. ( RR -cn-> CC ) )
63		cncfcdm 15808	. . . . . . . 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 415	. . . . . . 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 8319	. . . . . . . . 9 |- RR e. _V
68	67	mptex 5946	. . . . . . . 8 |- ( r e. RR |-> ( ( F ` r ) - U ) ) e. _V
69		eleq1 2301	. . . . . . . . 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 5697	. . . . . . . . . . . . . 14 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( f ` a ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) )
71	70	breq1d 4141	. . . . . . . . . . . . 13 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( ( f ` a ) < 0 <-> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` a ) < 0 ) )
72		fveq1 5697	. . . . . . . . . . . . . 14 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( f ` b ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) )
73	72	breq2d 4143	. . . . . . . . . . . . 13 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( 0 < ( f ` b ) <-> 0 < ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` b ) ) )
74	71, 73	3anbi23d 1356	. . . . . . . . . . . 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 5697	. . . . . . . . . . . . . . 15 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( f ` x ) = ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) )
76	75	eqeq1d 2247	. . . . . . . . . . . . . 14 |- ( f = ( r e. RR |-> ( ( F ` r ) - U ) ) -> ( ( f ` x ) = 0 <-> ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) )
77	76	3anbi3d 1359	. . . . . . . . . . . . 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 2551	. . . . . . . . . . . 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 2550	. . . . . . . . . 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 2550	. . . . . . . . 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 2919	. . . . . . 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 2934	. . . 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 2934	. . 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 8381	. . . . 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 8381	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> B e. RR* )
93		simprl 535	. . . . 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 1208	. . . 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 1076	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> A < x )
96		simprr2 1077	. . . . 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 10353	. . . 4 |- ( x e. ( A (,) B ) <-> ( ( A e. RR* /\ B e. RR* /\ x e. RR ) /\ ( A < x /\ x < B ) ) )
99	94, 97, 98	sylanbrc 421	. . 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 5847	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> ( F ` x ) e. RR )
102	101	recnd 8360	. . . 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 5698	. . . . . . 7 |- ( r = x -> ( F ` r ) = ( F ` x ) )
105	104	oveq1d 6103	. . . . . 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 8715	. . . . . 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 5802	. . . . 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 1078	. . . . 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 2273	. . . 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 8652	. . 3 |- ( ( ph /\ ( x e. RR /\ ( A < x /\ x < B /\ ( ( r e. RR |-> ( ( F ` r ) - U ) ) ` x ) = 0 ) ) ) -> ( F ` x ) = U )
112		fveqeq2 5707	. . . 4 |- ( c = x -> ( ( F ` c ) = U <-> ( F ` x ) = U ) )
113	112	rspcev 2929	. . 3 |- ( ( x e. ( A (,) B ) /\ ( F ` x ) = U ) -> E. c e. ( A (,) B ) ( F ` c ) = U )
114	99, 111, 113	syl2anc 415	. 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 2673	1 |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   A.wal 1400   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   C_ wss 3220   class class class wbr 4131   |-> cmpt 4193   -->wf 5376   ` cfv 5380  (class class class)co 6088   CCcc 8183   RRcr 8184   0cc0 8185   RR*cxr 8365   < clt 8366   - cmin 8504   (,)cioo 10307   -cn->ccncf 15796

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-map 6928  df-sup 7330  df-inf 7331  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-xneg 10191  df-xadd 10192  df-ioo 10311  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-rest 13654  df-topgen 13673  df-psmet 14989  df-xmet 14990  df-met 14991  df-bl 14992  df-mopn 14993  df-top 15224  df-topon 15237  df-bases 15269  df-cn 15414  df-cnp 15415  df-tx 15479  df-cncf 15797

This theorem is used by:  ivthdichlem  15877