Theorem dvivth 23773

Description: Darboux' theorem, or the intermediate value theorem for derivatives. A differentiable function's derivative satisfies the intermediate value property, even though it may not be continuous (so that ivthicc 23227 does not directly apply). (Contributed by Mario Carneiro, 24-Feb-2015.)

Hypotheses

Ref	Expression
dvivth.1	|- ( ph -> M e. ( A (,) B ) )
dvivth.2	|- ( ph -> N e. ( A (,) B ) )
dvivth.3	|- ( ph -> F e. ( ( A (,) B ) -cn-> RR ) )
dvivth.4	|- ( ph -> dom ( RR _D F ) = ( A (,) B ) )

Assertion

Ref	Expression
dvivth	|- ( ph -> ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) C_ ran ( RR _D F ) )

Proof of Theorem dvivth

Dummy variables x w y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvivth.1	. . . . . . . . . 10 |- ( ph -> M e. ( A (,) B ) )
2	1	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> M e. ( A (,) B ) )
3		dvivth.2	. . . . . . . . . 10 |- ( ph -> N e. ( A (,) B ) )
4	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> N e. ( A (,) B ) )
5		dvivth.3	. . . . . . . . . . . . . . 15 |- ( ph -> F e. ( ( A (,) B ) -cn-> RR ) )
6		cncff 22696	. . . . . . . . . . . . . . 15 |- ( F e. ( ( A (,) B ) -cn-> RR ) -> F : ( A (,) B ) --> RR )
7	5, 6	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> F : ( A (,) B ) --> RR )
8	7	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ph /\ w e. ( A (,) B ) ) -> ( F ` w ) e. RR )
9	8	renegcld 10457	. . . . . . . . . . . 12 |- ( ( ph /\ w e. ( A (,) B ) ) -> -u ( F ` w ) e. RR )
10		eqid 2622	. . . . . . . . . . . 12 |- ( w e. ( A (,) B ) |-> -u ( F ` w ) ) = ( w e. ( A (,) B ) |-> -u ( F ` w ) )
11	9, 10	fmptd 6385	. . . . . . . . . . 11 |- ( ph -> ( w e. ( A (,) B ) |-> -u ( F ` w ) ) : ( A (,) B ) --> RR )
12		ax-resscn 9993	. . . . . . . . . . . 12 |- RR C_ CC
13		ssid 3624	. . . . . . . . . . . . . . 15 |- CC C_ CC
14		cncfss 22702	. . . . . . . . . . . . . . 15 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( ( A (,) B ) -cn-> RR ) C_ ( ( A (,) B ) -cn-> CC ) )
15	12, 13, 14	mp2an 708	. . . . . . . . . . . . . 14 |- ( ( A (,) B ) -cn-> RR ) C_ ( ( A (,) B ) -cn-> CC )
16	15, 5	sseldi 3601	. . . . . . . . . . . . 13 |- ( ph -> F e. ( ( A (,) B ) -cn-> CC ) )
17	10	negfcncf 22722	. . . . . . . . . . . . 13 |- ( F e. ( ( A (,) B ) -cn-> CC ) -> ( w e. ( A (,) B ) |-> -u ( F ` w ) ) e. ( ( A (,) B ) -cn-> CC ) )
18	16, 17	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( w e. ( A (,) B ) |-> -u ( F ` w ) ) e. ( ( A (,) B ) -cn-> CC ) )
19		cncffvrn 22701	. . . . . . . . . . . 12 |- ( ( RR C_ CC /\ ( w e. ( A (,) B ) |-> -u ( F ` w ) ) e. ( ( A (,) B ) -cn-> CC ) ) -> ( ( w e. ( A (,) B ) |-> -u ( F ` w ) ) e. ( ( A (,) B ) -cn-> RR ) <-> ( w e. ( A (,) B ) |-> -u ( F ` w ) ) : ( A (,) B ) --> RR ) )
20	12, 18, 19	sylancr 695	. . . . . . . . . . 11 |- ( ph -> ( ( w e. ( A (,) B ) |-> -u ( F ` w ) ) e. ( ( A (,) B ) -cn-> RR ) <-> ( w e. ( A (,) B ) |-> -u ( F ` w ) ) : ( A (,) B ) --> RR ) )
21	11, 20	mpbird 247	. . . . . . . . . 10 |- ( ph -> ( w e. ( A (,) B ) |-> -u ( F ` w ) ) e. ( ( A (,) B ) -cn-> RR ) )
22	21	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( w e. ( A (,) B ) |-> -u ( F ` w ) ) e. ( ( A (,) B ) -cn-> RR ) )
23		reelprrecn 10028	. . . . . . . . . . . . 13 |- RR e. { RR , CC }
24	23	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> RR e. { RR , CC } )
25	7	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> F : ( A (,) B ) --> RR )
26	25	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) /\ w e. ( A (,) B ) ) -> ( F ` w ) e. RR )
27	26	recnd 10068	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) /\ w e. ( A (,) B ) ) -> ( F ` w ) e. CC )
28		fvexd 6203	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) /\ w e. ( A (,) B ) ) -> ( ( RR _D F ) ` w ) e. _V )
29	25	feqmptd 6249	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> F = ( w e. ( A (,) B ) |-> ( F ` w ) ) )
30	29	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( RR _D F ) = ( RR _D ( w e. ( A (,) B ) |-> ( F ` w ) ) ) )
31		ioossre 12235	. . . . . . . . . . . . . . . . 17 |- ( A (,) B ) C_ RR
32		dvfre 23714	. . . . . . . . . . . . . . . . 17 |- ( ( F : ( A (,) B ) --> RR /\ ( A (,) B ) C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )
33	7, 31, 32	sylancl 694	. . . . . . . . . . . . . . . 16 |- ( ph -> ( RR _D F ) : dom ( RR _D F ) --> RR )
34		dvivth.4	. . . . . . . . . . . . . . . . 17 |- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
35	34	feq2d 6031	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> RR <-> ( RR _D F ) : ( A (,) B ) --> RR ) )
36	33, 35	mpbid 222	. . . . . . . . . . . . . . 15 |- ( ph -> ( RR _D F ) : ( A (,) B ) --> RR )
37	36	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( RR _D F ) : ( A (,) B ) --> RR )
38	37	feqmptd 6249	. . . . . . . . . . . . 13 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( RR _D F ) = ( w e. ( A (,) B ) |-> ( ( RR _D F ) ` w ) ) )
39	30, 38	eqtr3d 2658	. . . . . . . . . . . 12 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( RR _D ( w e. ( A (,) B ) |-> ( F ` w ) ) ) = ( w e. ( A (,) B ) |-> ( ( RR _D F ) ` w ) ) )
40	24, 27, 28, 39	dvmptneg 23729	. . . . . . . . . . 11 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( RR _D ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ) = ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) )
41	40	dmeqd 5326	. . . . . . . . . 10 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> dom ( RR _D ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ) = dom ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) )
42		dmmptg 5632	. . . . . . . . . . 11 |- ( A. w e. ( A (,) B ) -u ( ( RR _D F ) ` w ) e. _V -> dom ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) = ( A (,) B ) )
43		negex 10279	. . . . . . . . . . . 12 |- -u ( ( RR _D F ) ` w ) e. _V
44	43	a1i 11	. . . . . . . . . . 11 |- ( w e. ( A (,) B ) -> -u ( ( RR _D F ) ` w ) e. _V )
45	42, 44	mprg 2926	. . . . . . . . . 10 |- dom ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) = ( A (,) B )
46	41, 45	syl6eq 2672	. . . . . . . . 9 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> dom ( RR _D ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ) = ( A (,) B ) )
47		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> M < N )
48		simprr 796	. . . . . . . . . . 11 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) )
49	36, 1	ffvelrnd 6360	. . . . . . . . . . . . 13 |- ( ph -> ( ( RR _D F ) ` M ) e. RR )
50	49	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( ( RR _D F ) ` M ) e. RR )
51	3, 34	eleqtrrd 2704	. . . . . . . . . . . . . 14 |- ( ph -> N e. dom ( RR _D F ) )
52	33, 51	ffvelrnd 6360	. . . . . . . . . . . . 13 |- ( ph -> ( ( RR _D F ) ` N ) e. RR )
53	52	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( ( RR _D F ) ` N ) e. RR )
54		iccssre 12255	. . . . . . . . . . . . . . 15 |- ( ( ( ( RR _D F ) ` M ) e. RR /\ ( ( RR _D F ) ` N ) e. RR ) -> ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) C_ RR )
55	49, 52, 54	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ph -> ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) C_ RR )
56	55	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) C_ RR )
57	56, 48	sseldd 3604	. . . . . . . . . . . 12 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> x e. RR )
58		iccneg 12293	. . . . . . . . . . . 12 |- ( ( ( ( RR _D F ) ` M ) e. RR /\ ( ( RR _D F ) ` N ) e. RR /\ x e. RR ) -> ( x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) <-> -u x e. ( -u ( ( RR _D F ) ` N ) [,] -u ( ( RR _D F ) ` M ) ) ) )
59	50, 53, 57, 58	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) <-> -u x e. ( -u ( ( RR _D F ) ` N ) [,] -u ( ( RR _D F ) ` M ) ) ) )
60	48, 59	mpbid 222	. . . . . . . . . 10 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> -u x e. ( -u ( ( RR _D F ) ` N ) [,] -u ( ( RR _D F ) ` M ) ) )
61	40	fveq1d 6193	. . . . . . . . . . . 12 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( ( RR _D ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ) ` N ) = ( ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) ` N ) )
62		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( w = N -> ( ( RR _D F ) ` w ) = ( ( RR _D F ) ` N ) )
63	62	negeqd 10275	. . . . . . . . . . . . . 14 |- ( w = N -> -u ( ( RR _D F ) ` w ) = -u ( ( RR _D F ) ` N ) )
64		eqid 2622	. . . . . . . . . . . . . 14 |- ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) = ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) )
65		negex 10279	. . . . . . . . . . . . . 14 |- -u ( ( RR _D F ) ` N ) e. _V
66	63, 64, 65	fvmpt 6282	. . . . . . . . . . . . 13 |- ( N e. ( A (,) B ) -> ( ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) ` N ) = -u ( ( RR _D F ) ` N ) )
67	4, 66	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) ` N ) = -u ( ( RR _D F ) ` N ) )
68	61, 67	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( ( RR _D ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ) ` N ) = -u ( ( RR _D F ) ` N ) )
69	40	fveq1d 6193	. . . . . . . . . . . 12 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( ( RR _D ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ) ` M ) = ( ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) ` M ) )
70		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( w = M -> ( ( RR _D F ) ` w ) = ( ( RR _D F ) ` M ) )
71	70	negeqd 10275	. . . . . . . . . . . . . 14 |- ( w = M -> -u ( ( RR _D F ) ` w ) = -u ( ( RR _D F ) ` M ) )
72		negex 10279	. . . . . . . . . . . . . 14 |- -u ( ( RR _D F ) ` M ) e. _V
73	71, 64, 72	fvmpt 6282	. . . . . . . . . . . . 13 |- ( M e. ( A (,) B ) -> ( ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) ` M ) = -u ( ( RR _D F ) ` M ) )
74	2, 73	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) ` M ) = -u ( ( RR _D F ) ` M ) )
75	69, 74	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( ( RR _D ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ) ` M ) = -u ( ( RR _D F ) ` M ) )
76	68, 75	oveq12d 6668	. . . . . . . . . 10 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( ( ( RR _D ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ) ` N ) [,] ( ( RR _D ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ) ` M ) ) = ( -u ( ( RR _D F ) ` N ) [,] -u ( ( RR _D F ) ` M ) ) )
77	60, 76	eleqtrrd 2704	. . . . . . . . 9 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> -u x e. ( ( ( RR _D ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ) ` N ) [,] ( ( RR _D ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ) ` M ) ) )
78		eqid 2622	. . . . . . . . 9 |- ( y e. ( A (,) B ) |-> ( ( ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ` y ) - ( -u x x. y ) ) ) = ( y e. ( A (,) B ) |-> ( ( ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ` y ) - ( -u x x. y ) ) )
79	2, 4, 22, 46, 47, 77, 78	dvivthlem2 23772	. . . . . . . 8 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> -u x e. ran ( RR _D ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ) )
80	40	rneqd 5353	. . . . . . . 8 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ran ( RR _D ( w e. ( A (,) B ) |-> -u ( F ` w ) ) ) = ran ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) )
81	79, 80	eleqtrd 2703	. . . . . . 7 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> -u x e. ran ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) )
82		negex 10279	. . . . . . . 8 |- -u x e. _V
83	64	elrnmpt 5372	. . . . . . . 8 |- ( -u x e. _V -> ( -u x e. ran ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) <-> E. w e. ( A (,) B ) -u x = -u ( ( RR _D F ) ` w ) ) )
84	82, 83	ax-mp 5	. . . . . . 7 |- ( -u x e. ran ( w e. ( A (,) B ) |-> -u ( ( RR _D F ) ` w ) ) <-> E. w e. ( A (,) B ) -u x = -u ( ( RR _D F ) ` w ) )
85	81, 84	sylib 208	. . . . . 6 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> E. w e. ( A (,) B ) -u x = -u ( ( RR _D F ) ` w ) )
86	57	recnd 10068	. . . . . . . . . 10 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> x e. CC )
87	86	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) /\ w e. ( A (,) B ) ) -> x e. CC )
88	24, 27, 28, 39	dvmptcl 23722	. . . . . . . . 9 |- ( ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) /\ w e. ( A (,) B ) ) -> ( ( RR _D F ) ` w ) e. CC )
89	87, 88	neg11ad 10388	. . . . . . . 8 |- ( ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) /\ w e. ( A (,) B ) ) -> ( -u x = -u ( ( RR _D F ) ` w ) <-> x = ( ( RR _D F ) ` w ) ) )
90		eqcom 2629	. . . . . . . 8 |- ( x = ( ( RR _D F ) ` w ) <-> ( ( RR _D F ) ` w ) = x )
91	89, 90	syl6bb 276	. . . . . . 7 |- ( ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) /\ w e. ( A (,) B ) ) -> ( -u x = -u ( ( RR _D F ) ` w ) <-> ( ( RR _D F ) ` w ) = x ) )
92	91	rexbidva 3049	. . . . . 6 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( E. w e. ( A (,) B ) -u x = -u ( ( RR _D F ) ` w ) <-> E. w e. ( A (,) B ) ( ( RR _D F ) ` w ) = x ) )
93	85, 92	mpbid 222	. . . . 5 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> E. w e. ( A (,) B ) ( ( RR _D F ) ` w ) = x )
94		ffn 6045	. . . . . . 7 |- ( ( RR _D F ) : ( A (,) B ) --> RR -> ( RR _D F ) Fn ( A (,) B ) )
95	37, 94	syl 17	. . . . . 6 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( RR _D F ) Fn ( A (,) B ) )
96		fvelrnb 6243	. . . . . 6 |- ( ( RR _D F ) Fn ( A (,) B ) -> ( x e. ran ( RR _D F ) <-> E. w e. ( A (,) B ) ( ( RR _D F ) ` w ) = x ) )
97	95, 96	syl 17	. . . . 5 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> ( x e. ran ( RR _D F ) <-> E. w e. ( A (,) B ) ( ( RR _D F ) ` w ) = x ) )
98	93, 97	mpbird 247	. . . 4 |- ( ( ph /\ ( M < N /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> x e. ran ( RR _D F ) )
99	98	expr 643	. . 3 |- ( ( ph /\ M < N ) -> ( x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) -> x e. ran ( RR _D F ) ) )
100	99	ssrdv 3609	. 2 |- ( ( ph /\ M < N ) -> ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) C_ ran ( RR _D F ) )
101		fveq2 6191	. . . . 5 |- ( M = N -> ( ( RR _D F ) ` M ) = ( ( RR _D F ) ` N ) )
102	101	oveq1d 6665	. . . 4 |- ( M = N -> ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) = ( ( ( RR _D F ) ` N ) [,] ( ( RR _D F ) ` N ) ) )
103	52	rexrd 10089	. . . . 5 |- ( ph -> ( ( RR _D F ) ` N ) e. RR* )
104		iccid 12220	. . . . 5 |- ( ( ( RR _D F ) ` N ) e. RR* -> ( ( ( RR _D F ) ` N ) [,] ( ( RR _D F ) ` N ) ) = { ( ( RR _D F ) ` N ) } )
105	103, 104	syl 17	. . . 4 |- ( ph -> ( ( ( RR _D F ) ` N ) [,] ( ( RR _D F ) ` N ) ) = { ( ( RR _D F ) ` N ) } )
106	102, 105	sylan9eqr 2678	. . 3 |- ( ( ph /\ M = N ) -> ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) = { ( ( RR _D F ) ` N ) } )
107		ffn 6045	. . . . . . 7 |- ( ( RR _D F ) : dom ( RR _D F ) --> RR -> ( RR _D F ) Fn dom ( RR _D F ) )
108	33, 107	syl 17	. . . . . 6 |- ( ph -> ( RR _D F ) Fn dom ( RR _D F ) )
109		fnfvelrn 6356	. . . . . 6 |- ( ( ( RR _D F ) Fn dom ( RR _D F ) /\ N e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` N ) e. ran ( RR _D F ) )
110	108, 51, 109	syl2anc 693	. . . . 5 |- ( ph -> ( ( RR _D F ) ` N ) e. ran ( RR _D F ) )
111	110	snssd 4340	. . . 4 |- ( ph -> { ( ( RR _D F ) ` N ) } C_ ran ( RR _D F ) )
112	111	adantr 481	. . 3 |- ( ( ph /\ M = N ) -> { ( ( RR _D F ) ` N ) } C_ ran ( RR _D F ) )
113	106, 112	eqsstrd 3639	. 2 |- ( ( ph /\ M = N ) -> ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) C_ ran ( RR _D F ) )
114	3	adantr 481	. . . . 5 |- ( ( ph /\ ( N < M /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> N e. ( A (,) B ) )
115	1	adantr 481	. . . . 5 |- ( ( ph /\ ( N < M /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> M e. ( A (,) B ) )
116	5	adantr 481	. . . . 5 |- ( ( ph /\ ( N < M /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> F e. ( ( A (,) B ) -cn-> RR ) )
117	34	adantr 481	. . . . 5 |- ( ( ph /\ ( N < M /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> dom ( RR _D F ) = ( A (,) B ) )
118		simprl 794	. . . . 5 |- ( ( ph /\ ( N < M /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> N < M )
119		simprr 796	. . . . 5 |- ( ( ph /\ ( N < M /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) )
120		eqid 2622	. . . . 5 |- ( y e. ( A (,) B ) |-> ( ( F ` y ) - ( x x. y ) ) ) = ( y e. ( A (,) B ) |-> ( ( F ` y ) - ( x x. y ) ) )
121	114, 115, 116, 117, 118, 119, 120	dvivthlem2 23772	. . . 4 |- ( ( ph /\ ( N < M /\ x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) ) ) -> x e. ran ( RR _D F ) )
122	121	expr 643	. . 3 |- ( ( ph /\ N < M ) -> ( x e. ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) -> x e. ran ( RR _D F ) ) )
123	122	ssrdv 3609	. 2 |- ( ( ph /\ N < M ) -> ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) C_ ran ( RR _D F ) )
124	31, 1	sseldi 3601	. . 3 |- ( ph -> M e. RR )
125	31, 3	sseldi 3601	. . 3 |- ( ph -> N e. RR )
126	124, 125	lttri4d 10178	. 2 |- ( ph -> ( M < N \/ M = N \/ N < M ) )
127	100, 113, 123, 126	mpjao3dan 1395	1 |- ( ph -> ( ( ( RR _D F ) ` M ) [,] ( ( RR _D F ) ` N ) ) C_ ran ( RR _D F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {csn 4177   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   x. cmul 9941   RR*cxr 10073   < clt 10074   - cmin 10266   -ucneg 10267   (,)cioo 12175   [,]cicc 12178   -cn->ccncf 22679   _D cdv 23627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  dvne0  23774