Theorem ivthle 23225

Description: The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-2014.)

Hypotheses

Ref	Expression
ivth.1	|- ( ph -> A e. RR )
ivth.2	|- ( ph -> B e. RR )
ivth.3	|- ( ph -> U e. RR )
ivth.4	|- ( ph -> A < B )
ivth.5	|- ( ph -> ( A [,] B ) C_ D )
ivth.7	|- ( ph -> F e. ( D -cn-> CC ) )
ivth.8	|- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
ivthle.9	|- ( ph -> ( ( F ` A ) <_ U /\ U <_ ( F ` B ) ) )

Assertion

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

Distinct variable groups:   x, c, B   D, c, x   F, c, x   ph, c, x   A, c, x   U, c, x

Proof of Theorem ivthle

Step	Hyp	Ref	Expression
1		ioossicc 12259	. . . . 5 |- ( A (,) B ) C_ ( A [,] B )
2		ivth.1	. . . . . . 7 |- ( ph -> A e. RR )
3	2	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( F ` A ) < U /\ U < ( F ` B ) ) ) -> A e. RR )
4		ivth.2	. . . . . . 7 |- ( ph -> B e. RR )
5	4	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( F ` A ) < U /\ U < ( F ` B ) ) ) -> B e. RR )
6		ivth.3	. . . . . . 7 |- ( ph -> U e. RR )
7	6	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( F ` A ) < U /\ U < ( F ` B ) ) ) -> U e. RR )
8		ivth.4	. . . . . . 7 |- ( ph -> A < B )
9	8	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( F ` A ) < U /\ U < ( F ` B ) ) ) -> A < B )
10		ivth.5	. . . . . . 7 |- ( ph -> ( A [,] B ) C_ D )
11	10	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( F ` A ) < U /\ U < ( F ` B ) ) ) -> ( A [,] B ) C_ D )
12		ivth.7	. . . . . . 7 |- ( ph -> F e. ( D -cn-> CC ) )
13	12	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( F ` A ) < U /\ U < ( F ` B ) ) ) -> F e. ( D -cn-> CC ) )
14		ivth.8	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
15	14	adantlr 751	. . . . . 6 |- ( ( ( ph /\ ( ( F ` A ) < U /\ U < ( F ` B ) ) ) /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
16		simpr 477	. . . . . 6 |- ( ( ph /\ ( ( F ` A ) < U /\ U < ( F ` B ) ) ) -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
17	3, 5, 7, 9, 11, 13, 15, 16	ivth 23223	. . . . 5 |- ( ( ph /\ ( ( F ` A ) < U /\ U < ( F ` B ) ) ) -> E. c e. ( A (,) B ) ( F ` c ) = U )
18		ssrexv 3667	. . . . 5 |- ( ( A (,) B ) C_ ( A [,] B ) -> ( E. c e. ( A (,) B ) ( F ` c ) = U -> E. c e. ( A [,] B ) ( F ` c ) = U ) )
19	1, 17, 18	mpsyl 68	. . . 4 |- ( ( ph /\ ( ( F ` A ) < U /\ U < ( F ` B ) ) ) -> E. c e. ( A [,] B ) ( F ` c ) = U )
20	19	anassrs 680	. . 3 |- ( ( ( ph /\ ( F ` A ) < U ) /\ U < ( F ` B ) ) -> E. c e. ( A [,] B ) ( F ` c ) = U )
21	2	rexrd 10089	. . . . . 6 |- ( ph -> A e. RR* )
22	4	rexrd 10089	. . . . . 6 |- ( ph -> B e. RR* )
23	2, 4, 8	ltled 10185	. . . . . 6 |- ( ph -> A <_ B )
24		ubicc2 12289	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
25	21, 22, 23, 24	syl3anc 1326	. . . . 5 |- ( ph -> B e. ( A [,] B ) )
26		eqcom 2629	. . . . . . 7 |- ( ( F ` c ) = U <-> U = ( F ` c ) )
27		fveq2 6191	. . . . . . . 8 |- ( c = B -> ( F ` c ) = ( F ` B ) )
28	27	eqeq2d 2632	. . . . . . 7 |- ( c = B -> ( U = ( F ` c ) <-> U = ( F ` B ) ) )
29	26, 28	syl5bb 272	. . . . . 6 |- ( c = B -> ( ( F ` c ) = U <-> U = ( F ` B ) ) )
30	29	rspcev 3309	. . . . 5 |- ( ( B e. ( A [,] B ) /\ U = ( F ` B ) ) -> E. c e. ( A [,] B ) ( F ` c ) = U )
31	25, 30	sylan 488	. . . 4 |- ( ( ph /\ U = ( F ` B ) ) -> E. c e. ( A [,] B ) ( F ` c ) = U )
32	31	adantlr 751	. . 3 |- ( ( ( ph /\ ( F ` A ) < U ) /\ U = ( F ` B ) ) -> E. c e. ( A [,] B ) ( F ` c ) = U )
33		ivthle.9	. . . . . 6 |- ( ph -> ( ( F ` A ) <_ U /\ U <_ ( F ` B ) ) )
34	33	simprd 479	. . . . 5 |- ( ph -> U <_ ( F ` B ) )
35	14	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
36		fveq2 6191	. . . . . . . . 9 |- ( x = B -> ( F ` x ) = ( F ` B ) )
37	36	eleq1d 2686	. . . . . . . 8 |- ( x = B -> ( ( F ` x ) e. RR <-> ( F ` B ) e. RR ) )
38	37	rspcv 3305	. . . . . . 7 |- ( B e. ( A [,] B ) -> ( A. x e. ( A [,] B ) ( F ` x ) e. RR -> ( F ` B ) e. RR ) )
39	25, 35, 38	sylc 65	. . . . . 6 |- ( ph -> ( F ` B ) e. RR )
40	6, 39	leloed 10180	. . . . 5 |- ( ph -> ( U <_ ( F ` B ) <-> ( U < ( F ` B ) \/ U = ( F ` B ) ) ) )
41	34, 40	mpbid 222	. . . 4 |- ( ph -> ( U < ( F ` B ) \/ U = ( F ` B ) ) )
42	41	adantr 481	. . 3 |- ( ( ph /\ ( F ` A ) < U ) -> ( U < ( F ` B ) \/ U = ( F ` B ) ) )
43	20, 32, 42	mpjaodan 827	. 2 |- ( ( ph /\ ( F ` A ) < U ) -> E. c e. ( A [,] B ) ( F ` c ) = U )
44		lbicc2 12288	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
45	21, 22, 23, 44	syl3anc 1326	. . 3 |- ( ph -> A e. ( A [,] B ) )
46		fveq2 6191	. . . . 5 |- ( c = A -> ( F ` c ) = ( F ` A ) )
47	46	eqeq1d 2624	. . . 4 |- ( c = A -> ( ( F ` c ) = U <-> ( F ` A ) = U ) )
48	47	rspcev 3309	. . 3 |- ( ( A e. ( A [,] B ) /\ ( F ` A ) = U ) -> E. c e. ( A [,] B ) ( F ` c ) = U )
49	45, 48	sylan 488	. 2 |- ( ( ph /\ ( F ` A ) = U ) -> E. c e. ( A [,] B ) ( F ` c ) = U )
50	33	simpld 475	. . 3 |- ( ph -> ( F ` A ) <_ U )
51		fveq2 6191	. . . . . . 7 |- ( x = A -> ( F ` x ) = ( F ` A ) )
52	51	eleq1d 2686	. . . . . 6 |- ( x = A -> ( ( F ` x ) e. RR <-> ( F ` A ) e. RR ) )
53	52	rspcv 3305	. . . . 5 |- ( A e. ( A [,] B ) -> ( A. x e. ( A [,] B ) ( F ` x ) e. RR -> ( F ` A ) e. RR ) )
54	45, 35, 53	sylc 65	. . . 4 |- ( ph -> ( F ` A ) e. RR )
55	54, 6	leloed 10180	. . 3 |- ( ph -> ( ( F ` A ) <_ U <-> ( ( F ` A ) < U \/ ( F ` A ) = U ) ) )
56	50, 55	mpbid 222	. 2 |- ( ph -> ( ( F ` A ) < U \/ ( F ` A ) = U ) )
57	43, 49, 56	mpjaodan 827	1 |- ( ph -> E. c e. ( A [,] B ) ( F ` c ) = U )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   [,]cicc 12178   -cn->ccncf 22679

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

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-iun 4522  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ioo 12179  df-icc 12182  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-cncf 22681

This theorem is referenced by:  ivthicc  23227  volivth  23375