ivthle2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ivthle2		Structured version Visualization version GIF version

Theorem ivthle2 24052

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

**Hypotheses**
Ref	Expression
ivth.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
ivth.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
ivth.3	⊢ (𝜑 → 𝑈 ∈ ℝ)
ivth.4	⊢ (𝜑 → 𝐴 < 𝐵)
ivth.5	⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
ivth.7	⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))
ivth.8	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)
ivthle2.9	⊢ (𝜑 → ((𝐹‘𝐵) ≤ 𝑈 ∧ 𝑈 ≤ (𝐹‘𝐴)))

**Assertion**
Ref	Expression
ivthle2	⊢ (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)

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

**Proof of Theorem ivthle2**
Step	Hyp	Ref	Expression
1		ioossicc 12816	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
2		ivth.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝐴 ∈ ℝ)
4		ivth.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	4	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝐵 ∈ ℝ)
6		ivth.3	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ)
7	6	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝑈 ∈ ℝ)
8		ivth.4	. . . . . . 7 ⊢ (𝜑 → 𝐴 < 𝐵)
9	8	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝐴 < 𝐵)
10		ivth.5	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
11	10	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → (𝐴[,]𝐵) ⊆ 𝐷)
12		ivth.7	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))
13	12	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝐹 ∈ (𝐷–cn→ℂ))
14		ivth.8	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)
15	14	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)
16		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴)))
17	3, 5, 7, 9, 11, 13, 15, 16	ivth2 24050	. . . . 5 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
18		ssrexv 4033	. . . . 5 ⊢ ((𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) → (∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈))
19	1, 17, 18	mpsyl 68	. . . 4 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
20	19	anassrs 470	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝐵) < 𝑈) ∧ 𝑈 < (𝐹‘𝐴)) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
21	2	rexrd 10685	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22	4	rexrd 10685	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
23	2, 4, 8	ltled 10782	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
24		lbicc2 12846	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
25	21, 22, 23, 24	syl3anc 1367	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
26		eqcom 2828	. . . . . . 7 ⊢ ((𝐹‘𝑐) = 𝑈 ↔ 𝑈 = (𝐹‘𝑐))
27		fveq2 6664	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → (𝐹‘𝑐) = (𝐹‘𝐴))
28	27	eqeq2d 2832	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (𝑈 = (𝐹‘𝑐) ↔ 𝑈 = (𝐹‘𝐴)))
29	26, 28	syl5bb 285	. . . . . 6 ⊢ (𝑐 = 𝐴 → ((𝐹‘𝑐) = 𝑈 ↔ 𝑈 = (𝐹‘𝐴)))
30	29	rspcev 3622	. . . . 5 ⊢ ((𝐴 ∈ (𝐴[,]𝐵) ∧ 𝑈 = (𝐹‘𝐴)) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
31	25, 30	sylan 582	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = (𝐹‘𝐴)) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
32	31	adantlr 713	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝐵) < 𝑈) ∧ 𝑈 = (𝐹‘𝐴)) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
33		ivthle2.9	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐵) ≤ 𝑈 ∧ 𝑈 ≤ (𝐹‘𝐴)))
34	33	simprd 498	. . . . 5 ⊢ (𝜑 → 𝑈 ≤ (𝐹‘𝐴))
35		fveq2 6664	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
36	35	eleq1d 2897	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐴) ∈ ℝ))
37	14	ralrimiva 3182	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
38	36, 37, 25	rspcdva 3624	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
39	6, 38	leloed 10777	. . . . 5 ⊢ (𝜑 → (𝑈 ≤ (𝐹‘𝐴) ↔ (𝑈 < (𝐹‘𝐴) ∨ 𝑈 = (𝐹‘𝐴))))
40	34, 39	mpbid 234	. . . 4 ⊢ (𝜑 → (𝑈 < (𝐹‘𝐴) ∨ 𝑈 = (𝐹‘𝐴)))
41	40	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐵) < 𝑈) → (𝑈 < (𝐹‘𝐴) ∨ 𝑈 = (𝐹‘𝐴)))
42	20, 32, 41	mpjaodan 955	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝐵) < 𝑈) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
43		ubicc2 12847	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
44	21, 22, 23, 43	syl3anc 1367	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
45		fveqeq2 6673	. . . 4 ⊢ (𝑐 = 𝐵 → ((𝐹‘𝑐) = 𝑈 ↔ (𝐹‘𝐵) = 𝑈))
46	45	rspcev 3622	. . 3 ⊢ ((𝐵 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝐵) = 𝑈) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
47	44, 46	sylan 582	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝐵) = 𝑈) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
48	33	simpld 497	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) ≤ 𝑈)
49		fveq2 6664	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
50	49	eleq1d 2897	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐵) ∈ ℝ))
51	50, 37, 44	rspcdva 3624	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
52	51, 6	leloed 10777	. . 3 ⊢ (𝜑 → ((𝐹‘𝐵) ≤ 𝑈 ↔ ((𝐹‘𝐵) < 𝑈 ∨ (𝐹‘𝐵) = 𝑈)))
53	48, 52	mpbid 234	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) < 𝑈 ∨ (𝐹‘𝐵) = 𝑈))
54	42, 47, 53	mpjaodan 955	1 ⊢ (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ⊆ wss 3935 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 ℝ^*cxr 10668 < clt 10669 ≤ cle 10670 (,)cioo 12732 [,]cicc 12735 –cn→ccncf 23478

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-mulf 10611

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cn 21829 df-cnp 21830 df-tx 22164 df-hmeo 22357 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480

This theorem is referenced by: ivthicc 24053 recosf1o 25113

W3C validator