ivthle2 - Metamath Proof Explorer

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Theorem ivthle2 25449

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 13384	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
2		ivth.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝐴 ∈ ℝ)
4		ivth.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝐵 ∈ ℝ)
6		ivth.3	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ)
7	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝑈 ∈ ℝ)
8		ivth.4	. . . . . . 7 ⊢ (𝜑 → 𝐴 < 𝐵)
9	8	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝐴 < 𝐵)
10		ivth.5	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
11	10	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → (𝐴[,]𝐵) ⊆ 𝐷)
12		ivth.7	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))
13	12	adantr 481	. . . . . 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 721	. . . . . 6 ⊢ (((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)
16		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴)))
17	3, 5, 7, 9, 11, 13, 15, 16	ivth2 25447	. . . . 5 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
18		ssrexv 3991	. . . . 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 468	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝐵) < 𝑈) ∧ 𝑈 < (𝐹‘𝐴)) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
21	2	rexrd 11193	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22	4	rexrd 11193	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
23	2, 4, 8	ltled 11292	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
24		lbicc2 13415	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
25	21, 22, 23, 24	syl3anc 1379	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
26		eqcom 2747	. . . . . . 7 ⊢ ((𝐹‘𝑐) = 𝑈 ↔ 𝑈 = (𝐹‘𝑐))
27		fveq2 6834	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → (𝐹‘𝑐) = (𝐹‘𝐴))
28	27	eqeq2d 2751	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (𝑈 = (𝐹‘𝑐) ↔ 𝑈 = (𝐹‘𝐴)))
29	26, 28	bitrid 284	. . . . . 6 ⊢ (𝑐 = 𝐴 → ((𝐹‘𝑐) = 𝑈 ↔ 𝑈 = (𝐹‘𝐴)))
30	29	rspcev 3567	. . . . 5 ⊢ ((𝐴 ∈ (𝐴[,]𝐵) ∧ 𝑈 = (𝐹‘𝐴)) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
31	25, 30	sylan 586	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = (𝐹‘𝐴)) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
32	31	adantlr 721	. . 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 496	. . . . 5 ⊢ (𝜑 → 𝑈 ≤ (𝐹‘𝐴))
35		fveq2 6834	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
36	35	eleq1d 2825	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐴) ∈ ℝ))
37	14	ralrimiva 3132	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
38	36, 37, 25	rspcdva 3568	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
39	6, 38	leloed 11287	. . . . 5 ⊢ (𝜑 → (𝑈 ≤ (𝐹‘𝐴) ↔ (𝑈 < (𝐹‘𝐴) ∨ 𝑈 = (𝐹‘𝐴))))
40	34, 39	mpbid 233	. . . 4 ⊢ (𝜑 → (𝑈 < (𝐹‘𝐴) ∨ 𝑈 = (𝐹‘𝐴)))
41	40	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐵) < 𝑈) → (𝑈 < (𝐹‘𝐴) ∨ 𝑈 = (𝐹‘𝐴)))
42	20, 32, 41	mpjaodan 966	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝐵) < 𝑈) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
43		ubicc2 13416	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
44	21, 22, 23, 43	syl3anc 1379	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
45		fveqeq2 6843	. . . 4 ⊢ (𝑐 = 𝐵 → ((𝐹‘𝑐) = 𝑈 ↔ (𝐹‘𝐵) = 𝑈))
46	45	rspcev 3567	. . 3 ⊢ ((𝐵 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝐵) = 𝑈) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
47	44, 46	sylan 586	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝐵) = 𝑈) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
48	33	simpld 495	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) ≤ 𝑈)
49		fveq2 6834	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
50	49	eleq1d 2825	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐵) ∈ ℝ))
51	50, 37, 44	rspcdva 3568	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
52	51, 6	leloed 11287	. . 3 ⊢ (𝜑 → ((𝐹‘𝐵) ≤ 𝑈 ↔ ((𝐹‘𝐵) < 𝑈 ∨ (𝐹‘𝐵) = 𝑈)))
53	48, 52	mpbid 233	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) < 𝑈 ∨ (𝐹‘𝐵) = 𝑈))
54	42, 47, 53	mpjaodan 966	1 ⊢ (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ∃wrex 3064 ⊆ wss 3890 class class class wbr 5079 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 ℝcr 11035 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 (,)cioo 13296 [,]cicc 13299 –cn→ccncf 24868

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-icc 13303 df-fz 13460 df-fzo 13607 df-seq 13962 df-exp 14022 df-hash 14291 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cn 23217 df-cnp 23218 df-tx 23552 df-hmeo 23745 df-xms 24310 df-ms 24311 df-tms 24312 df-cncf 24870

This theorem is referenced by: ivthicc 25450 recosf1o 26524

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