ivthle2 - Metamath Proof Explorer

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Theorem ivthle2 25808

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 13591	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
2		ivth.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝐴 ∈ ℝ)
4		ivth.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝐵 ∈ ℝ)
6		ivth.3	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ)
7	6	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝑈 ∈ ℝ)
8		ivth.4	. . . . . . 7 ⊢ (𝜑 → 𝐴 < 𝐵)
9	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝐴 < 𝐵)
10		ivth.5	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
11	10	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → (𝐴[,]𝐵) ⊆ 𝐷)
12		ivth.7	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))
13	12	adantr 486	. . . . . 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 728	. . . . . 6 ⊢ (((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)
16		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴)))
17	3, 5, 7, 9, 11, 13, 15, 16	ivth2 25806	. . . . 5 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
18		ssrexv 3983	. . . . 5 ⊢ ((𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) → (∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈))
19	1, 17, 18	mpsyl 69	. . . 4 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
20	19	anassrs 473	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝐵) < 𝑈) ∧ 𝑈 < (𝐹‘𝐴)) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
21	2	rexrd 11386	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22	4	rexrd 11386	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
23	2, 4, 8	ltled 11485	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
24		lbicc2 13622	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
25	21, 22, 23, 24	syl3anc 1399	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
26		eqcom 2770	. . . . . . 7 ⊢ ((𝐹‘𝑐) = 𝑈 ↔ 𝑈 = (𝐹‘𝑐))
27		fveq2 6908	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → (𝐹‘𝑐) = (𝐹‘𝐴))
28	27	eqeq2d 2774	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (𝑈 = (𝐹‘𝑐) ↔ 𝑈 = (𝐹‘𝐴)))
29	26, 28	bitrid 286	. . . . . 6 ⊢ (𝑐 = 𝐴 → ((𝐹‘𝑐) = 𝑈 ↔ 𝑈 = (𝐹‘𝐴)))
30	29	rspcev 3580	. . . . 5 ⊢ ((𝐴 ∈ (𝐴[,]𝐵) ∧ 𝑈 = (𝐹‘𝐴)) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
31	25, 30	sylan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = (𝐹‘𝐴)) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
32	31	adantlr 728	. . 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 501	. . . . 5 ⊢ (𝜑 → 𝑈 ≤ (𝐹‘𝐴))
35		fveq2 6908	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
36	35	eleq1d 2849	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐴) ∈ ℝ))
37	14	ralrimiva 3158	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
38	36, 37, 25	rspcdva 3581	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
39	6, 38	leloed 11480	. . . . 5 ⊢ (𝜑 → (𝑈 ≤ (𝐹‘𝐴) ↔ (𝑈 < (𝐹‘𝐴) ∨ 𝑈 = (𝐹‘𝐴))))
40	34, 39	mpbid 235	. . . 4 ⊢ (𝜑 → (𝑈 < (𝐹‘𝐴) ∨ 𝑈 = (𝐹‘𝐴)))
41	40	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐵) < 𝑈) → (𝑈 < (𝐹‘𝐴) ∨ 𝑈 = (𝐹‘𝐴)))
42	20, 32, 41	mpjaodan 973	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝐵) < 𝑈) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
43		ubicc2 13623	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
44	21, 22, 23, 43	syl3anc 1399	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
45		fveqeq2 6917	. . . 4 ⊢ (𝑐 = 𝐵 → ((𝐹‘𝑐) = 𝑈 ↔ (𝐹‘𝐵) = 𝑈))
46	45	rspcev 3580	. . 3 ⊢ ((𝐵 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝐵) = 𝑈) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
47	44, 46	sylan 592	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝐵) = 𝑈) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
48	33	simpld 500	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) ≤ 𝑈)
49		fveq2 6908	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
50	49	eleq1d 2849	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐵) ∈ ℝ))
51	50, 37, 44	rspcdva 3581	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
52	51, 6	leloed 11480	. . 3 ⊢ (𝜑 → ((𝐹‘𝐵) ≤ 𝑈 ↔ ((𝐹‘𝐵) < 𝑈 ∨ (𝐹‘𝐵) = 𝑈)))
53	48, 52	mpbid 235	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) < 𝑈 ∨ (𝐹‘𝐵) = 𝑈))
54	42, 47, 53	mpjaodan 973	1 ⊢ (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ∃wrex 3090 ⊆ wss 3900 class class class wbr 5106 ‘cfv 6558 (class class class)co 7443 ℂcc 11225 ℝcr 11226 ℝ^*cxr 11369 < clt 11370 ≤ cle 11371 (,)cioo 13503 [,]cicc 13506 –cn→ccncf 25227

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-om 7902 df-1st 8025 df-2nd 8026 df-supp 8205 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fsupp 9382 df-fi 9430 df-sup 9461 df-inf 9462 df-oi 9531 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-q 13103 df-rp 13148 df-xneg 13268 df-xadd 13269 df-xmul 13270 df-ioo 13507 df-icc 13510 df-fz 13667 df-fzo 13816 df-seq 14172 df-exp 14232 df-hash 14502 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-starv 17472 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-unif 17480 df-hom 17481 df-cco 17482 df-rest 17622 df-topn 17623 df-0g 17641 df-gsum 17642 df-topgen 17643 df-pt 17644 df-prds 17647 df-xrs 17703 df-qtop 17708 df-imas 17709 df-xps 17711 df-mre 17785 df-mrc 17786 df-acs 17788 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-submnd 19008 df-mulg 19307 df-cntz 19561 df-cmn 20026 df-psmet 21700 df-xmet 21701 df-met 21702 df-bl 21703 df-mopn 21704 df-cnfld 21709 df-top 23242 df-topon 23259 df-topsp 23281 df-bases 23294 df-cn 23575 df-cnp 23576 df-tx 23911 df-hmeo 24104 df-xms 24669 df-ms 24670 df-tms 24671 df-cncf 25229

This theorem is used by: ivthicc 25809 recosf1o 26893

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