ivthle2 - Metamath Proof Explorer

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Theorem ivthle2 25505

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 13469	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
2		ivth.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝐴 ∈ ℝ)
4		ivth.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝐵 ∈ ℝ)
6		ivth.3	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ)
7	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝑈 ∈ ℝ)
8		ivth.4	. . . . . . 7 ⊢ (𝜑 → 𝐴 < 𝐵)
9	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → 𝐴 < 𝐵)
10		ivth.5	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → (𝐴[,]𝐵) ⊆ 𝐷)
12		ivth.7	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))
13	12	adantr 480	. . . . . 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 715	. . . . . 6 ⊢ (((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)
16		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴)))
17	3, 5, 7, 9, 11, 13, 15, 16	ivth2 25503	. . . . 5 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
18		ssrexv 4064	. . . . 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 467	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝐵) < 𝑈) ∧ 𝑈 < (𝐹‘𝐴)) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
21	2	rexrd 11308	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22	4	rexrd 11308	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
23	2, 4, 8	ltled 11406	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
24		lbicc2 13500	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
25	21, 22, 23, 24	syl3anc 1370	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
26		eqcom 2741	. . . . . . 7 ⊢ ((𝐹‘𝑐) = 𝑈 ↔ 𝑈 = (𝐹‘𝑐))
27		fveq2 6906	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → (𝐹‘𝑐) = (𝐹‘𝐴))
28	27	eqeq2d 2745	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (𝑈 = (𝐹‘𝑐) ↔ 𝑈 = (𝐹‘𝐴)))
29	26, 28	bitrid 283	. . . . . 6 ⊢ (𝑐 = 𝐴 → ((𝐹‘𝑐) = 𝑈 ↔ 𝑈 = (𝐹‘𝐴)))
30	29	rspcev 3621	. . . . 5 ⊢ ((𝐴 ∈ (𝐴[,]𝐵) ∧ 𝑈 = (𝐹‘𝐴)) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
31	25, 30	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = (𝐹‘𝐴)) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
32	31	adantlr 715	. . 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 495	. . . . 5 ⊢ (𝜑 → 𝑈 ≤ (𝐹‘𝐴))
35		fveq2 6906	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
36	35	eleq1d 2823	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐴) ∈ ℝ))
37	14	ralrimiva 3143	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
38	36, 37, 25	rspcdva 3622	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
39	6, 38	leloed 11401	. . . . 5 ⊢ (𝜑 → (𝑈 ≤ (𝐹‘𝐴) ↔ (𝑈 < (𝐹‘𝐴) ∨ 𝑈 = (𝐹‘𝐴))))
40	34, 39	mpbid 232	. . . 4 ⊢ (𝜑 → (𝑈 < (𝐹‘𝐴) ∨ 𝑈 = (𝐹‘𝐴)))
41	40	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐵) < 𝑈) → (𝑈 < (𝐹‘𝐴) ∨ 𝑈 = (𝐹‘𝐴)))
42	20, 32, 41	mpjaodan 960	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝐵) < 𝑈) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
43		ubicc2 13501	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
44	21, 22, 23, 43	syl3anc 1370	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
45		fveqeq2 6915	. . . 4 ⊢ (𝑐 = 𝐵 → ((𝐹‘𝑐) = 𝑈 ↔ (𝐹‘𝐵) = 𝑈))
46	45	rspcev 3621	. . 3 ⊢ ((𝐵 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝐵) = 𝑈) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
47	44, 46	sylan 580	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝐵) = 𝑈) → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
48	33	simpld 494	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) ≤ 𝑈)
49		fveq2 6906	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
50	49	eleq1d 2823	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐵) ∈ ℝ))
51	50, 37, 44	rspcdva 3622	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
52	51, 6	leloed 11401	. . 3 ⊢ (𝜑 → ((𝐹‘𝐵) ≤ 𝑈 ↔ ((𝐹‘𝐵) < 𝑈 ∨ (𝐹‘𝐵) = 𝑈)))
53	48, 52	mpbid 232	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) < 𝑈 ∨ (𝐹‘𝐵) = 𝑈))
54	42, 47, 53	mpjaodan 960	1 ⊢ (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 ⊆ wss 3962 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 ℝ^*cxr 11291 < clt 11292 ≤ cle 11293 (,)cioo 13383 [,]cicc 13386 –cn→ccncf 24915

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cn 23250 df-cnp 23251 df-tx 23585 df-hmeo 23778 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917

This theorem is referenced by: ivthicc 25506 recosf1o 26591

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