ivthle2 - Metamath Proof Explorer

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Theorem ivthle2 25758

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 13545	. . . . 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 25756	. . . . 5 ⊢ ((𝜑 ∧ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
18		ssrexv 4001	. . . . 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 11340	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22	4	rexrd 11340	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
23	2, 4, 8	ltled 11439	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
24		lbicc2 13576	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
25	21, 22, 23, 24	syl3anc 1398	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
26		eqcom 2768	. . . . . . 7 ⊢ ((𝐹‘𝑐) = 𝑈 ↔ 𝑈 = (𝐹‘𝑐))
27		fveq2 6877	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → (𝐹‘𝑐) = (𝐹‘𝐴))
28	27	eqeq2d 2772	. . . . . . 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 3577	. . . . 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 6877	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
36	35	eleq1d 2846	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐴) ∈ ℝ))
37	14	ralrimiva 3155	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
38	36, 37, 25	rspcdva 3578	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
39	6, 38	leloed 11434	. . . . 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 13577	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
44	21, 22, 23, 43	syl3anc 1398	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
45		fveqeq2 6886	. . . 4 ⊢ (𝑐 = 𝐵 → ((𝐹‘𝑐) = 𝑈 ↔ (𝐹‘𝐵) = 𝑈))
46	45	rspcev 3577	. . 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 6877	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
50	49	eleq1d 2846	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐵) ∈ ℝ))
51	50, 37, 44	rspcdva 3578	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
52	51, 6	leloed 11434	. . 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 2145 ∃wrex 3087 ⊆ wss 3899 class class class wbr 5103 ‘cfv 6531 (class class class)co 7412 ℂcc 11179 ℝcr 11180 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 (,)cioo 13457 [,]cicc 13460 –cn→ccncf 25177

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ioo 13461 df-icc 13464 df-fz 13621 df-fzo 13769 df-seq 14125 df-exp 14185 df-hash 14455 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-rest 17573 df-topn 17574 df-0g 17592 df-gsum 17593 df-topgen 17594 df-pt 17595 df-prds 17598 df-xrs 17654 df-qtop 17659 df-imas 17660 df-xps 17662 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-mulg 19258 df-cntz 19511 df-cmn 19976 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-cnfld 21659 df-top 23192 df-topon 23209 df-topsp 23231 df-bases 23244 df-cn 23525 df-cnp 23526 df-tx 23861 df-hmeo 24054 df-xms 24619 df-ms 24620 df-tms 24621 df-cncf 25179

This theorem is used by: ivthicc 25759 recosf1o 26845

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