Theorem ivthdich 14973

Description: The intermediate value theorem implies real number dichotomy. Because real number dichotomy (also known as analytic LLPO) is a constructive taboo, this means we will be unable to prove the intermediate value theorem as stated here (although versions with additional conditions, such as ivthinc 14963 for strictly monotonic functions, can be proved).

The proof is via a function which we call the hover function and which is also described in Section 5.1 of [Bauer], p. 493. Consider any real number 𝑧. We want to show that 𝑧 ≤ 0 ∨ 0 ≤ 𝑧. Because of hovercncf 14966, hovera 14967, and hoverb 14968, we are able to apply the intermediate value theorem to get a value 𝑐 such that the hover function at 𝑐 equals 𝑧. By axltwlin 8111, 𝑐 < 1 or 0 < 𝑐, and that leads to 𝑧 ≤ 0 by hoverlt1 14969 or 0 ≤ 𝑧 by hovergt0 14970. (Contributed by Jim Kingdon and Mario Carneiro, 22-Jul-2025.)

Assertion

Ref	Expression
ivthdich	⊢ (∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))) → ∀𝑟 ∈ ℝ ∀𝑠 ∈ ℝ (𝑟 ≤ 𝑠 ∨ 𝑠 ≤ 𝑟))

Distinct variable groups:   𝑎,𝑏,𝑓,𝑥   𝑠,𝑟

Proof of Theorem ivthdich

Dummy variables 𝑞 𝑡 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4038	. . . . . . . . . 10 ⊢ (𝑥 = 𝑞 → (𝑎 < 𝑥 ↔ 𝑎 < 𝑞))
2		breq1 4037	. . . . . . . . . 10 ⊢ (𝑥 = 𝑞 → (𝑥 < 𝑏 ↔ 𝑞 < 𝑏))
3		fveqeq2 5570	. . . . . . . . . 10 ⊢ (𝑥 = 𝑞 → ((𝑓‘𝑥) = 0 ↔ (𝑓‘𝑞) = 0))
4	1, 2, 3	3anbi123d 1323	. . . . . . . . 9 ⊢ (𝑥 = 𝑞 → ((𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0) ↔ (𝑎 < 𝑞 ∧ 𝑞 < 𝑏 ∧ (𝑓‘𝑞) = 0)))
5	4	cbvrexv 2730	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0) ↔ ∃𝑞 ∈ ℝ (𝑎 < 𝑞 ∧ 𝑞 < 𝑏 ∧ (𝑓‘𝑞) = 0))
6	5	imbi2i 226	. . . . . . 7 ⊢ (((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0)) ↔ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑞 ∈ ℝ (𝑎 < 𝑞 ∧ 𝑞 < 𝑏 ∧ (𝑓‘𝑞) = 0)))
7	6	2ralbii 2505	. . . . . 6 ⊢ (∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0)) ↔ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑞 ∈ ℝ (𝑎 < 𝑞 ∧ 𝑞 < 𝑏 ∧ (𝑓‘𝑞) = 0)))
8	7	imbi2i 226	. . . . 5 ⊢ ((𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))) ↔ (𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑞 ∈ ℝ (𝑎 < 𝑞 ∧ 𝑞 < 𝑏 ∧ (𝑓‘𝑞) = 0))))
9	8	albii 1484	. . . 4 ⊢ (∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))) ↔ ∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑞 ∈ ℝ (𝑎 < 𝑞 ∧ 𝑞 < 𝑏 ∧ (𝑓‘𝑞) = 0))))
10		preq1 3700	. . . . . . . . 9 ⊢ (𝑡 = 𝑥 → {𝑡, 0} = {𝑥, 0})
11	10	infeq1d 7087	. . . . . . . 8 ⊢ (𝑡 = 𝑥 → inf({𝑡, 0}, ℝ, < ) = inf({𝑥, 0}, ℝ, < ))
12		oveq1 5932	. . . . . . . 8 ⊢ (𝑡 = 𝑥 → (𝑡 − 1) = (𝑥 − 1))
13	11, 12	preq12d 3708	. . . . . . 7 ⊢ (𝑡 = 𝑥 → {inf({𝑡, 0}, ℝ, < ), (𝑡 − 1)} = {inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)})
14	13	supeq1d 7062	. . . . . 6 ⊢ (𝑡 = 𝑥 → sup({inf({𝑡, 0}, ℝ, < ), (𝑡 − 1)}, ℝ, < ) = sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))
15	14	cbvmptv 4130	. . . . 5 ⊢ (𝑡 ∈ ℝ ↦ sup({inf({𝑡, 0}, ℝ, < ), (𝑡 − 1)}, ℝ, < )) = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))
16		simpr 110	. . . . 5 ⊢ ((∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑞 ∈ ℝ (𝑎 < 𝑞 ∧ 𝑞 < 𝑏 ∧ (𝑓‘𝑞) = 0))) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
17	9	biimpri 133	. . . . . 6 ⊢ (∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑞 ∈ ℝ (𝑎 < 𝑞 ∧ 𝑞 < 𝑏 ∧ (𝑓‘𝑞) = 0))) → ∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))))
18	17	adantr 276	. . . . 5 ⊢ ((∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑞 ∈ ℝ (𝑎 < 𝑞 ∧ 𝑞 < 𝑏 ∧ (𝑓‘𝑞) = 0))) ∧ 𝑧 ∈ ℝ) → ∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))))
19	15, 16, 18	ivthdichlem 14971	. . . 4 ⊢ ((∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑞 ∈ ℝ (𝑎 < 𝑞 ∧ 𝑞 < 𝑏 ∧ (𝑓‘𝑞) = 0))) ∧ 𝑧 ∈ ℝ) → (𝑧 ≤ 0 ∨ 0 ≤ 𝑧))
20	9, 19	sylanb 284	. . 3 ⊢ ((∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))) ∧ 𝑧 ∈ ℝ) → (𝑧 ≤ 0 ∨ 0 ≤ 𝑧))
21	20	ralrimiva 2570	. 2 ⊢ (∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))) → ∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧))
22		dich0 14972	. 2 ⊢ (∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ↔ ∀𝑟 ∈ ℝ ∀𝑠 ∈ ℝ (𝑟 ≤ 𝑠 ∨ 𝑠 ≤ 𝑟))
23	21, 22	sylib 122	1 ⊢ (∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))) → ∀𝑟 ∈ ℝ ∀𝑠 ∈ ℝ (𝑟 ≤ 𝑠 ∨ 𝑠 ≤ 𝑟))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   ∧ w3a 980  ∀wal 1362   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {cpr 3624   class class class wbr 4034   ↦ cmpt 4095  ‘cfv 5259  (class class class)co 5925  supcsup 7057  infcinf 7058  ℝcr 7895  0cc0 7896  1c1 7897   < clt 8078   ≤ cle 8079   − cmin 8214  –cn→ccncf 14890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016  ax-addf 8018

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-map 6718  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-xneg 9864  df-xadd 9865  df-ioo 9984  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-rest 12943  df-topgen 12962  df-psmet 14175  df-xmet 14176  df-met 14177  df-bl 14178  df-mopn 14179  df-top 14318  df-topon 14331  df-bases 14363  df-cn 14508  df-cnp 14509  df-tx 14573  df-cncf 14891

This theorem is referenced by: (None)