Theorem ivthdich 15756

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 15746 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 15749, hovera 15750, and hoverb 15751, 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 8393, 𝑐 < 1 or 0 < 𝑐, and that leads to 𝑧 ≤ 0 by hoverlt1 15752 or 0 ≤ 𝑧 by hovergt0 15753. (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 4134	. . . . . . . . . 10 ⊢ (𝑥 = 𝑞 → (𝑎 < 𝑥 ↔ 𝑎 < 𝑞))
2		breq1 4133	. . . . . . . . . 10 ⊢ (𝑥 = 𝑞 → (𝑥 < 𝑏 ↔ 𝑞 < 𝑏))
3		fveqeq2 5704	. . . . . . . . . 10 ⊢ (𝑥 = 𝑞 → ((𝑓‘𝑥) = 0 ↔ (𝑓‘𝑞) = 0))
4	1, 2, 3	3anbi123d 1353	. . . . . . . . 9 ⊢ (𝑥 = 𝑞 → ((𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0) ↔ (𝑎 < 𝑞 ∧ 𝑞 < 𝑏 ∧ (𝑓‘𝑞) = 0)))
5	4	cbvrexv 2787	. . . . . . . 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 2558	. . . . . 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 1523	. . . 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 3788	. . . . . . . . 9 ⊢ (𝑡 = 𝑥 → {𝑡, 0} = {𝑥, 0})
11	10	infeq1d 7352	. . . . . . . 8 ⊢ (𝑡 = 𝑥 → inf({𝑡, 0}, ℝ, < ) = inf({𝑥, 0}, ℝ, < ))
12		oveq1 6092	. . . . . . . 8 ⊢ (𝑡 = 𝑥 → (𝑡 − 1) = (𝑥 − 1))
13	11, 12	preq12d 3796	. . . . . . 7 ⊢ (𝑡 = 𝑥 → {inf({𝑡, 0}, ℝ, < ), (𝑡 − 1)} = {inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)})
14	13	supeq1d 7327	. . . . . 6 ⊢ (𝑡 = 𝑥 → sup({inf({𝑡, 0}, ℝ, < ), (𝑡 − 1)}, ℝ, < ) = sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))
15	14	cbvmptv 4227	. . . . 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 15754	. . . 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 2623	. 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 15755	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∨ wo 720   ∧ w3a 1009  ∀wal 1400   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {cpr 3710   class class class wbr 4130   ↦ cmpt 4192  ‘cfv 5377  (class class class)co 6085  supcsup 7322  infcinf 7323  ℝcr 8178  0cc0 8179  1c1 8180   < clt 8360   ≤ cle 8361   − cmin 8497  –cn→ccncf 15673

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-addf 8301

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-map 6924  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-xneg 10176  df-xadd 10177  df-ioo 10296  df-seqfrec 10887  df-exp 10978  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767  df-rest 13597  df-topgen 13616  df-psmet 14882  df-xmet 14883  df-met 14884  df-bl 14885  df-mopn 14886  df-top 15101  df-topon 15114  df-bases 15146  df-cn 15291  df-cnp 15292  df-tx 15356  df-cncf 15674

This theorem is used by: (None)