Theorem ivthdich 15169

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 15159 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 15162, hovera 15163, and hoverb 15164, 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 8147, 𝑐 < 1 or 0 < 𝑐, and that leads to 𝑧 ≤ 0 by hoverlt1 15165 or 0 ≤ 𝑧 by hovergt0 15166. (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 4051	. . . . . . . . . 10 ⊢ (𝑥 = 𝑞 → (𝑎 < 𝑥 ↔ 𝑎 < 𝑞))
2		breq1 4050	. . . . . . . . . 10 ⊢ (𝑥 = 𝑞 → (𝑥 < 𝑏 ↔ 𝑞 < 𝑏))
3		fveqeq2 5592	. . . . . . . . . 10 ⊢ (𝑥 = 𝑞 → ((𝑓‘𝑥) = 0 ↔ (𝑓‘𝑞) = 0))
4	1, 2, 3	3anbi123d 1325	. . . . . . . . 9 ⊢ (𝑥 = 𝑞 → ((𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0) ↔ (𝑎 < 𝑞 ∧ 𝑞 < 𝑏 ∧ (𝑓‘𝑞) = 0)))
5	4	cbvrexv 2740	. . . . . . . 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 2515	. . . . . 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 1494	. . . 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 3711	. . . . . . . . 9 ⊢ (𝑡 = 𝑥 → {𝑡, 0} = {𝑥, 0})
11	10	infeq1d 7121	. . . . . . . 8 ⊢ (𝑡 = 𝑥 → inf({𝑡, 0}, ℝ, < ) = inf({𝑥, 0}, ℝ, < ))
12		oveq1 5958	. . . . . . . 8 ⊢ (𝑡 = 𝑥 → (𝑡 − 1) = (𝑥 − 1))
13	11, 12	preq12d 3719	. . . . . . 7 ⊢ (𝑡 = 𝑥 → {inf({𝑡, 0}, ℝ, < ), (𝑡 − 1)} = {inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)})
14	13	supeq1d 7096	. . . . . 6 ⊢ (𝑡 = 𝑥 → sup({inf({𝑡, 0}, ℝ, < ), (𝑡 − 1)}, ℝ, < ) = sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))
15	14	cbvmptv 4144	. . . . 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 15167	. . . 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 2580	. 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 15168	. 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 710   ∧ w3a 981  ∀wal 1371   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  {cpr 3635   class class class wbr 4047   ↦ cmpt 4109  ‘cfv 5276  (class class class)co 5951  supcsup 7091  infcinf 7092  ℝcr 7931  0cc0 7932  1c1 7933   < clt 8114   ≤ cle 8115   − cmin 8250  –cn→ccncf 15086

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052  ax-addf 8054

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-frec 6484  df-map 6744  df-sup 7093  df-inf 7094  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-xneg 9901  df-xadd 9902  df-ioo 10021  df-seqfrec 10600  df-exp 10691  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-rest 13117  df-topgen 13136  df-psmet 14349  df-xmet 14350  df-met 14351  df-bl 14352  df-mopn 14353  df-top 14514  df-topon 14527  df-bases 14559  df-cn 14704  df-cnp 14705  df-tx 14769  df-cncf 15087

This theorem is referenced by: (None)