Theorem ivthdich 15912

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 15902 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 15905, hovera 15906, and hoverb 15907, 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 8400, 𝑐 < 1 or 0 < 𝑐, and that leads to 𝑧 ≤ 0 by hoverlt1 15908 or 0 ≤ 𝑧 by hovergt0 15909. (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 4135	. . . . . . . . . 10 ⊢ (𝑥 = 𝑞 → (𝑎 < 𝑥 ↔ 𝑎 < 𝑞))
2		breq1 4134	. . . . . . . . . 10 ⊢ (𝑥 = 𝑞 → (𝑥 < 𝑏 ↔ 𝑞 < 𝑏))
3		fveqeq2 5707	. . . . . . . . . 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 7359	. . . . . . . 8 ⊢ (𝑡 = 𝑥 → inf({𝑡, 0}, ℝ, < ) = inf({𝑥, 0}, ℝ, < ))
12		oveq1 6095	. . . . . . . 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 7334	. . . . . 6 ⊢ (𝑡 = 𝑥 → sup({inf({𝑡, 0}, ℝ, < ), (𝑡 − 1)}, ℝ, < ) = sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))
15	14	cbvmptv 4229	. . . . 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 15910	. . . 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 15911	. 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 4131   ↦ cmpt 4193  ‘cfv 5380  (class class class)co 6088  supcsup 7329  infcinf 7330  ℝcr 8185  0cc0 8186  1c1 8187   < clt 8367   ≤ cle 8368   − cmin 8505  –cn→ccncf 15829

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306  ax-addf 8308

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6581  df-frec 6667  df-map 6929  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9655  df-uz 9938  df-q 10036  df-rp 10072  df-xneg 10191  df-xadd 10192  df-ioo 10311  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-rest 13654  df-topgen 13673  df-psmet 14989  df-xmet 14990  df-met 14991  df-bl 14992  df-mopn 14993  df-top 15257  df-topon 15270  df-bases 15302  df-cn 15447  df-cnp 15448  df-tx 15512  df-cncf 15830

This theorem is used by: (None)