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Theorem ivthdichlem 15503

Description: Lemma for ivthdich 15505. The result, with a few notational conveniences. (Contributed by Jim Kingdon, 22-Jul-2025.)

**Hypotheses**
Ref	Expression
hover.f	⊢ 𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))
ivthdichlem.z	⊢ (𝜑 → 𝑍 ∈ ℝ)
ivthdichlem.i	⊢ (𝜑 → ∀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))))

**Assertion**
Ref	Expression
ivthdichlem	⊢ (𝜑 → (𝑍 ≤ 0 ∨ 0 ≤ 𝑍))

Distinct variable groups: 𝐹,𝑎,𝑏,𝑓,𝑥 𝑍,𝑎,𝑏,𝑓,𝑥 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑓,𝑎,𝑏)

Proof of Theorem ivthdichlem Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ivthdichlem.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ ℝ)
2		peano2rem 8536	. . . 4 ⊢ (𝑍 ∈ ℝ → (𝑍 − 1) ∈ ℝ)
3	1, 2	syl 14	. . 3 ⊢ (𝜑 → (𝑍 − 1) ∈ ℝ)
4		2re 9303	. . . . 5 ⊢ 2 ∈ ℝ
5	4	a1i 9	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
6	1, 5	readdcld 8299	. . 3 ⊢ (𝜑 → (𝑍 + 2) ∈ ℝ)
7	1	ltm1d 9202	. . . 4 ⊢ (𝜑 → (𝑍 − 1) < 𝑍)
8		2rp 9987	. . . . . 6 ⊢ 2 ∈ ℝ⁺
9	8	a1i 9	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ⁺)
10	1, 9	ltaddrpd 10059	. . . 4 ⊢ (𝜑 → 𝑍 < (𝑍 + 2))
11	3, 1, 6, 7, 10	lttrd 8395	. . 3 ⊢ (𝜑 → (𝑍 − 1) < (𝑍 + 2))
12		hover.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))
13	12	hovercncf 15498	. . . 4 ⊢ 𝐹 ∈ (ℝ–cn→ℝ)
14	13	a1i 9	. . 3 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ))
15	12	hovera 15499	. . . . 5 ⊢ (𝑍 ∈ ℝ → (𝐹‘(𝑍 − 1)) < 𝑍)
16	1, 15	syl 14	. . . 4 ⊢ (𝜑 → (𝐹‘(𝑍 − 1)) < 𝑍)
17	12	hoverb 15500	. . . . 5 ⊢ (𝑍 ∈ ℝ → 𝑍 < (𝐹‘(𝑍 + 2)))
18	1, 17	syl 14	. . . 4 ⊢ (𝜑 → 𝑍 < (𝐹‘(𝑍 + 2)))
19	16, 18	jca 306	. . 3 ⊢ (𝜑 → ((𝐹‘(𝑍 − 1)) < 𝑍 ∧ 𝑍 < (𝐹‘(𝑍 + 2))))
20		ivthdichlem.i	. . 3 ⊢ (𝜑 → ∀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))))
21	3, 6, 1, 11, 14, 19, 20	ivthreinc 15497	. 2 ⊢ (𝜑 → ∃𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2))(𝐹‘𝑐) = 𝑍)
22		0red 8271	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → 0 ∈ ℝ)
23		1red 8285	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → 1 ∈ ℝ)
24		elioore 10241	. . . . . 6 ⊢ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) → 𝑐 ∈ ℝ)
25	24	ad2antrl 490	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → 𝑐 ∈ ℝ)
26		0lt1 8396	. . . . . 6 ⊢ 0 < 1
27		axltwlin 8337	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (0 < 1 → (0 < 𝑐 ∨ 𝑐 < 1)))
28	26, 27	mpi 15	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (0 < 𝑐 ∨ 𝑐 < 1))
29	22, 23, 25, 28	syl3anc 1274	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → (0 < 𝑐 ∨ 𝑐 < 1))
30	29	orcomd 737	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → (𝑐 < 1 ∨ 0 < 𝑐))
31		simplrr 538	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) ∧ 𝑐 < 1) → (𝐹‘𝑐) = 𝑍)
32	12	hoverlt1 15501	. . . . . . 7 ⊢ ((𝑐 ∈ ℝ ∧ 𝑐 < 1) → (𝐹‘𝑐) ≤ 0)
33	25, 32	sylan 283	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) ∧ 𝑐 < 1) → (𝐹‘𝑐) ≤ 0)
34	31, 33	eqbrtrrd 4132	. . . . 5 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) ∧ 𝑐 < 1) → 𝑍 ≤ 0)
35	34	ex 115	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → (𝑐 < 1 → 𝑍 ≤ 0))
36	12	hovergt0 15502	. . . . . . 7 ⊢ ((𝑐 ∈ ℝ ∧ 0 < 𝑐) → 0 ≤ (𝐹‘𝑐))
37	25, 36	sylan 283	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) ∧ 0 < 𝑐) → 0 ≤ (𝐹‘𝑐))
38		simplrr 538	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) ∧ 0 < 𝑐) → (𝐹‘𝑐) = 𝑍)
39	37, 38	breqtrd 4134	. . . . 5 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) ∧ 0 < 𝑐) → 0 ≤ 𝑍)
40	39	ex 115	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → (0 < 𝑐 → 0 ≤ 𝑍))
41	35, 40	orim12d 794	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → ((𝑐 < 1 ∨ 0 < 𝑐) → (𝑍 ≤ 0 ∨ 0 ≤ 𝑍)))
42	30, 41	mpd 13	. 2 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → (𝑍 ≤ 0 ∨ 0 ≤ 𝑍))
43	21, 42	rexlimddv 2665	1 ⊢ (𝜑 → (𝑍 ≤ 0 ∨ 0 ≤ 𝑍))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∨ wo 716 ∧ w3a 1005 ∀wal 1396 = wceq 1398 ∈ wcel 2203 ∀wral 2520 ∃wrex 2521 {cpr 3689 class class class wbr 4108 ↦ cmpt 4170 ‘cfv 5351 (class class class)co 6049 supcsup 7272 infcinf 7273 ℝcr 8122 0cc0 8123 1c1 8124 + caddc 8126 < clt 8304 ≤ cle 8305 − cmin 8440 2c2 9284 ℝ⁺crp 9982 (,)cioo 10217 –cn→ccncf 15422

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8214 ax-resscn 8215 ax-1cn 8216 ax-1re 8217 ax-icn 8218 ax-addcl 8219 ax-addrcl 8220 ax-mulcl 8221 ax-mulrcl 8222 ax-addcom 8223 ax-mulcom 8224 ax-addass 8225 ax-mulass 8226 ax-distr 8227 ax-i2m1 8228 ax-0lt1 8229 ax-1rid 8230 ax-0id 8231 ax-rnegex 8232 ax-precex 8233 ax-cnre 8234 ax-pre-ltirr 8235 ax-pre-ltwlin 8236 ax-pre-lttrn 8237 ax-pre-apti 8238 ax-pre-ltadd 8239 ax-pre-mulgt0 8240 ax-pre-mulext 8241 ax-arch 8242 ax-caucvg 8243 ax-addf 8245

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-isom 5360 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-frec 6621 df-map 6883 df-sup 7274 df-inf 7275 df-pnf 8306 df-mnf 8307 df-xr 8308 df-ltxr 8309 df-le 8310 df-sub 8442 df-neg 8443 df-reap 8845 df-ap 8852 df-div 8943 df-inn 9234 df-2 9292 df-3 9293 df-4 9294 df-n0 9493 df-z 9574 df-uz 9850 df-q 9948 df-rp 9983 df-xneg 10101 df-xadd 10102 df-ioo 10221 df-seqfrec 10806 df-exp 10897 df-cj 11520 df-re 11521 df-im 11522 df-rsqrt 11676 df-abs 11677 df-rest 13443 df-topgen 13462 df-psmet 14678 df-xmet 14679 df-met 14680 df-bl 14681 df-mopn 14682 df-top 14850 df-topon 14863 df-bases 14895 df-cn 15040 df-cnp 15041 df-tx 15105 df-cncf 15423

This theorem is referenced by: ivthdich 15505

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