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Theorem ivthdichlem 15333

Description: Lemma for ivthdich 15335. 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 8421	. . . 4 ⊢ (𝑍 ∈ ℝ → (𝑍 − 1) ∈ ℝ)
3	1, 2	syl 14	. . 3 ⊢ (𝜑 → (𝑍 − 1) ∈ ℝ)
4		2re 9188	. . . . 5 ⊢ 2 ∈ ℝ
5	4	a1i 9	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
6	1, 5	readdcld 8184	. . 3 ⊢ (𝜑 → (𝑍 + 2) ∈ ℝ)
7	1	ltm1d 9087	. . . 4 ⊢ (𝜑 → (𝑍 − 1) < 𝑍)
8		2rp 9862	. . . . . 6 ⊢ 2 ∈ ℝ⁺
9	8	a1i 9	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ⁺)
10	1, 9	ltaddrpd 9934	. . . 4 ⊢ (𝜑 → 𝑍 < (𝑍 + 2))
11	3, 1, 6, 7, 10	lttrd 8280	. . 3 ⊢ (𝜑 → (𝑍 − 1) < (𝑍 + 2))
12		hover.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))
13	12	hovercncf 15328	. . . 4 ⊢ 𝐹 ∈ (ℝ–cn→ℝ)
14	13	a1i 9	. . 3 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ))
15	12	hovera 15329	. . . . 5 ⊢ (𝑍 ∈ ℝ → (𝐹‘(𝑍 − 1)) < 𝑍)
16	1, 15	syl 14	. . . 4 ⊢ (𝜑 → (𝐹‘(𝑍 − 1)) < 𝑍)
17	12	hoverb 15330	. . . . 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 15327	. 2 ⊢ (𝜑 → ∃𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2))(𝐹‘𝑐) = 𝑍)
22		0red 8155	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → 0 ∈ ℝ)
23		1red 8169	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → 1 ∈ ℝ)
24		elioore 10116	. . . . . 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 8281	. . . . . 6 ⊢ 0 < 1
27		axltwlin 8222	. . . . . 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 1271	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → (0 < 𝑐 ∨ 𝑐 < 1))
30	29	orcomd 734	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → (𝑐 < 1 ∨ 0 < 𝑐))
31		simplrr 536	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) ∧ 𝑐 < 1) → (𝐹‘𝑐) = 𝑍)
32	12	hoverlt1 15331	. . . . . . 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 4107	. . . . 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 15332	. . . . . . 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 536	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) ∧ 0 < 𝑐) → (𝐹‘𝑐) = 𝑍)
39	37, 38	breqtrd 4109	. . . . 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 791	. . 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 2653	1 ⊢ (𝜑 → (𝑍 ≤ 0 ∨ 0 ≤ 𝑍))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∨ wo 713 ∧ w3a 1002 ∀wal 1393 = wceq 1395 ∈ wcel 2200 ∀wral 2508 ∃wrex 2509 {cpr 3667 class class class wbr 4083 ↦ cmpt 4145 ‘cfv 5318 (class class class)co 6007 supcsup 7157 infcinf 7158 ℝcr 8006 0cc0 8007 1c1 8008 + caddc 8010 < clt 8189 ≤ cle 8190 − cmin 8325 2c2 9169 ℝ⁺crp 9857 (,)cioo 10092 –cn→ccncf 15252

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125 ax-arch 8126 ax-caucvg 8127 ax-addf 8129

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-map 6805 df-sup 7159 df-inf 7160 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-n0 9378 df-z 9455 df-uz 9731 df-q 9823 df-rp 9858 df-xneg 9976 df-xadd 9977 df-ioo 10096 df-seqfrec 10678 df-exp 10769 df-cj 11361 df-re 11362 df-im 11363 df-rsqrt 11517 df-abs 11518 df-rest 13282 df-topgen 13301 df-psmet 14515 df-xmet 14516 df-met 14517 df-bl 14518 df-mopn 14519 df-top 14680 df-topon 14693 df-bases 14725 df-cn 14870 df-cnp 14871 df-tx 14935 df-cncf 15253

This theorem is referenced by: ivthdich 15335

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