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Theorem ivthdichlem 15167

Description: Lemma for ivthdich 15169. 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 8346	. . . 4 ⊢ (𝑍 ∈ ℝ → (𝑍 − 1) ∈ ℝ)
3	1, 2	syl 14	. . 3 ⊢ (𝜑 → (𝑍 − 1) ∈ ℝ)
4		2re 9113	. . . . 5 ⊢ 2 ∈ ℝ
5	4	a1i 9	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
6	1, 5	readdcld 8109	. . 3 ⊢ (𝜑 → (𝑍 + 2) ∈ ℝ)
7	1	ltm1d 9012	. . . 4 ⊢ (𝜑 → (𝑍 − 1) < 𝑍)
8		2rp 9787	. . . . . 6 ⊢ 2 ∈ ℝ⁺
9	8	a1i 9	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ⁺)
10	1, 9	ltaddrpd 9859	. . . 4 ⊢ (𝜑 → 𝑍 < (𝑍 + 2))
11	3, 1, 6, 7, 10	lttrd 8205	. . 3 ⊢ (𝜑 → (𝑍 − 1) < (𝑍 + 2))
12		hover.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))
13	12	hovercncf 15162	. . . 4 ⊢ 𝐹 ∈ (ℝ–cn→ℝ)
14	13	a1i 9	. . 3 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ))
15	12	hovera 15163	. . . . 5 ⊢ (𝑍 ∈ ℝ → (𝐹‘(𝑍 − 1)) < 𝑍)
16	1, 15	syl 14	. . . 4 ⊢ (𝜑 → (𝐹‘(𝑍 − 1)) < 𝑍)
17	12	hoverb 15164	. . . . 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 15161	. 2 ⊢ (𝜑 → ∃𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2))(𝐹‘𝑐) = 𝑍)
22		0red 8080	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → 0 ∈ ℝ)
23		1red 8094	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → 1 ∈ ℝ)
24		elioore 10041	. . . . . 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 8206	. . . . . 6 ⊢ 0 < 1
27		axltwlin 8147	. . . . . 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 1250	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → (0 < 𝑐 ∨ 𝑐 < 1))
30	29	orcomd 731	. . 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 15165	. . . . . . 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 4071	. . . . 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 15166	. . . . . . 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 4073	. . . . 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 788	. . 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 2629	1 ⊢ (𝜑 → (𝑍 ≤ 0 ∨ 0 ≤ 𝑍))

Colors of variables: wff set class

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 + caddc 7935 < clt 8114 ≤ cle 8115 − cmin 8250 2c2 9094 ℝ⁺crp 9782 (,)cioo 10017 –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: ivthdich 15169

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