ivthdichlem - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > ivthdichlem		GIF version

Theorem ivthdichlem 14971

Description: Lemma for ivthdich 14973. 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 8310	. . . 4 ⊢ (𝑍 ∈ ℝ → (𝑍 − 1) ∈ ℝ)
3	1, 2	syl 14	. . 3 ⊢ (𝜑 → (𝑍 − 1) ∈ ℝ)
4		2re 9077	. . . . 5 ⊢ 2 ∈ ℝ
5	4	a1i 9	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
6	1, 5	readdcld 8073	. . 3 ⊢ (𝜑 → (𝑍 + 2) ∈ ℝ)
7	1	ltm1d 8976	. . . 4 ⊢ (𝜑 → (𝑍 − 1) < 𝑍)
8		2rp 9750	. . . . . 6 ⊢ 2 ∈ ℝ⁺
9	8	a1i 9	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ⁺)
10	1, 9	ltaddrpd 9822	. . . 4 ⊢ (𝜑 → 𝑍 < (𝑍 + 2))
11	3, 1, 6, 7, 10	lttrd 8169	. . 3 ⊢ (𝜑 → (𝑍 − 1) < (𝑍 + 2))
12		hover.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))
13	12	hovercncf 14966	. . . 4 ⊢ 𝐹 ∈ (ℝ–cn→ℝ)
14	13	a1i 9	. . 3 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ))
15	12	hovera 14967	. . . . 5 ⊢ (𝑍 ∈ ℝ → (𝐹‘(𝑍 − 1)) < 𝑍)
16	1, 15	syl 14	. . . 4 ⊢ (𝜑 → (𝐹‘(𝑍 − 1)) < 𝑍)
17	12	hoverb 14968	. . . . 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 14965	. 2 ⊢ (𝜑 → ∃𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2))(𝐹‘𝑐) = 𝑍)
22		0red 8044	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → 0 ∈ ℝ)
23		1red 8058	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → 1 ∈ ℝ)
24		elioore 10004	. . . . . 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 8170	. . . . . 6 ⊢ 0 < 1
27		axltwlin 8111	. . . . . 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 1249	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → (0 < 𝑐 ∨ 𝑐 < 1))
30	29	orcomd 730	. . 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 14969	. . . . . . 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 4058	. . . . 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 14970	. . . . . . 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 4060	. . . . 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 787	. . 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 2619	1 ⊢ (𝜑 → (𝑍 ≤ 0 ∨ 0 ≤ 𝑍))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∨ wo 709 ∧ w3a 980 ∀wal 1362 = wceq 1364 ∈ wcel 2167 ∀wral 2475 ∃wrex 2476 {cpr 3624 class class class wbr 4034 ↦ cmpt 4095 ‘cfv 5259 (class class class)co 5925 supcsup 7057 infcinf 7058 ℝcr 7895 0cc0 7896 1c1 7897 + caddc 7899 < clt 8078 ≤ cle 8079 − cmin 8214 2c2 9058 ℝ⁺crp 9745 (,)cioo 9980 –cn→ccncf 14890

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016 ax-addf 8018

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-map 6718 df-sup 7059 df-inf 7060 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-n0 9267 df-z 9344 df-uz 9619 df-q 9711 df-rp 9746 df-xneg 9864 df-xadd 9865 df-ioo 9984 df-seqfrec 10557 df-exp 10648 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181 df-rest 12943 df-topgen 12962 df-psmet 14175 df-xmet 14176 df-met 14177 df-bl 14178 df-mopn 14179 df-top 14318 df-topon 14331 df-bases 14363 df-cn 14508 df-cnp 14509 df-tx 14573 df-cncf 14891

This theorem is referenced by: ivthdich 14973

W3C validator