ivthdichlem - Intuitionistic Logic Explorer

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

Theorem ivthdichlem 15889

Description: Lemma for ivthdich 15891. 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 8601	. . . 4 ⊢ (𝑍 ∈ ℝ → (𝑍 − 1) ∈ ℝ)
3	1, 2	syl 14	. . 3 ⊢ (𝜑 → (𝑍 − 1) ∈ ℝ)
4		2re 9384	. . . . 5 ⊢ 2 ∈ ℝ
5	4	a1i 9	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
6	1, 5	readdcld 8362	. . 3 ⊢ (𝜑 → (𝑍 + 2) ∈ ℝ)
7	1	ltm1d 9271	. . . 4 ⊢ (𝜑 → (𝑍 − 1) < 𝑍)
8		2rp 10077	. . . . . 6 ⊢ 2 ∈ ℝ⁺
9	8	a1i 9	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ⁺)
10	1, 9	ltaddrpd 10149	. . . 4 ⊢ (𝜑 → 𝑍 < (𝑍 + 2))
11	3, 1, 6, 7, 10	lttrd 8460	. . 3 ⊢ (𝜑 → (𝑍 − 1) < (𝑍 + 2))
12		hover.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))
13	12	hovercncf 15884	. . . 4 ⊢ 𝐹 ∈ (ℝ–cn→ℝ)
14	13	a1i 9	. . 3 ⊢ (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ))
15	12	hovera 15885	. . . . 5 ⊢ (𝑍 ∈ ℝ → (𝐹‘(𝑍 − 1)) < 𝑍)
16	1, 15	syl 14	. . . 4 ⊢ (𝜑 → (𝐹‘(𝑍 − 1)) < 𝑍)
17	12	hoverb 15886	. . . . 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 15883	. 2 ⊢ (𝜑 → ∃𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2))(𝐹‘𝑐) = 𝑍)
22		0red 8334	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → 0 ∈ ℝ)
23		1red 8348	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → 1 ∈ ℝ)
24		elioore 10332	. . . . . 6 ⊢ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) → 𝑐 ∈ ℝ)
25	24	ad2antrl 494	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → 𝑐 ∈ ℝ)
26		0lt1 8461	. . . . . 6 ⊢ 0 < 1
27		axltwlin 8400	. . . . . 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 1278	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → (0 < 𝑐 ∨ 𝑐 < 1))
30	29	orcomd 741	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) → (𝑐 < 1 ∨ 0 < 𝑐))
31		simplrr 542	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) ∧ 𝑐 < 1) → (𝐹‘𝑐) = 𝑍)
32	12	hoverlt1 15887	. . . . . . 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 4155	. . . . 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 15888	. . . . . . 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 542	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝑍 − 1)(,)(𝑍 + 2)) ∧ (𝐹‘𝑐) = 𝑍)) ∧ 0 < 𝑐) → (𝐹‘𝑐) = 𝑍)
39	37, 38	breqtrd 4157	. . . . 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 798	. . 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 2673	1 ⊢ (𝜑 → (𝑍 ≤ 0 ∨ 0 ≤ 𝑍))

Colors of variables: wff set class

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 + caddc 8189 < clt 8367 ≤ cle 8368 − cmin 8505 2c2 9365 ℝ⁺crp 10072 (,)cioo 10308 –cn→ccncf 15808

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 6580 df-frec 6666 df-map 6928 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 9657 df-uz 9939 df-q 10037 df-rp 10073 df-xneg 10192 df-xadd 10193 df-ioo 10312 df-seqfrec 10907 df-exp 10998 df-cj 11630 df-re 11631 df-im 11632 df-rsqrt 11787 df-abs 11788 df-rest 13655 df-topgen 13674 df-psmet 14990 df-xmet 14991 df-met 14992 df-bl 14993 df-mopn 14994 df-top 15236 df-topon 15249 df-bases 15281 df-cn 15426 df-cnp 15427 df-tx 15491 df-cncf 15809

This theorem is used by: ivthdich 15891

W3C validator