hoiqssbllem1 - Mathbox for Glauco Siliprandi

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Theorem hoiqssbllem1 45423

Description: The center of the n-dimensional ball belongs to the half-open interval. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypotheses**
Ref	Expression
hoiqssbllem1.i	⊢ Ⅎ𝑖𝜑
hoiqssbllem1.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hoiqssbllem1.n	⊢ (𝜑 → 𝑋 ≠ ∅)
hoiqssbllem1.y	⊢ (𝜑 → 𝑌 ∈ (ℝ ↑_m 𝑋))
hoiqssbllem1.c	⊢ (𝜑 → 𝐶:𝑋⟶ℝ)
hoiqssbllem1.d	⊢ (𝜑 → 𝐷:𝑋⟶ℝ)
hoiqssbllem1.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
hoiqssbllem1.l	⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
hoiqssbllem1.r	⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))

**Assertion**
Ref	Expression
hoiqssbllem1	⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))

Distinct variable groups: 𝑖,𝑋 𝑖,𝑌 Allowed substitution hints: 𝜑(𝑖) 𝐶(𝑖) 𝐷(𝑖) 𝐸(𝑖)

**Proof of Theorem hoiqssbllem1**
Step	Hyp	Ref	Expression
1		hoiqssbllem1.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑_m 𝑋))
2	1	elexd 3494	. . 3 ⊢ (𝜑 → 𝑌 ∈ V)
3		elmapfn 8861	. . . 4 ⊢ (𝑌 ∈ (ℝ ↑_m 𝑋) → 𝑌 Fn 𝑋)
4	1, 3	syl 17	. . 3 ⊢ (𝜑 → 𝑌 Fn 𝑋)
5		hoiqssbllem1.i	. . . 4 ⊢ Ⅎ𝑖𝜑
6		hoiqssbllem1.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶:𝑋⟶ℝ)
7	6	ffvelcdmda 7086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ)
8	7	rexrd 11266	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ^*)
9		hoiqssbllem1.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷:𝑋⟶ℝ)
10	9	ffvelcdmda 7086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ)
11	10	rexrd 11266	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ^*)
12		elmapi 8845	. . . . . . . . 9 ⊢ (𝑌 ∈ (ℝ ↑_m 𝑋) → 𝑌:𝑋⟶ℝ)
13	1, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌:𝑋⟶ℝ)
14	13	ffvelcdmda 7086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ)
15	14	rexrd 11266	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ^*)
16		hoiqssbllem1.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
17		2rp 12981	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ⁺
18	17	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ⁺)
19		hoiqssbllem1.n	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ≠ ∅)
20		hoiqssbllem1.x	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 ∈ Fin)
21		hashnncl 14328	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
22	20, 21	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
23	19, 22	mpbird 256	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
24	23	nnred 12229	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ)
25	23	nngt0d 12263	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < (♯‘𝑋))
26	24, 25	elrpd 13015	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ⁺)
27	26	rpsqrtcld 15360	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℝ⁺)
28	18, 27	rpmulcld 13034	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · (√‘(♯‘𝑋))) ∈ ℝ⁺)
29	16, 28	rpdivcld 13035	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ⁺)
30	29	rpred 13018	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
31	30	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
32	14, 31	resubcld 11644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
33	32	rexrd 11266	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^*)
34		hoiqssbllem1.l	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
35		iooltub 44308	. . . . . . . 8 ⊢ ((((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^* ∧ (𝑌‘𝑖) ∈ ℝ^* ∧ (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → (𝐶‘𝑖) < (𝑌‘𝑖))
36	33, 15, 34, 35	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) < (𝑌‘𝑖))
37	7, 14, 36	ltled 11364	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ≤ (𝑌‘𝑖))
38	14, 31	readdcld 11245	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
39	38	rexrd 11266	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^*)
40		hoiqssbllem1.r	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
41		ioogtlb 44293	. . . . . . 7 ⊢ (((𝑌‘𝑖) ∈ ℝ^* ∧ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^* ∧ (𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → (𝑌‘𝑖) < (𝐷‘𝑖))
42	15, 39, 40, 41	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) < (𝐷‘𝑖))
43	8, 11, 15, 37, 42	elicod 13376	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
44	43	ex 413	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑋 → (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖))))
45	5, 44	ralrimi 3254	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
46	2, 4, 45	3jca 1128	. 2 ⊢ (𝜑 → (𝑌 ∈ V ∧ 𝑌 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖))))
47		elixp2 8897	. 2 ⊢ (𝑌 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ↔ (𝑌 ∈ V ∧ 𝑌 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖))))
48	46, 47	sylibr 233	1 ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 Ⅎwnf 1785 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 Vcvv 3474 ∅c0 4322 class class class wbr 5148 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ↑_m cmap 8822 Xcixp 8893 Fincfn 8941 ℝcr 11111 + caddc 11115 · cmul 11117 ℝ^*cxr 11249 < clt 11250 − cmin 11446 / cdiv 11873 ℕcn 12214 2c2 12269 ℝ⁺crp 12976 (,)cioo 13326 [,)cico 13328 ♯chash 14292 √csqrt 15182

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-rp 12977 df-ioo 13330 df-ico 13332 df-fz 13487 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184

This theorem is referenced by: hoiqssbllem3 45425

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