hoiqssbllem1 - Mathbox for Glauco Siliprandi

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Theorem hoiqssbllem1 45638

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 8862	. . . 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 7087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ)
8	7	rexrd 11269	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ^*)
9		hoiqssbllem1.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷:𝑋⟶ℝ)
10	9	ffvelcdmda 7087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ)
11	10	rexrd 11269	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ^*)
12		elmapi 8846	. . . . . . . . 9 ⊢ (𝑌 ∈ (ℝ ↑_m 𝑋) → 𝑌:𝑋⟶ℝ)
13	1, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌:𝑋⟶ℝ)
14	13	ffvelcdmda 7087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ)
15	14	rexrd 11269	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ^*)
16		hoiqssbllem1.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
17		2rp 12984	. . . . . . . . . . . . . . 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 14331	. . . . . . . . . . . . . . . . . . 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 12232	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ)
25	23	nngt0d 12266	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < (♯‘𝑋))
26	24, 25	elrpd 13018	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ⁺)
27	26	rpsqrtcld 15363	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℝ⁺)
28	18, 27	rpmulcld 13037	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · (√‘(♯‘𝑋))) ∈ ℝ⁺)
29	16, 28	rpdivcld 13038	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ⁺)
30	29	rpred 13021	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
31	30	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
32	14, 31	resubcld 11647	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
33	32	rexrd 11269	. . . . . . . 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 44523	. . . . . . . 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 1370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) < (𝑌‘𝑖))
37	7, 14, 36	ltled 11367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ≤ (𝑌‘𝑖))
38	14, 31	readdcld 11248	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
39	38	rexrd 11269	. . . . . . 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 44508	. . . . . . 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 1370	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) < (𝐷‘𝑖))
43	8, 11, 15, 37, 42	elicod 13379	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
44	43	ex 412	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑋 → (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖))))
45	5, 44	ralrimi 3253	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
46	2, 4, 45	3jca 1127	. 2 ⊢ (𝜑 → (𝑌 ∈ V ∧ 𝑌 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖))))
47		elixp2 8898	. 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 395 ∧ w3a 1086 Ⅎwnf 1784 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 Vcvv 3473 ∅c0 4323 class class class wbr 5149 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7412 ↑_m cmap 8823 Xcixp 8894 Fincfn 8942 ℝcr 11112 + caddc 11116 · cmul 11118 ℝ^*cxr 11252 < clt 11253 − cmin 11449 / cdiv 11876 ℕcn 12217 2c2 12272 ℝ⁺crp 12979 (,)cioo 13329 [,)cico 13331 ♯chash 14295 √csqrt 15185

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9440 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-ioo 13333 df-ico 13335 df-fz 13490 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187

This theorem is referenced by: hoiqssbllem3 45640

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