ioorrnopnxr - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ioorrnopnxr		Structured version Visualization version GIF version

Theorem ioorrnopnxr 46284

Description: The indexed product of open intervals is an open set in (ℝ^‘𝑋). Similar to ioorrnopn 46282 but here unbounded intervals are allowed. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
ioorrnopnxr.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ioorrnopnxr.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ^*)
ioorrnopnxr.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ^*)

**Assertion**
Ref	Expression
ioorrnopnxr	⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))

Distinct variable groups: 𝐴,𝑖 𝐵,𝑖 𝑖,𝑋 𝜑,𝑖

Proof of Theorem ioorrnopnxr Dummy variables 𝑓 𝑗 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		p0ex 5354	. . . . . 6 ⊢ {∅} ∈ V
2	1	prid2 4739	. . . . 5 ⊢ {∅} ∈ {∅, {∅}}
3	2	a1i 11	. . . 4 ⊢ (𝑋 = ∅ → {∅} ∈ {∅, {∅}})
4		ixpeq1 8920	. . . . . 6 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = X𝑖 ∈ ∅ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
5		ixp0x 8938	. . . . . . 7 ⊢ X𝑖 ∈ ∅ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = {∅}
6	5	a1i 11	. . . . . 6 ⊢ (𝑋 = ∅ → X𝑖 ∈ ∅ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = {∅})
7	4, 6	eqtrd 2770	. . . . 5 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = {∅})
8		2fveq3 6880	. . . . . 6 ⊢ (𝑋 = ∅ → (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(ℝ^‘∅)))
9		rrxtopn0b 46273	. . . . . . 7 ⊢ (TopOpen‘(ℝ^‘∅)) = {∅, {∅}}
10	9	a1i 11	. . . . . 6 ⊢ (𝑋 = ∅ → (TopOpen‘(ℝ^‘∅)) = {∅, {∅}})
11	8, 10	eqtrd 2770	. . . . 5 ⊢ (𝑋 = ∅ → (TopOpen‘(ℝ^‘𝑋)) = {∅, {∅}})
12	7, 11	eleq12d 2828	. . . 4 ⊢ (𝑋 = ∅ → (X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)) ↔ {∅} ∈ {∅, {∅}}))
13	3, 12	mpbird 257	. . 3 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
14	13	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
15		neqne 2940	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
16	15	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
17		fveq2 6875	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝐴‘𝑖) = (𝐴‘𝑗))
18		fveq2 6875	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝐵‘𝑖) = (𝐵‘𝑗))
19	17, 18	oveq12d 7421	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = ((𝐴‘𝑗)(,)(𝐵‘𝑗)))
20	19	cbvixpv 8927	. . . . . . . . 9 ⊢ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))
21	20	eleq2i 2826	. . . . . . . 8 ⊢ (𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗)))
22	21	biimpi 216	. . . . . . 7 ⊢ (𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗)))
23	22	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗)))
24		ioorrnopnxr.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
25	24	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → 𝑋 ∈ Fin)
26		ioorrnopnxr.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ^*)
27	26	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → 𝐴:𝑋⟶ℝ^*)
28		ioorrnopnxr.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ^*)
29	28	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → 𝐵:𝑋⟶ℝ^*)
30	21	biimpri 228	. . . . . . . 8 ⊢ (𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗)) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
31	30	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
32		fveq2 6875	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝐴‘𝑗) = (𝐴‘𝑖))
33	32	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝐴‘𝑗) = -∞ ↔ (𝐴‘𝑖) = -∞))
34		fveq2 6875	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑓‘𝑗) = (𝑓‘𝑖))
35	34	oveq1d 7418	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝑓‘𝑗) − 1) = ((𝑓‘𝑖) − 1))
36	33, 35, 32	ifbieq12d 4529	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → if((𝐴‘𝑗) = -∞, ((𝑓‘𝑗) − 1), (𝐴‘𝑗)) = if((𝐴‘𝑖) = -∞, ((𝑓‘𝑖) − 1), (𝐴‘𝑖)))
37	36	cbvmptv 5225	. . . . . . 7 ⊢ (𝑗 ∈ 𝑋 ↦ if((𝐴‘𝑗) = -∞, ((𝑓‘𝑗) − 1), (𝐴‘𝑗))) = (𝑖 ∈ 𝑋 ↦ if((𝐴‘𝑖) = -∞, ((𝑓‘𝑖) − 1), (𝐴‘𝑖)))
38		fveq2 6875	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝐵‘𝑗) = (𝐵‘𝑖))
39	38	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝐵‘𝑗) = +∞ ↔ (𝐵‘𝑖) = +∞))
40	34	oveq1d 7418	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝑓‘𝑗) + 1) = ((𝑓‘𝑖) + 1))
41	39, 40, 38	ifbieq12d 4529	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → if((𝐵‘𝑗) = +∞, ((𝑓‘𝑗) + 1), (𝐵‘𝑗)) = if((𝐵‘𝑖) = +∞, ((𝑓‘𝑖) + 1), (𝐵‘𝑖)))
42	41	cbvmptv 5225	. . . . . . 7 ⊢ (𝑗 ∈ 𝑋 ↦ if((𝐵‘𝑗) = +∞, ((𝑓‘𝑗) + 1), (𝐵‘𝑗))) = (𝑖 ∈ 𝑋 ↦ if((𝐵‘𝑖) = +∞, ((𝑓‘𝑖) + 1), (𝐵‘𝑖)))
43		eqid 2735	. . . . . . 7 ⊢ X𝑖 ∈ 𝑋 (((𝑗 ∈ 𝑋 ↦ if((𝐴‘𝑗) = -∞, ((𝑓‘𝑗) − 1), (𝐴‘𝑗)))‘𝑖)(,)((𝑗 ∈ 𝑋 ↦ if((𝐵‘𝑗) = +∞, ((𝑓‘𝑗) + 1), (𝐵‘𝑗)))‘𝑖)) = X𝑖 ∈ 𝑋 (((𝑗 ∈ 𝑋 ↦ if((𝐴‘𝑗) = -∞, ((𝑓‘𝑗) − 1), (𝐴‘𝑗)))‘𝑖)(,)((𝑗 ∈ 𝑋 ↦ if((𝐵‘𝑗) = +∞, ((𝑓‘𝑗) + 1), (𝐵‘𝑗)))‘𝑖))
44	25, 27, 29, 31, 37, 42, 43	ioorrnopnxrlem 46283	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝑓 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
45	23, 44	syldan 591	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝑓 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
46	45	ralrimiva 3132	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∀𝑓 ∈ X 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝑓 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
47		eqid 2735	. . . . . . . 8 ⊢ (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(ℝ^‘𝑋))
48	47	rrxtop 46266	. . . . . . 7 ⊢ (𝑋 ∈ Fin → (TopOpen‘(ℝ^‘𝑋)) ∈ Top)
49	24, 48	syl 17	. . . . . 6 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) ∈ Top)
50	49	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (TopOpen‘(ℝ^‘𝑋)) ∈ Top)
51		eltop2 22911	. . . . 5 ⊢ ((TopOpen‘(ℝ^‘𝑋)) ∈ Top → (X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)) ↔ ∀𝑓 ∈ X 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝑓 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))))
52	50, 51	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)) ↔ ∀𝑓 ∈ X 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝑓 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))))
53	46, 52	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
54	16, 53	syldan 591	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
55	14, 54	pm2.61dan 812	1 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 ⊆ wss 3926 ∅c0 4308 ifcif 4500 {csn 4601 {cpr 4603 ↦ cmpt 5201 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 Xcixp 8909 Fincfn 8957 1c1 11128 + caddc 11130 +∞cpnf 11264 -∞cmnf 11265 ℝ^*cxr 11266 − cmin 11464 (,)cioo 13360 TopOpenctopn 17433 Topctop 22829 ℝ^crrx 25333

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-inf2 9653 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206 ax-mulf 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8717 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-sup 9452 df-inf 9453 df-oi 9522 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ioo 13364 df-ico 13366 df-fz 13523 df-fzo 13670 df-seq 14018 df-exp 14078 df-hash 14347 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-clim 15502 df-sum 15701 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-prds 17459 df-pws 17461 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-mhm 18759 df-submnd 18760 df-grp 18917 df-minusg 18918 df-sbg 18919 df-subg 19104 df-ghm 19194 df-cntz 19298 df-cmn 19761 df-abl 19762 df-mgp 20099 df-rng 20111 df-ur 20140 df-ring 20193 df-cring 20194 df-oppr 20295 df-dvdsr 20315 df-unit 20316 df-invr 20346 df-dvr 20359 df-rhm 20430 df-subrng 20504 df-subrg 20528 df-drng 20689 df-field 20690 df-abv 20767 df-staf 20797 df-srng 20798 df-lmod 20817 df-lss 20887 df-lmhm 20978 df-lvec 21059 df-sra 21129 df-rgmod 21130 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-cnfld 21314 df-refld 21563 df-phl 21584 df-dsmm 21690 df-frlm 21705 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-xms 24257 df-ms 24258 df-nm 24519 df-ngp 24520 df-tng 24521 df-nrg 24522 df-nlm 24523 df-clm 25012 df-cph 25118 df-tcph 25119 df-rrx 25335

This theorem is referenced by: ioovonmbl 46654

W3C validator