ioorrnopnxr - Mathbox for Glauco Siliprandi

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Theorem ioorrnopnxr 47004

Description: The indexed product of open intervals is an open set in (ℝ^‘𝑋). Similar to ioorrnopn 47002 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 5357	. . . . . 6 ⊢ {∅} ∈ V
2	1	prid2 4730	. . . . 5 ⊢ {∅} ∈ {∅, {∅}}
3	2	a1i 11	. . . 4 ⊢ (𝑋 = ∅ → {∅} ∈ {∅, {∅}})
4		ixpeq1 8907	. . . . . 6 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = X𝑖 ∈ ∅ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
5		ixp0x 8925	. . . . . . 7 ⊢ X𝑖 ∈ ∅ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = {∅}
6	5	a1i 11	. . . . . 6 ⊢ (𝑋 = ∅ → X𝑖 ∈ ∅ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = {∅})
7	4, 6	eqtrd 2798	. . . . 5 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = {∅})
8		2fveq3 6888	. . . . . 6 ⊢ (𝑋 = ∅ → (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(ℝ^‘∅)))
9		rrxtopn0b 46993	. . . . . . 7 ⊢ (TopOpen‘(ℝ^‘∅)) = {∅, {∅}}
10	9	a1i 11	. . . . . 6 ⊢ (𝑋 = ∅ → (TopOpen‘(ℝ^‘∅)) = {∅, {∅}})
11	8, 10	eqtrd 2798	. . . . 5 ⊢ (𝑋 = ∅ → (TopOpen‘(ℝ^‘𝑋)) = {∅, {∅}})
12	7, 11	eleq12d 2857	. . . 4 ⊢ (𝑋 = ∅ → (X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)) ↔ {∅} ∈ {∅, {∅}}))
13	3, 12	mpbird 260	. . 3 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
14	13	adantl 486	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
15		neqne 2966	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
16	15	adantl 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
17		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝐴‘𝑖) = (𝐴‘𝑗))
18		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝐵‘𝑖) = (𝐵‘𝑗))
19	17, 18	oveq12d 7430	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = ((𝐴‘𝑗)(,)(𝐵‘𝑗)))
20	19	cbvixpv 8914	. . . . . . . 8 ⊢ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))
21	20	eleq2i 2855	. . . . . . 7 ⊢ (𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗)))
22	21	bilani 509	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗)))
23		ioorrnopnxr.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
24	23	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → 𝑋 ∈ Fin)
25		ioorrnopnxr.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ^*)
26	25	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → 𝐴:𝑋⟶ℝ^*)
27		ioorrnopnxr.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ^*)
28	27	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → 𝐵:𝑋⟶ℝ^*)
29	21	bilanri 511	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
30		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝐴‘𝑗) = (𝐴‘𝑖))
31	30	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝐴‘𝑗) = -∞ ↔ (𝐴‘𝑖) = -∞))
32		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑓‘𝑗) = (𝑓‘𝑖))
33	32	oveq1d 7427	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝑓‘𝑗) − 1) = ((𝑓‘𝑖) − 1))
34	31, 33, 30	ifbieq12d 4517	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → if((𝐴‘𝑗) = -∞, ((𝑓‘𝑗) − 1), (𝐴‘𝑗)) = if((𝐴‘𝑖) = -∞, ((𝑓‘𝑖) − 1), (𝐴‘𝑖)))
35	34	cbvmptv 5216	. . . . . . 7 ⊢ (𝑗 ∈ 𝑋 ↦ if((𝐴‘𝑗) = -∞, ((𝑓‘𝑗) − 1), (𝐴‘𝑗))) = (𝑖 ∈ 𝑋 ↦ if((𝐴‘𝑖) = -∞, ((𝑓‘𝑖) − 1), (𝐴‘𝑖)))
36		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝐵‘𝑗) = (𝐵‘𝑖))
37	36	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝐵‘𝑗) = +∞ ↔ (𝐵‘𝑖) = +∞))
38	32	oveq1d 7427	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝑓‘𝑗) + 1) = ((𝑓‘𝑖) + 1))
39	37, 38, 36	ifbieq12d 4517	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → if((𝐵‘𝑗) = +∞, ((𝑓‘𝑗) + 1), (𝐵‘𝑗)) = if((𝐵‘𝑖) = +∞, ((𝑓‘𝑖) + 1), (𝐵‘𝑖)))
40	39	cbvmptv 5216	. . . . . . 7 ⊢ (𝑗 ∈ 𝑋 ↦ if((𝐵‘𝑗) = +∞, ((𝑓‘𝑗) + 1), (𝐵‘𝑗))) = (𝑖 ∈ 𝑋 ↦ if((𝐵‘𝑖) = +∞, ((𝑓‘𝑖) + 1), (𝐵‘𝑖)))
41		eqid 2763	. . . . . . 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))
42	24, 26, 28, 29, 35, 40, 41	ioorrnopnxrlem 47003	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))) → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝑓 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
43	22, 42	syldan 602	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝑓 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
44	43	ralrimiva 3157	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∀𝑓 ∈ X 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝑓 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
45		eqid 2763	. . . . . . . 8 ⊢ (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(ℝ^‘𝑋))
46	45	rrxtop 46986	. . . . . . 7 ⊢ (𝑋 ∈ Fin → (TopOpen‘(ℝ^‘𝑋)) ∈ Top)
47	23, 46	syl 18	. . . . . 6 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) ∈ Top)
48	47	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (TopOpen‘(ℝ^‘𝑋)) ∈ Top)
49		eltop2 23113	. . . . 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)))))
50	48, 49	syl 18	. . . 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)))))
51	44, 50	mpbird 260	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
52	16, 51	syldan 602	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
53	14, 52	pm2.61dan 824	1 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 ⊆ wss 3906 ∅c0 4287 ifcif 4488 {csn 4590 {cpr 4592 ↦ cmpt 5193 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 Xcixp 8896 Fincfn 8944 1c1 11102 + caddc 11104 +∞cpnf 11241 -∞cmnf 11242 ℝ^*cxr 11243 − cmin 11442 (,)cioo 13373 TopOpenctopn 17475 Topctop 23031 ℝ^crrx 25523

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180 ax-mulf 11181

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8158 df-tpos 8223 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-map 8827 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fsupp 9323 df-sup 9403 df-inf 9404 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-ico 13379 df-fz 13537 df-fzo 13685 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-sum 15740 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17476 df-topn 17477 df-0g 17495 df-gsum 17496 df-topgen 17497 df-prds 17501 df-pws 17503 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-mhm 18842 df-submnd 18843 df-grp 19004 df-minusg 19005 df-sbg 19006 df-subg 19190 df-ghm 19285 df-cntz 19388 df-cmn 19853 df-abl 19854 df-mgp 20218 df-rng 20232 df-ur 20265 df-ring 20318 df-cring 20319 df-oppr 20420 df-dvdsr 20440 df-unit 20441 df-invr 20471 df-dvr 20484 df-rhm 20555 df-subrng 20632 df-subrg 20656 df-drng 20816 df-field 20817 df-abv 20893 df-staf 20923 df-srng 20924 df-lmod 20964 df-lss 21034 df-lmhm 21124 df-lvec 21205 df-sra 21275 df-rgmod 21276 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-cnfld 21504 df-refld 21736 df-phl 21757 df-dsmm 21863 df-frlm 21878 df-top 23032 df-topon 23049 df-topsp 23071 df-bases 23084 df-xms 24458 df-ms 24459 df-nm 24720 df-ngp 24721 df-tng 24722 df-nrg 24723 df-nlm 24724 df-clm 25203 df-cph 25308 df-tcph 25309 df-rrx 25525

This theorem is referenced by: ioovonmbl 47374

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