ioorrnopnxr - Mathbox for Glauco Siliprandi

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Theorem ioorrnopnxr 46429

Description: The indexed product of open intervals is an open set in (ℝ^‘𝑋). Similar to ioorrnopn 46427 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 5324	. . . . . 6 ⊢ {∅} ∈ V
2	1	prid2 4715	. . . . 5 ⊢ {∅} ∈ {∅, {∅}}
3	2	a1i 11	. . . 4 ⊢ (𝑋 = ∅ → {∅} ∈ {∅, {∅}})
4		ixpeq1 8838	. . . . . 6 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = X𝑖 ∈ ∅ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
5		ixp0x 8856	. . . . . . 7 ⊢ X𝑖 ∈ ∅ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = {∅}
6	5	a1i 11	. . . . . 6 ⊢ (𝑋 = ∅ → X𝑖 ∈ ∅ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = {∅})
7	4, 6	eqtrd 2768	. . . . 5 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = {∅})
8		2fveq3 6833	. . . . . 6 ⊢ (𝑋 = ∅ → (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(ℝ^‘∅)))
9		rrxtopn0b 46418	. . . . . . 7 ⊢ (TopOpen‘(ℝ^‘∅)) = {∅, {∅}}
10	9	a1i 11	. . . . . 6 ⊢ (𝑋 = ∅ → (TopOpen‘(ℝ^‘∅)) = {∅, {∅}})
11	8, 10	eqtrd 2768	. . . . 5 ⊢ (𝑋 = ∅ → (TopOpen‘(ℝ^‘𝑋)) = {∅, {∅}})
12	7, 11	eleq12d 2827	. . . 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 2937	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
16	15	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
17		fveq2 6828	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝐴‘𝑖) = (𝐴‘𝑗))
18		fveq2 6828	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝐵‘𝑖) = (𝐵‘𝑗))
19	17, 18	oveq12d 7370	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = ((𝐴‘𝑗)(,)(𝐵‘𝑗)))
20	19	cbvixpv 8845	. . . . . . . . 9 ⊢ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) = X𝑗 ∈ 𝑋 ((𝐴‘𝑗)(,)(𝐵‘𝑗))
21	20	eleq2i 2825	. . . . . . . 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 6828	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝐴‘𝑗) = (𝐴‘𝑖))
33	32	eqeq1d 2735	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝐴‘𝑗) = -∞ ↔ (𝐴‘𝑖) = -∞))
34		fveq2 6828	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑓‘𝑗) = (𝑓‘𝑖))
35	34	oveq1d 7367	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝑓‘𝑗) − 1) = ((𝑓‘𝑖) − 1))
36	33, 35, 32	ifbieq12d 4503	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → if((𝐴‘𝑗) = -∞, ((𝑓‘𝑗) − 1), (𝐴‘𝑗)) = if((𝐴‘𝑖) = -∞, ((𝑓‘𝑖) − 1), (𝐴‘𝑖)))
37	36	cbvmptv 5197	. . . . . . 7 ⊢ (𝑗 ∈ 𝑋 ↦ if((𝐴‘𝑗) = -∞, ((𝑓‘𝑗) − 1), (𝐴‘𝑗))) = (𝑖 ∈ 𝑋 ↦ if((𝐴‘𝑖) = -∞, ((𝑓‘𝑖) − 1), (𝐴‘𝑖)))
38		fveq2 6828	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝐵‘𝑗) = (𝐵‘𝑖))
39	38	eqeq1d 2735	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝐵‘𝑗) = +∞ ↔ (𝐵‘𝑖) = +∞))
40	34	oveq1d 7367	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝑓‘𝑗) + 1) = ((𝑓‘𝑖) + 1))
41	39, 40, 38	ifbieq12d 4503	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → if((𝐵‘𝑗) = +∞, ((𝑓‘𝑗) + 1), (𝐵‘𝑗)) = if((𝐵‘𝑖) = +∞, ((𝑓‘𝑖) + 1), (𝐵‘𝑖)))
42	41	cbvmptv 5197	. . . . . . 7 ⊢ (𝑗 ∈ 𝑋 ↦ if((𝐵‘𝑗) = +∞, ((𝑓‘𝑗) + 1), (𝐵‘𝑗))) = (𝑖 ∈ 𝑋 ↦ if((𝐵‘𝑖) = +∞, ((𝑓‘𝑖) + 1), (𝐵‘𝑖)))
43		eqid 2733	. . . . . . 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 46428	. . . . . 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 3125	. . . 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 2733	. . . . . . . 8 ⊢ (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(ℝ^‘𝑋))
48	47	rrxtop 46411	. . . . . . 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 22891	. . . . 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 1541 ∈ wcel 2113 ≠ wne 2929 ∀wral 3048 ∃wrex 3057 ⊆ wss 3898 ∅c0 4282 ifcif 4474 {csn 4575 {cpr 4577 ↦ cmpt 5174 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 Xcixp 8827 Fincfn 8875 1c1 11014 + caddc 11016 +∞cpnf 11150 -∞cmnf 11151 ℝ^*cxr 11152 − cmin 11351 (,)cioo 13247 TopOpenctopn 17327 Topctop 22809 ℝ^crrx 25311

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9538 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091 ax-addf 11092 ax-mulf 11093

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-map 8758 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-sup 9333 df-inf 9334 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-q 12849 df-rp 12893 df-xneg 13013 df-xadd 13014 df-xmul 13015 df-ioo 13251 df-ico 13253 df-fz 13410 df-fzo 13557 df-seq 13911 df-exp 13971 df-hash 14240 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-clim 15397 df-sum 15596 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-starv 17178 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-unif 17186 df-hom 17187 df-cco 17188 df-rest 17328 df-topn 17329 df-0g 17347 df-gsum 17348 df-topgen 17349 df-prds 17353 df-pws 17355 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-mhm 18693 df-submnd 18694 df-grp 18851 df-minusg 18852 df-sbg 18853 df-subg 19038 df-ghm 19127 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-cring 20156 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-dvr 20321 df-rhm 20392 df-subrng 20463 df-subrg 20487 df-drng 20648 df-field 20649 df-abv 20726 df-staf 20756 df-srng 20757 df-lmod 20797 df-lss 20867 df-lmhm 20958 df-lvec 21039 df-sra 21109 df-rgmod 21110 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-cnfld 21294 df-refld 21544 df-phl 21565 df-dsmm 21671 df-frlm 21686 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-xms 24236 df-ms 24237 df-nm 24498 df-ngp 24499 df-tng 24500 df-nrg 24501 df-nlm 24502 df-clm 24991 df-cph 25096 df-tcph 25097 df-rrx 25313

This theorem is referenced by: ioovonmbl 46799

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