Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrxtoponfi | Structured version Visualization version GIF version |
Description: The topology on n-dimensional Euclidean real spaces. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
rrxtopfi.1 | ⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼)) |
Ref | Expression |
---|---|
rrxtoponfi | ⊢ (𝐼 ∈ Fin → 𝐽 ∈ (TopOn‘(ℝ ↑m 𝐼))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxtopfi.1 | . . 3 ⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼)) | |
2 | 1 | rrxtopon 42647 | . 2 ⊢ (𝐼 ∈ Fin → 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼)))) |
3 | id 22 | . . . 4 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin) | |
4 | eqid 2820 | . . . 4 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼) | |
5 | eqid 2820 | . . . 4 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼)) | |
6 | 3, 4, 5 | rrxbasefi 24008 | . . 3 ⊢ (𝐼 ∈ Fin → (Base‘(ℝ^‘𝐼)) = (ℝ ↑m 𝐼)) |
7 | 6 | fveq2d 6667 | . 2 ⊢ (𝐼 ∈ Fin → (TopOn‘(Base‘(ℝ^‘𝐼))) = (TopOn‘(ℝ ↑m 𝐼))) |
8 | 2, 7 | eleqtrd 2914 | 1 ⊢ (𝐼 ∈ Fin → 𝐽 ∈ (TopOn‘(ℝ ↑m 𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 ↑m cmap 8399 Fincfn 8502 ℝcr 10529 Basecbs 16478 TopOpenctopn 16690 TopOnctopon 21513 ℝ^crrx 23981 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-sup 8899 df-inf 8900 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ico 12738 df-fz 12890 df-fzo 13031 df-seq 13367 df-exp 13427 df-hash 13688 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-clim 14840 df-sum 15038 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-starv 16575 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-unif 16583 df-hom 16584 df-cco 16585 df-rest 16691 df-topn 16692 df-0g 16710 df-gsum 16711 df-topgen 16712 df-prds 16716 df-pws 16718 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-mhm 17951 df-submnd 17952 df-grp 18101 df-minusg 18102 df-sbg 18103 df-subg 18271 df-ghm 18351 df-cntz 18442 df-cmn 18903 df-abl 18904 df-mgp 19235 df-ur 19247 df-ring 19294 df-cring 19295 df-oppr 19368 df-dvdsr 19386 df-unit 19387 df-invr 19417 df-dvr 19428 df-rnghom 19462 df-drng 19499 df-field 19500 df-subrg 19528 df-abv 19583 df-staf 19611 df-srng 19612 df-lmod 19631 df-lss 19699 df-lmhm 19789 df-lvec 19870 df-sra 19939 df-rgmod 19940 df-psmet 20532 df-xmet 20533 df-met 20534 df-bl 20535 df-mopn 20536 df-cnfld 20541 df-refld 20744 df-phl 20765 df-dsmm 20871 df-frlm 20886 df-top 21497 df-topon 21514 df-topsp 21536 df-bases 21549 df-xms 22925 df-ms 22926 df-nm 23187 df-ngp 23188 df-tng 23189 df-nrg 23190 df-nlm 23191 df-clm 23662 df-cph 23767 df-tcph 23768 df-rrx 23983 |
This theorem is referenced by: rrxunitopnfi 42651 rrxtopn0 42652 opnvonmbllem2 42989 |
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