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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 | β’ π½ = (TopOpenβ(β^βπΌ)) |
Ref | Expression |
---|---|
rrxtoponfi | β’ (πΌ β Fin β π½ β (TopOnβ(β βm πΌ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxtopfi.1 | . . 3 β’ π½ = (TopOpenβ(β^βπΌ)) | |
2 | 1 | rrxtopon 45304 | . 2 β’ (πΌ β Fin β π½ β (TopOnβ(Baseβ(β^βπΌ)))) |
3 | id 22 | . . . 4 β’ (πΌ β Fin β πΌ β Fin) | |
4 | eqid 2731 | . . . 4 β’ (β^βπΌ) = (β^βπΌ) | |
5 | eqid 2731 | . . . 4 β’ (Baseβ(β^βπΌ)) = (Baseβ(β^βπΌ)) | |
6 | 3, 4, 5 | rrxbasefi 25159 | . . 3 β’ (πΌ β Fin β (Baseβ(β^βπΌ)) = (β βm πΌ)) |
7 | 6 | fveq2d 6896 | . 2 β’ (πΌ β Fin β (TopOnβ(Baseβ(β^βπΌ))) = (TopOnβ(β βm πΌ))) |
8 | 2, 7 | eleqtrd 2834 | 1 β’ (πΌ β Fin β π½ β (TopOnβ(β βm πΌ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 βcfv 6544 (class class class)co 7412 βm cmap 8823 Fincfn 8942 βcr 11112 Basecbs 17149 TopOpenctopn 17372 TopOnctopon 22633 β^crrx 25132 |
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-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 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 ax-addf 11192 ax-mulf 11193 |
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-tp 4634 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-se 5633 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-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-tpos 8214 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-fsupp 9365 df-sup 9440 df-inf 9441 df-oi 9508 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-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ico 13335 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-prds 17398 df-pws 17400 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-rhm 20364 df-subrng 20435 df-subrg 20460 df-drng 20503 df-field 20504 df-abv 20569 df-staf 20597 df-srng 20598 df-lmod 20617 df-lss 20688 df-lmhm 20778 df-lvec 20859 df-sra 20931 df-rgmod 20932 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-cnfld 21146 df-refld 21378 df-phl 21399 df-dsmm 21507 df-frlm 21522 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-xms 24047 df-ms 24048 df-nm 24312 df-ngp 24313 df-tng 24314 df-nrg 24315 df-nlm 24316 df-clm 24811 df-cph 24917 df-tcph 24918 df-rrx 25134 |
This theorem is referenced by: rrxunitopnfi 45308 rrxtopn0 45309 opnvonmbllem2 45649 |
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