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| Mirrors > Home > MPE Home > Th. List > rrxbasefi | Structured version Visualization version GIF version | ||
| Description: The base of the generalized real Euclidean space, when the dimension of the space is finite. This justifies the use of (β βm π) for the development of the Lebesgue measure theory for n-dimensional real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
| Ref | Expression |
|---|---|
| rrxbasefi.x | β’ (π β π β Fin) |
| rrxbasefi.h | β’ π» = (β^βπ) |
| rrxbasefi.b | β’ π΅ = (Baseβπ») |
| Ref | Expression |
|---|---|
| rrxbasefi | β’ (π β π΅ = (β βm π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rrxbasefi.x | . . . 4 β’ (π β π β Fin) | |
| 2 | rrxbasefi.h | . . . . 5 β’ π» = (β^βπ) | |
| 3 | rrxbasefi.b | . . . . 5 β’ π΅ = (Baseβπ») | |
| 4 | 2, 3 | rrxbase 24896 | . . . 4 β’ (π β Fin β π΅ = {π β (β βm π) β£ π finSupp 0}) |
| 5 | 1, 4 | syl 17 | . . 3 β’ (π β π΅ = {π β (β βm π) β£ π finSupp 0}) |
| 6 | ssrab2 4076 | . . 3 β’ {π β (β βm π) β£ π finSupp 0} β (β βm π) | |
| 7 | 5, 6 | eqsstrdi 4035 | . 2 β’ (π β π΅ β (β βm π)) |
| 8 | simpr 485 | . . . 4 β’ ((π β§ π β (β βm π)) β π β (β βm π)) | |
| 9 | elmapi 8839 | . . . . . 6 β’ (π β (β βm π) β π:πβΆβ) | |
| 10 | 9 | adantl 482 | . . . . 5 β’ ((π β§ π β (β βm π)) β π:πβΆβ) |
| 11 | 1 | adantr 481 | . . . . 5 β’ ((π β§ π β (β βm π)) β π β Fin) |
| 12 | c0ex 11204 | . . . . . 6 β’ 0 β V | |
| 13 | 12 | a1i 11 | . . . . 5 β’ ((π β§ π β (β βm π)) β 0 β V) |
| 14 | 10, 11, 13 | fdmfifsupp 9369 | . . . 4 β’ ((π β§ π β (β βm π)) β π finSupp 0) |
| 15 | rabid 3452 | . . . 4 β’ (π β {π β (β βm π) β£ π finSupp 0} β (π β (β βm π) β§ π finSupp 0)) | |
| 16 | 8, 14, 15 | sylanbrc 583 | . . 3 β’ ((π β§ π β (β βm π)) β π β {π β (β βm π) β£ π finSupp 0}) |
| 17 | 5 | eqcomd 2738 | . . . 4 β’ (π β {π β (β βm π) β£ π finSupp 0} = π΅) |
| 18 | 17 | adantr 481 | . . 3 β’ ((π β§ π β (β βm π)) β {π β (β βm π) β£ π finSupp 0} = π΅) |
| 19 | 16, 18 | eleqtrd 2835 | . 2 β’ ((π β§ π β (β βm π)) β π β π΅) |
| 20 | 7, 19 | eqelssd 4002 | 1 β’ (π β π΅ = (β βm π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 {crab 3432 Vcvv 3474 class class class wbr 5147 βΆwf 6536 βcfv 6540 (class class class)co 7405 βm cmap 8816 Fincfn 8935 finSupp cfsupp 9357 βcr 11105 0cc0 11106 Basecbs 17140 β^crrx 24891 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-0g 17383 df-prds 17389 df-pws 17391 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-subg 18997 df-cmn 19644 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-drng 20309 df-field 20310 df-subrg 20353 df-sra 20777 df-rgmod 20778 df-cnfld 20937 df-refld 21149 df-dsmm 21278 df-frlm 21293 df-tng 24084 df-tcph 24677 df-rrx 24893 |
| This theorem is referenced by: rrxdsfi 24919 rrxmetfi 24920 rrxtopnfi 44989 rrxtoponfi 44993 qndenserrnopnlem 44999 qndenserrn 45001 rrnprjdstle 45003 rrxlines 47372 rrxlinesc 47374 rrxlinec 47375 rrxsphere 47387 |
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