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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 25226 | . . . 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 4069 | . . 3 β’ {π β (β βm π) β£ π finSupp 0} β (β βm π) | |
| 7 | 5, 6 | eqsstrdi 4028 | . 2 β’ (π β π΅ β (β βm π)) |
| 8 | simpr 484 | . . . 4 β’ ((π β§ π β (β βm π)) β π β (β βm π)) | |
| 9 | elmapi 8838 | . . . . . 6 β’ (π β (β βm π) β π:πβΆβ) | |
| 10 | 9 | adantl 481 | . . . . 5 β’ ((π β§ π β (β βm π)) β π:πβΆβ) |
| 11 | 1 | adantr 480 | . . . . 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 9368 | . . . 4 β’ ((π β§ π β (β βm π)) β π finSupp 0) |
| 15 | rabid 3444 | . . . 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 582 | . . 3 β’ ((π β§ π β (β βm π)) β π β {π β (β βm π) β£ π finSupp 0}) |
| 17 | 5 | eqcomd 2730 | . . . 4 β’ (π β {π β (β βm π) β£ π finSupp 0} = π΅) |
| 18 | 17 | adantr 480 | . . 3 β’ ((π β§ π β (β βm π)) β {π β (β βm π) β£ π finSupp 0} = π΅) |
| 19 | 16, 18 | eleqtrd 2827 | . 2 β’ ((π β§ π β (β βm π)) β π β π΅) |
| 20 | 7, 19 | eqelssd 3995 | 1 β’ (π β π΅ = (β βm π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {crab 3424 Vcvv 3466 class class class wbr 5138 βΆwf 6529 βcfv 6533 (class class class)co 7401 βm cmap 8815 Fincfn 8934 finSupp cfsupp 9356 βcr 11104 0cc0 11105 Basecbs 17140 β^crrx 25221 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 ax-mulf 11185 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-sup 9432 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 18560 df-sgrp 18639 df-mnd 18655 df-grp 18853 df-minusg 18854 df-subg 19035 df-cmn 19687 df-abl 19688 df-mgp 20025 df-rng 20043 df-ur 20072 df-ring 20125 df-cring 20126 df-oppr 20221 df-dvdsr 20244 df-unit 20245 df-invr 20275 df-dvr 20288 df-subrng 20431 df-subrg 20456 df-drng 20574 df-field 20575 df-sra 21006 df-rgmod 21007 df-cnfld 21224 df-refld 21458 df-dsmm 21587 df-frlm 21602 df-tng 24403 df-tcph 25007 df-rrx 25223 |
| This theorem is referenced by: rrxdsfi 25249 rrxmetfi 25250 rrxtopnfi 45454 rrxtoponfi 45458 qndenserrnopnlem 45464 qndenserrn 45466 rrnprjdstle 45468 rrxlines 47573 rrxlinesc 47575 rrxlinec 47576 rrxsphere 47588 |
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