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Mirrors > Home > MPE Home > Th. List > cardfz | Structured version Visualization version GIF version |
Description: The cardinality of a finite set of sequential integers. (See om2uz0i 13861 for a description of the hypothesis.) (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
fzennn.1 | β’ πΊ = (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) |
Ref | Expression |
---|---|
cardfz | β’ (π β β0 β (cardβ(1...π)) = (β‘πΊβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzennn.1 | . . . 4 β’ πΊ = (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) | |
2 | 1 | fzennn 13882 | . . 3 β’ (π β β0 β (1...π) β (β‘πΊβπ)) |
3 | carden2b 9911 | . . 3 β’ ((1...π) β (β‘πΊβπ) β (cardβ(1...π)) = (cardβ(β‘πΊβπ))) | |
4 | 2, 3 | syl 17 | . 2 β’ (π β β0 β (cardβ(1...π)) = (cardβ(β‘πΊβπ))) |
5 | 0z 12518 | . . . . 5 β’ 0 β β€ | |
6 | 5, 1 | om2uzf1oi 13867 | . . . 4 β’ πΊ:Οβ1-1-ontoβ(β€β₯β0) |
7 | elnn0uz 12816 | . . . . 5 β’ (π β β0 β π β (β€β₯β0)) | |
8 | 7 | biimpi 215 | . . . 4 β’ (π β β0 β π β (β€β₯β0)) |
9 | f1ocnvdm 7235 | . . . 4 β’ ((πΊ:Οβ1-1-ontoβ(β€β₯β0) β§ π β (β€β₯β0)) β (β‘πΊβπ) β Ο) | |
10 | 6, 8, 9 | sylancr 588 | . . 3 β’ (π β β0 β (β‘πΊβπ) β Ο) |
11 | cardnn 9907 | . . 3 β’ ((β‘πΊβπ) β Ο β (cardβ(β‘πΊβπ)) = (β‘πΊβπ)) | |
12 | 10, 11 | syl 17 | . 2 β’ (π β β0 β (cardβ(β‘πΊβπ)) = (β‘πΊβπ)) |
13 | 4, 12 | eqtrd 2773 | 1 β’ (π β β0 β (cardβ(1...π)) = (β‘πΊβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3447 class class class wbr 5109 β¦ cmpt 5192 β‘ccnv 5636 βΎ cres 5639 β1-1-ontoβwf1o 6499 βcfv 6500 (class class class)co 7361 Οcom 7806 reccrdg 8359 β cen 8886 cardccrd 9879 0cc0 11059 1c1 11060 + caddc 11062 β0cn0 12421 β€β₯cuz 12771 ...cfz 13433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 |
This theorem is referenced by: hashfz1 14255 |
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