![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > isnumbasabl | Structured version Visualization version GIF version |
Description: A set is numerable iff it and its Hartogs number can be jointly given the structure of an Abelian group. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
Ref | Expression |
---|---|
isnumbasabl | β’ (π β dom card β (π βͺ (harβπ)) β (Base β Abel)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | harcl 9502 | . . . . 5 β’ (harβπ) β On | |
2 | onenon 9892 | . . . . 5 β’ ((harβπ) β On β (harβπ) β dom card) | |
3 | 1, 2 | ax-mp 5 | . . . 4 β’ (harβπ) β dom card |
4 | unnum 10139 | . . . 4 β’ ((π β dom card β§ (harβπ) β dom card) β (π βͺ (harβπ)) β dom card) | |
5 | 3, 4 | mpan2 690 | . . 3 β’ (π β dom card β (π βͺ (harβπ)) β dom card) |
6 | ssun2 4138 | . . . 4 β’ (harβπ) β (π βͺ (harβπ)) | |
7 | harn0 41458 | . . . 4 β’ (π β dom card β (harβπ) β β ) | |
8 | ssn0 4365 | . . . 4 β’ (((harβπ) β (π βͺ (harβπ)) β§ (harβπ) β β ) β (π βͺ (harβπ)) β β ) | |
9 | 6, 7, 8 | sylancr 588 | . . 3 β’ (π β dom card β (π βͺ (harβπ)) β β ) |
10 | isnumbasgrplem3 41461 | . . 3 β’ (((π βͺ (harβπ)) β dom card β§ (π βͺ (harβπ)) β β ) β (π βͺ (harβπ)) β (Base β Abel)) | |
11 | 5, 9, 10 | syl2anc 585 | . 2 β’ (π β dom card β (π βͺ (harβπ)) β (Base β Abel)) |
12 | ablgrp 19574 | . . . . . 6 β’ (π₯ β Abel β π₯ β Grp) | |
13 | 12 | ssriv 3953 | . . . . 5 β’ Abel β Grp |
14 | imass2 6059 | . . . . 5 β’ (Abel β Grp β (Base β Abel) β (Base β Grp)) | |
15 | 13, 14 | ax-mp 5 | . . . 4 β’ (Base β Abel) β (Base β Grp) |
16 | 15 | sseli 3945 | . . 3 β’ ((π βͺ (harβπ)) β (Base β Abel) β (π βͺ (harβπ)) β (Base β Grp)) |
17 | isnumbasgrplem2 41460 | . . 3 β’ ((π βͺ (harβπ)) β (Base β Grp) β π β dom card) | |
18 | 16, 17 | syl 17 | . 2 β’ ((π βͺ (harβπ)) β (Base β Abel) β π β dom card) |
19 | 11, 18 | impbii 208 | 1 β’ (π β dom card β (π βͺ (harβπ)) β (Base β Abel)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β wcel 2107 β wne 2944 βͺ cun 3913 β wss 3915 β c0 4287 dom cdm 5638 β cima 5641 Oncon0 6322 βcfv 6501 harchar 9499 cardccrd 9878 Basecbs 17090 Grpcgrp 18755 Abelcabl 19570 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-seqom 8399 df-1o 8417 df-2o 8418 df-oadd 8421 df-omul 8422 df-er 8655 df-ec 8657 df-qs 8661 df-map 8774 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-sup 9385 df-inf 9386 df-oi 9453 df-har 9500 df-wdom 9508 df-dju 9844 df-card 9882 df-acn 9885 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-rp 12923 df-fz 13432 df-fzo 13575 df-fl 13704 df-mod 13782 df-seq 13914 df-hash 14238 df-dvds 16144 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-hom 17164 df-cco 17165 df-0g 17330 df-prds 17336 df-pws 17338 df-imas 17397 df-qus 17398 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-grp 18758 df-minusg 18759 df-sbg 18760 df-mulg 18880 df-subg 18932 df-nsg 18933 df-eqg 18934 df-ghm 19013 df-gim 19056 df-gic 19057 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-cring 19974 df-oppr 20056 df-dvdsr 20077 df-rnghom 20155 df-subrg 20236 df-lmod 20340 df-lss 20409 df-lsp 20449 df-sra 20649 df-rgmod 20650 df-lidl 20651 df-rsp 20652 df-2idl 20718 df-cnfld 20813 df-zring 20886 df-zrh 20920 df-zn 20923 df-dsmm 21154 df-frlm 21169 |
This theorem is referenced by: isnumbasgrp 41463 |
Copyright terms: Public domain | W3C validator |