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Mathbox for Stefan O'Rear |
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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 9568 | . . . . 5 β’ (harβπ) β On | |
2 | onenon 9958 | . . . . 5 β’ ((harβπ) β On β (harβπ) β dom card) | |
3 | 1, 2 | ax-mp 5 | . . . 4 β’ (harβπ) β dom card |
4 | unnum 10205 | . . . 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 4169 | . . . 4 β’ (harβπ) β (π βͺ (harβπ)) | |
7 | harn0 42438 | . . . 4 β’ (π β dom card β (harβπ) β β ) | |
8 | ssn0 4396 | . . . 4 β’ (((harβπ) β (π βͺ (harβπ)) β§ (harβπ) β β ) β (π βͺ (harβπ)) β β ) | |
9 | 6, 7, 8 | sylancr 586 | . . 3 β’ (π β dom card β (π βͺ (harβπ)) β β ) |
10 | isnumbasgrplem3 42441 | . . 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 583 | . 2 β’ (π β dom card β (π βͺ (harβπ)) β (Base β Abel)) |
12 | ablgrp 19724 | . . . . . 6 β’ (π₯ β Abel β π₯ β Grp) | |
13 | 12 | ssriv 3982 | . . . . 5 β’ Abel β Grp |
14 | imass2 6100 | . . . . 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 3974 | . . 3 β’ ((π βͺ (harβπ)) β (Base β Abel) β (π βͺ (harβπ)) β (Base β Grp)) |
17 | isnumbasgrplem2 42440 | . . 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 2099 β wne 2935 βͺ cun 3942 β wss 3944 β c0 4318 dom cdm 5672 β cima 5675 Oncon0 6363 βcfv 6542 harchar 9565 cardccrd 9944 Basecbs 17165 Grpcgrp 18875 Abelcabl 19720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203 ax-mulf 11204 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-seqom 8460 df-1o 8478 df-2o 8479 df-oadd 8482 df-omul 8483 df-er 8716 df-ec 8718 df-qs 8722 df-map 8836 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-sup 9451 df-inf 9452 df-oi 9519 df-har 9566 df-wdom 9574 df-dju 9910 df-card 9948 df-acn 9951 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-rp 12993 df-fz 13503 df-fzo 13646 df-fl 13775 df-mod 13853 df-seq 13985 df-hash 14308 df-dvds 16217 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-0g 17408 df-prds 17414 df-pws 17416 df-imas 17475 df-qus 17476 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19008 df-subg 19062 df-nsg 19063 df-eqg 19064 df-ghm 19152 df-gim 19197 df-gic 19198 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-cring 20160 df-oppr 20255 df-dvdsr 20278 df-rhm 20393 df-subrng 20465 df-subrg 20490 df-lmod 20727 df-lss 20798 df-lsp 20838 df-sra 21040 df-rgmod 21041 df-lidl 21086 df-rsp 21087 df-2idl 21126 df-cnfld 21260 df-zring 21353 df-zrh 21409 df-zn 21412 df-dsmm 21646 df-frlm 21661 |
This theorem is referenced by: isnumbasgrp 42443 |
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