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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 | ⊢ (𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | harcl 8736 | . . . . 5 ⊢ (har‘𝑆) ∈ On | |
2 | onenon 9089 | . . . . 5 ⊢ ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (har‘𝑆) ∈ dom card |
4 | unnum 9338 | . . . 4 ⊢ ((𝑆 ∈ dom card ∧ (har‘𝑆) ∈ dom card) → (𝑆 ∪ (har‘𝑆)) ∈ dom card) | |
5 | 3, 4 | mpan2 684 | . . 3 ⊢ (𝑆 ∈ dom card → (𝑆 ∪ (har‘𝑆)) ∈ dom card) |
6 | ssun2 4005 | . . . 4 ⊢ (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆)) | |
7 | harn0 38516 | . . . 4 ⊢ (𝑆 ∈ dom card → (har‘𝑆) ≠ ∅) | |
8 | ssn0 4202 | . . . 4 ⊢ (((har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆)) ∧ (har‘𝑆) ≠ ∅) → (𝑆 ∪ (har‘𝑆)) ≠ ∅) | |
9 | 6, 7, 8 | sylancr 583 | . . 3 ⊢ (𝑆 ∈ dom card → (𝑆 ∪ (har‘𝑆)) ≠ ∅) |
10 | isnumbasgrplem3 38519 | . . 3 ⊢ (((𝑆 ∪ (har‘𝑆)) ∈ dom card ∧ (𝑆 ∪ (har‘𝑆)) ≠ ∅) → (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) | |
11 | 5, 9, 10 | syl2anc 581 | . 2 ⊢ (𝑆 ∈ dom card → (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) |
12 | ablgrp 18552 | . . . . . 6 ⊢ (𝑥 ∈ Abel → 𝑥 ∈ Grp) | |
13 | 12 | ssriv 3832 | . . . . 5 ⊢ Abel ⊆ Grp |
14 | imass2 5743 | . . . . 5 ⊢ (Abel ⊆ Grp → (Base “ Abel) ⊆ (Base “ Grp)) | |
15 | 13, 14 | ax-mp 5 | . . . 4 ⊢ (Base “ Abel) ⊆ (Base “ Grp) |
16 | 15 | sseli 3824 | . . 3 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel) → (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp)) |
17 | isnumbasgrplem2 38518 | . . 3 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card) | |
18 | 16, 17 | syl 17 | . 2 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel) → 𝑆 ∈ dom card) |
19 | 11, 18 | impbii 201 | 1 ⊢ (𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2166 ≠ wne 3000 ∪ cun 3797 ⊆ wss 3799 ∅c0 4145 dom cdm 5343 “ cima 5346 Oncon0 5964 ‘cfv 6124 harchar 8731 cardccrd 9075 Basecbs 16223 Grpcgrp 17777 Abelcabl 18548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 ax-addf 10332 ax-mulf 10333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-supp 7561 df-tpos 7618 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-seqom 7810 df-1o 7827 df-2o 7828 df-oadd 7831 df-omul 7832 df-er 8010 df-ec 8012 df-qs 8016 df-map 8125 df-ixp 8177 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-fsupp 8546 df-sup 8618 df-inf 8619 df-oi 8685 df-har 8733 df-wdom 8734 df-card 9079 df-acn 9082 df-cda 9306 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-uz 11970 df-rp 12114 df-fz 12621 df-fzo 12762 df-fl 12889 df-mod 12965 df-seq 13097 df-hash 13412 df-dvds 15359 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-starv 16321 df-sca 16322 df-vsca 16323 df-ip 16324 df-tset 16325 df-ple 16326 df-ds 16328 df-unif 16329 df-hom 16330 df-cco 16331 df-0g 16456 df-prds 16462 df-pws 16464 df-imas 16522 df-qus 16523 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-mhm 17689 df-grp 17780 df-minusg 17781 df-sbg 17782 df-mulg 17896 df-subg 17943 df-nsg 17944 df-eqg 17945 df-ghm 18010 df-gim 18053 df-gic 18054 df-cmn 18549 df-abl 18550 df-mgp 18845 df-ur 18857 df-ring 18904 df-cring 18905 df-oppr 18978 df-dvdsr 18996 df-rnghom 19072 df-subrg 19135 df-lmod 19222 df-lss 19290 df-lsp 19332 df-sra 19534 df-rgmod 19535 df-lidl 19536 df-rsp 19537 df-2idl 19594 df-cnfld 20108 df-zring 20180 df-zrh 20213 df-zn 20216 df-dsmm 20440 df-frlm 20455 |
This theorem is referenced by: isnumbasgrp 38521 |
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