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 | ⊢ (𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | harcl 9025 | . . . . 5 ⊢ (har‘𝑆) ∈ On | |
2 | onenon 9378 | . . . . 5 ⊢ ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (har‘𝑆) ∈ dom card |
4 | unnum 9622 | . . . 4 ⊢ ((𝑆 ∈ dom card ∧ (har‘𝑆) ∈ dom card) → (𝑆 ∪ (har‘𝑆)) ∈ dom card) | |
5 | 3, 4 | mpan2 689 | . . 3 ⊢ (𝑆 ∈ dom card → (𝑆 ∪ (har‘𝑆)) ∈ dom card) |
6 | ssun2 4149 | . . . 4 ⊢ (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆)) | |
7 | harn0 39722 | . . . 4 ⊢ (𝑆 ∈ dom card → (har‘𝑆) ≠ ∅) | |
8 | ssn0 4354 | . . . 4 ⊢ (((har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆)) ∧ (har‘𝑆) ≠ ∅) → (𝑆 ∪ (har‘𝑆)) ≠ ∅) | |
9 | 6, 7, 8 | sylancr 589 | . . 3 ⊢ (𝑆 ∈ dom card → (𝑆 ∪ (har‘𝑆)) ≠ ∅) |
10 | isnumbasgrplem3 39725 | . . 3 ⊢ (((𝑆 ∪ (har‘𝑆)) ∈ dom card ∧ (𝑆 ∪ (har‘𝑆)) ≠ ∅) → (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) | |
11 | 5, 9, 10 | syl2anc 586 | . 2 ⊢ (𝑆 ∈ dom card → (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) |
12 | ablgrp 18911 | . . . . . 6 ⊢ (𝑥 ∈ Abel → 𝑥 ∈ Grp) | |
13 | 12 | ssriv 3971 | . . . . 5 ⊢ Abel ⊆ Grp |
14 | imass2 5965 | . . . . 5 ⊢ (Abel ⊆ Grp → (Base “ Abel) ⊆ (Base “ Grp)) | |
15 | 13, 14 | ax-mp 5 | . . . 4 ⊢ (Base “ Abel) ⊆ (Base “ Grp) |
16 | 15 | sseli 3963 | . . 3 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel) → (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp)) |
17 | isnumbasgrplem2 39724 | . . 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 211 | 1 ⊢ (𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 ≠ wne 3016 ∪ cun 3934 ⊆ wss 3936 ∅c0 4291 dom cdm 5555 “ cima 5558 Oncon0 6191 ‘cfv 6355 harchar 9020 cardccrd 9364 Basecbs 16483 Grpcgrp 18103 Abelcabl 18907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-seqom 8084 df-1o 8102 df-2o 8103 df-oadd 8106 df-omul 8107 df-er 8289 df-ec 8291 df-qs 8295 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-inf 8907 df-oi 8974 df-har 9022 df-wdom 9023 df-dju 9330 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-hash 13692 df-dvds 15608 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-0g 16715 df-prds 16721 df-pws 16723 df-imas 16781 df-qus 16782 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-nsg 18277 df-eqg 18278 df-ghm 18356 df-gim 18399 df-gic 18400 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-rnghom 19467 df-subrg 19533 df-lmod 19636 df-lss 19704 df-lsp 19744 df-sra 19944 df-rgmod 19945 df-lidl 19946 df-rsp 19947 df-2idl 20005 df-cnfld 20546 df-zring 20618 df-zrh 20651 df-zn 20654 df-dsmm 20876 df-frlm 20891 |
This theorem is referenced by: isnumbasgrp 39727 |
Copyright terms: Public domain | W3C validator |