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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 9027 | . . . . 5 ⊢ (har‘𝑆) ∈ On | |
2 | onenon 9380 | . . . . 5 ⊢ ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (har‘𝑆) ∈ dom card |
4 | unnum 9624 | . . . 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 4151 | . . . 4 ⊢ (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆)) | |
7 | harn0 39709 | . . . 4 ⊢ (𝑆 ∈ dom card → (har‘𝑆) ≠ ∅) | |
8 | ssn0 4356 | . . . 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 39712 | . . 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 18913 | . . . . . 6 ⊢ (𝑥 ∈ Abel → 𝑥 ∈ Grp) | |
13 | 12 | ssriv 3973 | . . . . 5 ⊢ Abel ⊆ Grp |
14 | imass2 5967 | . . . . 5 ⊢ (Abel ⊆ Grp → (Base “ Abel) ⊆ (Base “ Grp)) | |
15 | 13, 14 | ax-mp 5 | . . . 4 ⊢ (Base “ Abel) ⊆ (Base “ Grp) |
16 | 15 | sseli 3965 | . . 3 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel) → (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp)) |
17 | isnumbasgrplem2 39711 | . . 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 3018 ∪ cun 3936 ⊆ wss 3938 ∅c0 4293 dom cdm 5557 “ cima 5560 Oncon0 6193 ‘cfv 6357 harchar 9022 cardccrd 9366 Basecbs 16485 Grpcgrp 18105 Abelcabl 18909 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-seqom 8086 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-inf 8909 df-oi 8976 df-har 9024 df-wdom 9025 df-dju 9332 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-hash 13694 df-dvds 15610 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-0g 16717 df-prds 16723 df-pws 16725 df-imas 16783 df-qus 16784 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-nsg 18279 df-eqg 18280 df-ghm 18358 df-gim 18401 df-gic 18402 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-rnghom 19469 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-sra 19946 df-rgmod 19947 df-lidl 19948 df-rsp 19949 df-2idl 20007 df-cnfld 20548 df-zring 20620 df-zrh 20653 df-zn 20656 df-dsmm 20878 df-frlm 20893 |
This theorem is referenced by: isnumbasgrp 39714 |
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