Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfacbasgrp | Structured version Visualization version GIF version |
Description: A choice equivalent in abstract algebra: All nonempty sets admit a group structure. From http://mathoverflow.net/a/12988. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
Ref | Expression |
---|---|
dfacbasgrp | ⊢ (CHOICE ↔ (Base “ Grp) = (V ∖ {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfac10 9551 | . 2 ⊢ (CHOICE ↔ dom card = V) | |
2 | basfn 16491 | . . . . . . . . . 10 ⊢ Base Fn V | |
3 | ssv 3988 | . . . . . . . . . 10 ⊢ Grp ⊆ V | |
4 | fvelimab 6730 | . . . . . . . . . 10 ⊢ ((Base Fn V ∧ Grp ⊆ V) → (𝑥 ∈ (Base “ Grp) ↔ ∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥)) | |
5 | 2, 3, 4 | mp2an 688 | . . . . . . . . 9 ⊢ (𝑥 ∈ (Base “ Grp) ↔ ∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥) |
6 | eqid 2818 | . . . . . . . . . . . 12 ⊢ (Base‘𝑦) = (Base‘𝑦) | |
7 | 6 | grpbn0 18070 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ Grp → (Base‘𝑦) ≠ ∅) |
8 | neeq1 3075 | . . . . . . . . . . 11 ⊢ ((Base‘𝑦) = 𝑥 → ((Base‘𝑦) ≠ ∅ ↔ 𝑥 ≠ ∅)) | |
9 | 7, 8 | syl5ibcom 246 | . . . . . . . . . 10 ⊢ (𝑦 ∈ Grp → ((Base‘𝑦) = 𝑥 → 𝑥 ≠ ∅)) |
10 | 9 | rexlimiv 3277 | . . . . . . . . 9 ⊢ (∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥 → 𝑥 ≠ ∅) |
11 | 5, 10 | sylbi 218 | . . . . . . . 8 ⊢ (𝑥 ∈ (Base “ Grp) → 𝑥 ≠ ∅) |
12 | 11 | adantl 482 | . . . . . . 7 ⊢ ((dom card = V ∧ 𝑥 ∈ (Base “ Grp)) → 𝑥 ≠ ∅) |
13 | vex 3495 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
14 | 12, 13 | jctil 520 | . . . . . 6 ⊢ ((dom card = V ∧ 𝑥 ∈ (Base “ Grp)) → (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) |
15 | ablgrp 18840 | . . . . . . . . 9 ⊢ (𝑥 ∈ Abel → 𝑥 ∈ Grp) | |
16 | 15 | ssriv 3968 | . . . . . . . 8 ⊢ Abel ⊆ Grp |
17 | imass2 5958 | . . . . . . . 8 ⊢ (Abel ⊆ Grp → (Base “ Abel) ⊆ (Base “ Grp)) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (Base “ Abel) ⊆ (Base “ Grp) |
19 | simprl 767 | . . . . . . . . 9 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ V) | |
20 | simpl 483 | . . . . . . . . 9 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → dom card = V) | |
21 | 19, 20 | eleqtrrd 2913 | . . . . . . . 8 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ dom card) |
22 | simprr 769 | . . . . . . . 8 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ≠ ∅) | |
23 | isnumbasgrplem3 39583 | . . . . . . . 8 ⊢ ((𝑥 ∈ dom card ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (Base “ Abel)) | |
24 | 21, 22, 23 | syl2anc 584 | . . . . . . 7 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (Base “ Abel)) |
25 | 18, 24 | sseldi 3962 | . . . . . 6 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (Base “ Grp)) |
26 | 14, 25 | impbida 797 | . . . . 5 ⊢ (dom card = V → (𝑥 ∈ (Base “ Grp) ↔ (𝑥 ∈ V ∧ 𝑥 ≠ ∅))) |
27 | eldifsn 4711 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ {∅}) ↔ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) | |
28 | 26, 27 | syl6bbr 290 | . . . 4 ⊢ (dom card = V → (𝑥 ∈ (Base “ Grp) ↔ 𝑥 ∈ (V ∖ {∅}))) |
29 | 28 | eqrdv 2816 | . . 3 ⊢ (dom card = V → (Base “ Grp) = (V ∖ {∅})) |
30 | fvex 6676 | . . . . . . . . . 10 ⊢ (har‘𝑥) ∈ V | |
31 | 13, 30 | unex 7458 | . . . . . . . . 9 ⊢ (𝑥 ∪ (har‘𝑥)) ∈ V |
32 | ssun2 4146 | . . . . . . . . . 10 ⊢ (har‘𝑥) ⊆ (𝑥 ∪ (har‘𝑥)) | |
33 | harn0 39580 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ V → (har‘𝑥) ≠ ∅) | |
34 | 13, 33 | ax-mp 5 | . . . . . . . . . 10 ⊢ (har‘𝑥) ≠ ∅ |
35 | ssn0 4351 | . . . . . . . . . 10 ⊢ (((har‘𝑥) ⊆ (𝑥 ∪ (har‘𝑥)) ∧ (har‘𝑥) ≠ ∅) → (𝑥 ∪ (har‘𝑥)) ≠ ∅) | |
36 | 32, 34, 35 | mp2an 688 | . . . . . . . . 9 ⊢ (𝑥 ∪ (har‘𝑥)) ≠ ∅ |
37 | eldifsn 4711 | . . . . . . . . 9 ⊢ ((𝑥 ∪ (har‘𝑥)) ∈ (V ∖ {∅}) ↔ ((𝑥 ∪ (har‘𝑥)) ∈ V ∧ (𝑥 ∪ (har‘𝑥)) ≠ ∅)) | |
38 | 31, 36, 37 | mpbir2an 707 | . . . . . . . 8 ⊢ (𝑥 ∪ (har‘𝑥)) ∈ (V ∖ {∅}) |
39 | 38 | a1i 11 | . . . . . . 7 ⊢ ((Base “ Grp) = (V ∖ {∅}) → (𝑥 ∪ (har‘𝑥)) ∈ (V ∖ {∅})) |
40 | id 22 | . . . . . . 7 ⊢ ((Base “ Grp) = (V ∖ {∅}) → (Base “ Grp) = (V ∖ {∅})) | |
41 | 39, 40 | eleqtrrd 2913 | . . . . . 6 ⊢ ((Base “ Grp) = (V ∖ {∅}) → (𝑥 ∪ (har‘𝑥)) ∈ (Base “ Grp)) |
42 | isnumbasgrp 39585 | . . . . . 6 ⊢ (𝑥 ∈ dom card ↔ (𝑥 ∪ (har‘𝑥)) ∈ (Base “ Grp)) | |
43 | 41, 42 | sylibr 235 | . . . . 5 ⊢ ((Base “ Grp) = (V ∖ {∅}) → 𝑥 ∈ dom card) |
44 | 13 | a1i 11 | . . . . 5 ⊢ ((Base “ Grp) = (V ∖ {∅}) → 𝑥 ∈ V) |
45 | 43, 44 | 2thd 266 | . . . 4 ⊢ ((Base “ Grp) = (V ∖ {∅}) → (𝑥 ∈ dom card ↔ 𝑥 ∈ V)) |
46 | 45 | eqrdv 2816 | . . 3 ⊢ ((Base “ Grp) = (V ∖ {∅}) → dom card = V) |
47 | 29, 46 | impbii 210 | . 2 ⊢ (dom card = V ↔ (Base “ Grp) = (V ∖ {∅})) |
48 | 1, 47 | bitri 276 | 1 ⊢ (CHOICE ↔ (Base “ Grp) = (V ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∃wrex 3136 Vcvv 3492 ∖ cdif 3930 ∪ cun 3931 ⊆ wss 3933 ∅c0 4288 {csn 4557 dom cdm 5548 “ cima 5551 Fn wfn 6343 ‘cfv 6348 harchar 9008 cardccrd 9352 CHOICEwac 9529 Basecbs 16471 Grpcgrp 18041 Abelcabl 18836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-seqom 8073 df-1o 8091 df-2o 8092 df-oadd 8095 df-omul 8096 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-inf 8895 df-oi 8962 df-har 9010 df-wdom 9011 df-dju 9318 df-card 9356 df-acn 9359 df-ac 9530 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-hash 13679 df-dvds 15596 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-0g 16703 df-prds 16709 df-pws 16711 df-imas 16769 df-qus 16770 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-nsg 18215 df-eqg 18216 df-ghm 18294 df-gim 18337 df-gic 18338 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-rnghom 19396 df-subrg 19462 df-lmod 19565 df-lss 19633 df-lsp 19673 df-sra 19873 df-rgmod 19874 df-lidl 19875 df-rsp 19876 df-2idl 19933 df-cnfld 20474 df-zring 20546 df-zrh 20579 df-zn 20582 df-dsmm 20804 df-frlm 20819 |
This theorem is referenced by: (None) |
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