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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 9548 | . 2 ⊢ (CHOICE ↔ dom card = V) | |
2 | basfn 16495 | . . . . . . . . . 10 ⊢ Base Fn V | |
3 | ssv 3939 | . . . . . . . . . 10 ⊢ Grp ⊆ V | |
4 | fvelimab 6712 | . . . . . . . . . 10 ⊢ ((Base Fn V ∧ Grp ⊆ V) → (𝑥 ∈ (Base “ Grp) ↔ ∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥)) | |
5 | 2, 3, 4 | mp2an 691 | . . . . . . . . 9 ⊢ (𝑥 ∈ (Base “ Grp) ↔ ∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥) |
6 | eqid 2798 | . . . . . . . . . . . 12 ⊢ (Base‘𝑦) = (Base‘𝑦) | |
7 | 6 | grpbn0 18124 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ Grp → (Base‘𝑦) ≠ ∅) |
8 | neeq1 3049 | . . . . . . . . . . 11 ⊢ ((Base‘𝑦) = 𝑥 → ((Base‘𝑦) ≠ ∅ ↔ 𝑥 ≠ ∅)) | |
9 | 7, 8 | syl5ibcom 248 | . . . . . . . . . 10 ⊢ (𝑦 ∈ Grp → ((Base‘𝑦) = 𝑥 → 𝑥 ≠ ∅)) |
10 | 9 | rexlimiv 3239 | . . . . . . . . 9 ⊢ (∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥 → 𝑥 ≠ ∅) |
11 | 5, 10 | sylbi 220 | . . . . . . . 8 ⊢ (𝑥 ∈ (Base “ Grp) → 𝑥 ≠ ∅) |
12 | 11 | adantl 485 | . . . . . . 7 ⊢ ((dom card = V ∧ 𝑥 ∈ (Base “ Grp)) → 𝑥 ≠ ∅) |
13 | vex 3444 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
14 | 12, 13 | jctil 523 | . . . . . 6 ⊢ ((dom card = V ∧ 𝑥 ∈ (Base “ Grp)) → (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) |
15 | ablgrp 18903 | . . . . . . . . 9 ⊢ (𝑥 ∈ Abel → 𝑥 ∈ Grp) | |
16 | 15 | ssriv 3919 | . . . . . . . 8 ⊢ Abel ⊆ Grp |
17 | imass2 5932 | . . . . . . . 8 ⊢ (Abel ⊆ Grp → (Base “ Abel) ⊆ (Base “ Grp)) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (Base “ Abel) ⊆ (Base “ Grp) |
19 | simprl 770 | . . . . . . . . 9 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ V) | |
20 | simpl 486 | . . . . . . . . 9 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → dom card = V) | |
21 | 19, 20 | eleqtrrd 2893 | . . . . . . . 8 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ dom card) |
22 | simprr 772 | . . . . . . . 8 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ≠ ∅) | |
23 | isnumbasgrplem3 40049 | . . . . . . . 8 ⊢ ((𝑥 ∈ dom card ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (Base “ Abel)) | |
24 | 21, 22, 23 | syl2anc 587 | . . . . . . 7 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (Base “ Abel)) |
25 | 18, 24 | sseldi 3913 | . . . . . 6 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (Base “ Grp)) |
26 | 14, 25 | impbida 800 | . . . . 5 ⊢ (dom card = V → (𝑥 ∈ (Base “ Grp) ↔ (𝑥 ∈ V ∧ 𝑥 ≠ ∅))) |
27 | eldifsn 4680 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ {∅}) ↔ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) | |
28 | 26, 27 | syl6bbr 292 | . . . 4 ⊢ (dom card = V → (𝑥 ∈ (Base “ Grp) ↔ 𝑥 ∈ (V ∖ {∅}))) |
29 | 28 | eqrdv 2796 | . . 3 ⊢ (dom card = V → (Base “ Grp) = (V ∖ {∅})) |
30 | fvex 6658 | . . . . . . . . . 10 ⊢ (har‘𝑥) ∈ V | |
31 | 13, 30 | unex 7449 | . . . . . . . . 9 ⊢ (𝑥 ∪ (har‘𝑥)) ∈ V |
32 | ssun2 4100 | . . . . . . . . . 10 ⊢ (har‘𝑥) ⊆ (𝑥 ∪ (har‘𝑥)) | |
33 | harn0 40046 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ V → (har‘𝑥) ≠ ∅) | |
34 | 13, 33 | ax-mp 5 | . . . . . . . . . 10 ⊢ (har‘𝑥) ≠ ∅ |
35 | ssn0 4308 | . . . . . . . . . 10 ⊢ (((har‘𝑥) ⊆ (𝑥 ∪ (har‘𝑥)) ∧ (har‘𝑥) ≠ ∅) → (𝑥 ∪ (har‘𝑥)) ≠ ∅) | |
36 | 32, 34, 35 | mp2an 691 | . . . . . . . . 9 ⊢ (𝑥 ∪ (har‘𝑥)) ≠ ∅ |
37 | eldifsn 4680 | . . . . . . . . 9 ⊢ ((𝑥 ∪ (har‘𝑥)) ∈ (V ∖ {∅}) ↔ ((𝑥 ∪ (har‘𝑥)) ∈ V ∧ (𝑥 ∪ (har‘𝑥)) ≠ ∅)) | |
38 | 31, 36, 37 | mpbir2an 710 | . . . . . . . 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 2893 | . . . . . 6 ⊢ ((Base “ Grp) = (V ∖ {∅}) → (𝑥 ∪ (har‘𝑥)) ∈ (Base “ Grp)) |
42 | isnumbasgrp 40051 | . . . . . 6 ⊢ (𝑥 ∈ dom card ↔ (𝑥 ∪ (har‘𝑥)) ∈ (Base “ Grp)) | |
43 | 41, 42 | sylibr 237 | . . . . 5 ⊢ ((Base “ Grp) = (V ∖ {∅}) → 𝑥 ∈ dom card) |
44 | 13 | a1i 11 | . . . . 5 ⊢ ((Base “ Grp) = (V ∖ {∅}) → 𝑥 ∈ V) |
45 | 43, 44 | 2thd 268 | . . . 4 ⊢ ((Base “ Grp) = (V ∖ {∅}) → (𝑥 ∈ dom card ↔ 𝑥 ∈ V)) |
46 | 45 | eqrdv 2796 | . . 3 ⊢ ((Base “ Grp) = (V ∖ {∅}) → dom card = V) |
47 | 29, 46 | impbii 212 | . 2 ⊢ (dom card = V ↔ (Base “ Grp) = (V ∖ {∅})) |
48 | 1, 47 | bitri 278 | 1 ⊢ (CHOICE ↔ (Base “ Grp) = (V ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ⊆ wss 3881 ∅c0 4243 {csn 4525 dom cdm 5519 “ cima 5522 Fn wfn 6319 ‘cfv 6324 harchar 9004 cardccrd 9348 CHOICEwac 9526 Basecbs 16475 Grpcgrp 18095 Abelcabl 18899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-seqom 8067 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-inf 8891 df-oi 8958 df-har 9005 df-wdom 9013 df-dju 9314 df-card 9352 df-acn 9355 df-ac 9527 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-hash 13687 df-dvds 15600 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-prds 16713 df-pws 16715 df-imas 16773 df-qus 16774 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-nsg 18269 df-eqg 18270 df-ghm 18348 df-gim 18391 df-gic 18392 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-rnghom 19463 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-sra 19937 df-rgmod 19938 df-lidl 19939 df-rsp 19940 df-2idl 19998 df-cnfld 20092 df-zring 20164 df-zrh 20197 df-zn 20200 df-dsmm 20421 df-frlm 20436 |
This theorem is referenced by: (None) |
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