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 9650 | . 2 ⊢ (CHOICE ↔ dom card = V) | |
2 | basfn 16619 | . . . . . . . . . 10 ⊢ Base Fn V | |
3 | ssv 3911 | . . . . . . . . . 10 ⊢ Grp ⊆ V | |
4 | fvelimab 6754 | . . . . . . . . . 10 ⊢ ((Base Fn V ∧ Grp ⊆ V) → (𝑥 ∈ (Base “ Grp) ↔ ∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥)) | |
5 | 2, 3, 4 | mp2an 692 | . . . . . . . . 9 ⊢ (𝑥 ∈ (Base “ Grp) ↔ ∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥) |
6 | eqid 2739 | . . . . . . . . . . . 12 ⊢ (Base‘𝑦) = (Base‘𝑦) | |
7 | 6 | grpbn0 18263 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ Grp → (Base‘𝑦) ≠ ∅) |
8 | neeq1 2997 | . . . . . . . . . . 11 ⊢ ((Base‘𝑦) = 𝑥 → ((Base‘𝑦) ≠ ∅ ↔ 𝑥 ≠ ∅)) | |
9 | 7, 8 | syl5ibcom 248 | . . . . . . . . . 10 ⊢ (𝑦 ∈ Grp → ((Base‘𝑦) = 𝑥 → 𝑥 ≠ ∅)) |
10 | 9 | rexlimiv 3191 | . . . . . . . . 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 3404 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
14 | 12, 13 | jctil 523 | . . . . . 6 ⊢ ((dom card = V ∧ 𝑥 ∈ (Base “ Grp)) → (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) |
15 | ablgrp 19042 | . . . . . . . . 9 ⊢ (𝑥 ∈ Abel → 𝑥 ∈ Grp) | |
16 | 15 | ssriv 3891 | . . . . . . . 8 ⊢ Abel ⊆ Grp |
17 | imass2 5949 | . . . . . . . 8 ⊢ (Abel ⊆ Grp → (Base “ Abel) ⊆ (Base “ Grp)) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (Base “ Abel) ⊆ (Base “ Grp) |
19 | simprl 771 | . . . . . . . . 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 2837 | . . . . . . . 8 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ dom card) |
22 | simprr 773 | . . . . . . . 8 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ≠ ∅) | |
23 | isnumbasgrplem3 40543 | . . . . . . . 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 3885 | . . . . . 6 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (Base “ Grp)) |
26 | 14, 25 | impbida 801 | . . . . 5 ⊢ (dom card = V → (𝑥 ∈ (Base “ Grp) ↔ (𝑥 ∈ V ∧ 𝑥 ≠ ∅))) |
27 | eldifsn 4685 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ {∅}) ↔ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) | |
28 | 26, 27 | bitr4di 292 | . . . 4 ⊢ (dom card = V → (𝑥 ∈ (Base “ Grp) ↔ 𝑥 ∈ (V ∖ {∅}))) |
29 | 28 | eqrdv 2737 | . . 3 ⊢ (dom card = V → (Base “ Grp) = (V ∖ {∅})) |
30 | fvex 6700 | . . . . . . . . . 10 ⊢ (har‘𝑥) ∈ V | |
31 | 13, 30 | unex 7500 | . . . . . . . . 9 ⊢ (𝑥 ∪ (har‘𝑥)) ∈ V |
32 | ssun2 4073 | . . . . . . . . . 10 ⊢ (har‘𝑥) ⊆ (𝑥 ∪ (har‘𝑥)) | |
33 | harn0 40540 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ V → (har‘𝑥) ≠ ∅) | |
34 | 13, 33 | ax-mp 5 | . . . . . . . . . 10 ⊢ (har‘𝑥) ≠ ∅ |
35 | ssn0 4299 | . . . . . . . . . 10 ⊢ (((har‘𝑥) ⊆ (𝑥 ∪ (har‘𝑥)) ∧ (har‘𝑥) ≠ ∅) → (𝑥 ∪ (har‘𝑥)) ≠ ∅) | |
36 | 32, 34, 35 | mp2an 692 | . . . . . . . . 9 ⊢ (𝑥 ∪ (har‘𝑥)) ≠ ∅ |
37 | eldifsn 4685 | . . . . . . . . 9 ⊢ ((𝑥 ∪ (har‘𝑥)) ∈ (V ∖ {∅}) ↔ ((𝑥 ∪ (har‘𝑥)) ∈ V ∧ (𝑥 ∪ (har‘𝑥)) ≠ ∅)) | |
38 | 31, 36, 37 | mpbir2an 711 | . . . . . . . 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 2837 | . . . . . 6 ⊢ ((Base “ Grp) = (V ∖ {∅}) → (𝑥 ∪ (har‘𝑥)) ∈ (Base “ Grp)) |
42 | isnumbasgrp 40545 | . . . . . 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 2737 | . . 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 1542 ∈ wcel 2114 ≠ wne 2935 ∃wrex 3055 Vcvv 3400 ∖ cdif 3850 ∪ cun 3851 ⊆ wss 3853 ∅c0 4221 {csn 4526 dom cdm 5535 “ cima 5538 Fn wfn 6345 ‘cfv 6350 harchar 9106 cardccrd 9450 CHOICEwac 9628 Basecbs 16599 Grpcgrp 18232 Abelcabl 19038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-inf2 9190 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-pre-sup 10706 ax-addf 10707 ax-mulf 10708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-supp 7870 df-tpos 7934 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-seqom 8126 df-1o 8144 df-2o 8145 df-oadd 8148 df-omul 8149 df-er 8333 df-ec 8335 df-qs 8339 df-map 8452 df-ixp 8521 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-fsupp 8920 df-sup 8992 df-inf 8993 df-oi 9060 df-har 9107 df-wdom 9115 df-dju 9416 df-card 9454 df-acn 9457 df-ac 9629 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-7 11797 df-8 11798 df-9 11799 df-n0 11990 df-z 12076 df-dec 12193 df-uz 12338 df-rp 12486 df-fz 12995 df-fzo 13138 df-fl 13266 df-mod 13342 df-seq 13474 df-hash 13796 df-dvds 15713 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-mulr 16695 df-starv 16696 df-sca 16697 df-vsca 16698 df-ip 16699 df-tset 16700 df-ple 16701 df-ds 16703 df-unif 16704 df-hom 16705 df-cco 16706 df-0g 16831 df-prds 16837 df-pws 16839 df-imas 16897 df-qus 16898 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-mhm 18085 df-grp 18235 df-minusg 18236 df-sbg 18237 df-mulg 18356 df-subg 18407 df-nsg 18408 df-eqg 18409 df-ghm 18487 df-gim 18530 df-gic 18531 df-cmn 19039 df-abl 19040 df-mgp 19372 df-ur 19384 df-ring 19431 df-cring 19432 df-oppr 19508 df-dvdsr 19526 df-rnghom 19602 df-subrg 19665 df-lmod 19768 df-lss 19836 df-lsp 19876 df-sra 20076 df-rgmod 20077 df-lidl 20078 df-rsp 20079 df-2idl 20137 df-cnfld 20231 df-zring 20303 df-zrh 20337 df-zn 20340 df-dsmm 20561 df-frlm 20576 |
This theorem is referenced by: (None) |
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