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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 10158 | . 2 ⊢ (CHOICE ↔ dom card = V) | |
2 | basfn 17181 | . . . . . . . . . 10 ⊢ Base Fn V | |
3 | ssv 3997 | . . . . . . . . . 10 ⊢ Grp ⊆ V | |
4 | fvelimab 6965 | . . . . . . . . . 10 ⊢ ((Base Fn V ∧ Grp ⊆ V) → (𝑥 ∈ (Base “ Grp) ↔ ∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥)) | |
5 | 2, 3, 4 | mp2an 690 | . . . . . . . . 9 ⊢ (𝑥 ∈ (Base “ Grp) ↔ ∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥) |
6 | eqid 2725 | . . . . . . . . . . . 12 ⊢ (Base‘𝑦) = (Base‘𝑦) | |
7 | 6 | grpbn0 18925 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ Grp → (Base‘𝑦) ≠ ∅) |
8 | neeq1 2993 | . . . . . . . . . . 11 ⊢ ((Base‘𝑦) = 𝑥 → ((Base‘𝑦) ≠ ∅ ↔ 𝑥 ≠ ∅)) | |
9 | 7, 8 | syl5ibcom 244 | . . . . . . . . . 10 ⊢ (𝑦 ∈ Grp → ((Base‘𝑦) = 𝑥 → 𝑥 ≠ ∅)) |
10 | 9 | rexlimiv 3138 | . . . . . . . . 9 ⊢ (∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥 → 𝑥 ≠ ∅) |
11 | 5, 10 | sylbi 216 | . . . . . . . 8 ⊢ (𝑥 ∈ (Base “ Grp) → 𝑥 ≠ ∅) |
12 | 11 | adantl 480 | . . . . . . 7 ⊢ ((dom card = V ∧ 𝑥 ∈ (Base “ Grp)) → 𝑥 ≠ ∅) |
13 | vex 3467 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
14 | 12, 13 | jctil 518 | . . . . . 6 ⊢ ((dom card = V ∧ 𝑥 ∈ (Base “ Grp)) → (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) |
15 | ablgrp 19742 | . . . . . . . . 9 ⊢ (𝑥 ∈ Abel → 𝑥 ∈ Grp) | |
16 | 15 | ssriv 3976 | . . . . . . . 8 ⊢ Abel ⊆ Grp |
17 | imass2 6101 | . . . . . . . 8 ⊢ (Abel ⊆ Grp → (Base “ Abel) ⊆ (Base “ Grp)) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (Base “ Abel) ⊆ (Base “ Grp) |
19 | simprl 769 | . . . . . . . . 9 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ V) | |
20 | simpl 481 | . . . . . . . . 9 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → dom card = V) | |
21 | 19, 20 | eleqtrrd 2828 | . . . . . . . 8 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ dom card) |
22 | simprr 771 | . . . . . . . 8 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ≠ ∅) | |
23 | isnumbasgrplem3 42593 | . . . . . . . 8 ⊢ ((𝑥 ∈ dom card ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (Base “ Abel)) | |
24 | 21, 22, 23 | syl2anc 582 | . . . . . . 7 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (Base “ Abel)) |
25 | 18, 24 | sselid 3970 | . . . . . 6 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (Base “ Grp)) |
26 | 14, 25 | impbida 799 | . . . . 5 ⊢ (dom card = V → (𝑥 ∈ (Base “ Grp) ↔ (𝑥 ∈ V ∧ 𝑥 ≠ ∅))) |
27 | eldifsn 4786 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ {∅}) ↔ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) | |
28 | 26, 27 | bitr4di 288 | . . . 4 ⊢ (dom card = V → (𝑥 ∈ (Base “ Grp) ↔ 𝑥 ∈ (V ∖ {∅}))) |
29 | 28 | eqrdv 2723 | . . 3 ⊢ (dom card = V → (Base “ Grp) = (V ∖ {∅})) |
30 | fvex 6904 | . . . . . . . . . 10 ⊢ (har‘𝑥) ∈ V | |
31 | 13, 30 | unex 7745 | . . . . . . . . 9 ⊢ (𝑥 ∪ (har‘𝑥)) ∈ V |
32 | ssun2 4167 | . . . . . . . . . 10 ⊢ (har‘𝑥) ⊆ (𝑥 ∪ (har‘𝑥)) | |
33 | harn0 42590 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ V → (har‘𝑥) ≠ ∅) | |
34 | 13, 33 | ax-mp 5 | . . . . . . . . . 10 ⊢ (har‘𝑥) ≠ ∅ |
35 | ssn0 4396 | . . . . . . . . . 10 ⊢ (((har‘𝑥) ⊆ (𝑥 ∪ (har‘𝑥)) ∧ (har‘𝑥) ≠ ∅) → (𝑥 ∪ (har‘𝑥)) ≠ ∅) | |
36 | 32, 34, 35 | mp2an 690 | . . . . . . . . 9 ⊢ (𝑥 ∪ (har‘𝑥)) ≠ ∅ |
37 | eldifsn 4786 | . . . . . . . . 9 ⊢ ((𝑥 ∪ (har‘𝑥)) ∈ (V ∖ {∅}) ↔ ((𝑥 ∪ (har‘𝑥)) ∈ V ∧ (𝑥 ∪ (har‘𝑥)) ≠ ∅)) | |
38 | 31, 36, 37 | mpbir2an 709 | . . . . . . . 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 2828 | . . . . . 6 ⊢ ((Base “ Grp) = (V ∖ {∅}) → (𝑥 ∪ (har‘𝑥)) ∈ (Base “ Grp)) |
42 | isnumbasgrp 42595 | . . . . . 6 ⊢ (𝑥 ∈ dom card ↔ (𝑥 ∪ (har‘𝑥)) ∈ (Base “ Grp)) | |
43 | 41, 42 | sylibr 233 | . . . . 5 ⊢ ((Base “ Grp) = (V ∖ {∅}) → 𝑥 ∈ dom card) |
44 | 13 | a1i 11 | . . . . 5 ⊢ ((Base “ Grp) = (V ∖ {∅}) → 𝑥 ∈ V) |
45 | 43, 44 | 2thd 264 | . . . 4 ⊢ ((Base “ Grp) = (V ∖ {∅}) → (𝑥 ∈ dom card ↔ 𝑥 ∈ V)) |
46 | 45 | eqrdv 2723 | . . 3 ⊢ ((Base “ Grp) = (V ∖ {∅}) → dom card = V) |
47 | 29, 46 | impbii 208 | . 2 ⊢ (dom card = V ↔ (Base “ Grp) = (V ∖ {∅})) |
48 | 1, 47 | bitri 274 | 1 ⊢ (CHOICE ↔ (Base “ Grp) = (V ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∃wrex 3060 Vcvv 3463 ∖ cdif 3937 ∪ cun 3938 ⊆ wss 3940 ∅c0 4318 {csn 4624 dom cdm 5672 “ cima 5675 Fn wfn 6537 ‘cfv 6542 harchar 9577 cardccrd 9956 CHOICEwac 10136 Basecbs 17177 Grpcgrp 18892 Abelcabl 19738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215 ax-mulf 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-seqom 8465 df-1o 8483 df-2o 8484 df-oadd 8487 df-omul 8488 df-er 8721 df-ec 8723 df-qs 8727 df-map 8843 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-inf 9464 df-oi 9531 df-har 9578 df-wdom 9586 df-dju 9922 df-card 9960 df-acn 9963 df-ac 10137 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-rp 13005 df-fz 13515 df-fzo 13658 df-fl 13787 df-mod 13865 df-seq 13997 df-hash 14320 df-dvds 16229 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-0g 17420 df-prds 17426 df-pws 17428 df-imas 17487 df-qus 17488 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-nsg 19081 df-eqg 19082 df-ghm 19170 df-gim 19215 df-gic 19216 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-cring 20178 df-oppr 20275 df-dvdsr 20298 df-rhm 20413 df-subrng 20485 df-subrg 20510 df-lmod 20747 df-lss 20818 df-lsp 20858 df-sra 21060 df-rgmod 21061 df-lidl 21106 df-rsp 21107 df-2idl 21146 df-cnfld 21282 df-zring 21375 df-zrh 21431 df-zn 21434 df-dsmm 21668 df-frlm 21683 |
This theorem is referenced by: (None) |
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