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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 10138 | . 2 ⊢ (CHOICE ↔ dom card = V) | |
2 | basfn 17155 | . . . . . . . . . 10 ⊢ Base Fn V | |
3 | ssv 4006 | . . . . . . . . . 10 ⊢ Grp ⊆ V | |
4 | fvelimab 6964 | . . . . . . . . . 10 ⊢ ((Base Fn V ∧ Grp ⊆ V) → (𝑥 ∈ (Base “ Grp) ↔ ∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥)) | |
5 | 2, 3, 4 | mp2an 689 | . . . . . . . . 9 ⊢ (𝑥 ∈ (Base “ Grp) ↔ ∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥) |
6 | eqid 2731 | . . . . . . . . . . . 12 ⊢ (Base‘𝑦) = (Base‘𝑦) | |
7 | 6 | grpbn0 18894 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ Grp → (Base‘𝑦) ≠ ∅) |
8 | neeq1 3002 | . . . . . . . . . . 11 ⊢ ((Base‘𝑦) = 𝑥 → ((Base‘𝑦) ≠ ∅ ↔ 𝑥 ≠ ∅)) | |
9 | 7, 8 | syl5ibcom 244 | . . . . . . . . . 10 ⊢ (𝑦 ∈ Grp → ((Base‘𝑦) = 𝑥 → 𝑥 ≠ ∅)) |
10 | 9 | rexlimiv 3147 | . . . . . . . . 9 ⊢ (∃𝑦 ∈ Grp (Base‘𝑦) = 𝑥 → 𝑥 ≠ ∅) |
11 | 5, 10 | sylbi 216 | . . . . . . . 8 ⊢ (𝑥 ∈ (Base “ Grp) → 𝑥 ≠ ∅) |
12 | 11 | adantl 481 | . . . . . . 7 ⊢ ((dom card = V ∧ 𝑥 ∈ (Base “ Grp)) → 𝑥 ≠ ∅) |
13 | vex 3477 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
14 | 12, 13 | jctil 519 | . . . . . 6 ⊢ ((dom card = V ∧ 𝑥 ∈ (Base “ Grp)) → (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) |
15 | ablgrp 19701 | . . . . . . . . 9 ⊢ (𝑥 ∈ Abel → 𝑥 ∈ Grp) | |
16 | 15 | ssriv 3986 | . . . . . . . 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 768 | . . . . . . . . 9 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ V) | |
20 | simpl 482 | . . . . . . . . 9 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → dom card = V) | |
21 | 19, 20 | eleqtrrd 2835 | . . . . . . . 8 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ dom card) |
22 | simprr 770 | . . . . . . . 8 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ≠ ∅) | |
23 | isnumbasgrplem3 42313 | . . . . . . . 8 ⊢ ((𝑥 ∈ dom card ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (Base “ Abel)) | |
24 | 21, 22, 23 | syl2anc 583 | . . . . . . 7 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (Base “ Abel)) |
25 | 18, 24 | sselid 3980 | . . . . . 6 ⊢ ((dom card = V ∧ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ (Base “ Grp)) |
26 | 14, 25 | impbida 798 | . . . . 5 ⊢ (dom card = V → (𝑥 ∈ (Base “ Grp) ↔ (𝑥 ∈ V ∧ 𝑥 ≠ ∅))) |
27 | eldifsn 4790 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ {∅}) ↔ (𝑥 ∈ V ∧ 𝑥 ≠ ∅)) | |
28 | 26, 27 | bitr4di 289 | . . . 4 ⊢ (dom card = V → (𝑥 ∈ (Base “ Grp) ↔ 𝑥 ∈ (V ∖ {∅}))) |
29 | 28 | eqrdv 2729 | . . 3 ⊢ (dom card = V → (Base “ Grp) = (V ∖ {∅})) |
30 | fvex 6904 | . . . . . . . . . 10 ⊢ (har‘𝑥) ∈ V | |
31 | 13, 30 | unex 7737 | . . . . . . . . 9 ⊢ (𝑥 ∪ (har‘𝑥)) ∈ V |
32 | ssun2 4173 | . . . . . . . . . 10 ⊢ (har‘𝑥) ⊆ (𝑥 ∪ (har‘𝑥)) | |
33 | harn0 42310 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ V → (har‘𝑥) ≠ ∅) | |
34 | 13, 33 | ax-mp 5 | . . . . . . . . . 10 ⊢ (har‘𝑥) ≠ ∅ |
35 | ssn0 4400 | . . . . . . . . . 10 ⊢ (((har‘𝑥) ⊆ (𝑥 ∪ (har‘𝑥)) ∧ (har‘𝑥) ≠ ∅) → (𝑥 ∪ (har‘𝑥)) ≠ ∅) | |
36 | 32, 34, 35 | mp2an 689 | . . . . . . . . 9 ⊢ (𝑥 ∪ (har‘𝑥)) ≠ ∅ |
37 | eldifsn 4790 | . . . . . . . . 9 ⊢ ((𝑥 ∪ (har‘𝑥)) ∈ (V ∖ {∅}) ↔ ((𝑥 ∪ (har‘𝑥)) ∈ V ∧ (𝑥 ∪ (har‘𝑥)) ≠ ∅)) | |
38 | 31, 36, 37 | mpbir2an 708 | . . . . . . . 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 2835 | . . . . . 6 ⊢ ((Base “ Grp) = (V ∖ {∅}) → (𝑥 ∪ (har‘𝑥)) ∈ (Base “ Grp)) |
42 | isnumbasgrp 42315 | . . . . . 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 265 | . . . 4 ⊢ ((Base “ Grp) = (V ∖ {∅}) → (𝑥 ∈ dom card ↔ 𝑥 ∈ V)) |
46 | 45 | eqrdv 2729 | . . 3 ⊢ ((Base “ Grp) = (V ∖ {∅}) → dom card = V) |
47 | 29, 46 | impbii 208 | . 2 ⊢ (dom card = V ↔ (Base “ Grp) = (V ∖ {∅})) |
48 | 1, 47 | bitri 275 | 1 ⊢ (CHOICE ↔ (Base “ Grp) = (V ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∃wrex 3069 Vcvv 3473 ∖ cdif 3945 ∪ cun 3946 ⊆ wss 3948 ∅c0 4322 {csn 4628 dom cdm 5676 “ cima 5679 Fn wfn 6538 ‘cfv 6543 harchar 9557 cardccrd 9936 CHOICEwac 10116 Basecbs 17151 Grpcgrp 18861 Abelcabl 19697 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-seqom 8454 df-1o 8472 df-2o 8473 df-oadd 8476 df-omul 8477 df-er 8709 df-ec 8711 df-qs 8715 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-inf 9444 df-oi 9511 df-har 9558 df-wdom 9566 df-dju 9902 df-card 9940 df-acn 9943 df-ac 10117 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-rp 12982 df-fz 13492 df-fzo 13635 df-fl 13764 df-mod 13842 df-seq 13974 df-hash 14298 df-dvds 16205 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-0g 17394 df-prds 17400 df-pws 17402 df-imas 17461 df-qus 17462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19046 df-nsg 19047 df-eqg 19048 df-ghm 19135 df-gim 19180 df-gic 19181 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-lmod 20704 df-lss 20775 df-lsp 20815 df-sra 21017 df-rgmod 21018 df-lidl 21063 df-rsp 21064 df-2idl 21103 df-cnfld 21235 df-zring 21308 df-zrh 21364 df-zn 21367 df-dsmm 21598 df-frlm 21613 |
This theorem is referenced by: (None) |
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