![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > isnumbasgrplem3 | Structured version Visualization version GIF version |
Description: Every nonempty numerable set can be given the structure of an Abelian group, either a finite cyclic group or a vector space over Z/2Z. (Contributed by Stefan O'Rear, 10-Jul-2015.) |
Ref | Expression |
---|---|
isnumbasgrplem3 | ⊢ ((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (Base “ Abel)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashcl 13393 | . . . . . 6 ⊢ (𝑆 ∈ Fin → (♯‘𝑆) ∈ ℕ0) | |
2 | 1 | adantl 474 | . . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) ∈ ℕ0) |
3 | eqid 2797 | . . . . . 6 ⊢ (ℤ/nℤ‘(♯‘𝑆)) = (ℤ/nℤ‘(♯‘𝑆)) | |
4 | 3 | zncrng 20211 | . . . . 5 ⊢ ((♯‘𝑆) ∈ ℕ0 → (ℤ/nℤ‘(♯‘𝑆)) ∈ CRing) |
5 | crngring 18871 | . . . . 5 ⊢ ((ℤ/nℤ‘(♯‘𝑆)) ∈ CRing → (ℤ/nℤ‘(♯‘𝑆)) ∈ Ring) | |
6 | ringabl 18893 | . . . . 5 ⊢ ((ℤ/nℤ‘(♯‘𝑆)) ∈ Ring → (ℤ/nℤ‘(♯‘𝑆)) ∈ Abel) | |
7 | 2, 4, 5, 6 | 4syl 19 | . . . 4 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (ℤ/nℤ‘(♯‘𝑆)) ∈ Abel) |
8 | hashnncl 13403 | . . . . . . . 8 ⊢ (𝑆 ∈ Fin → ((♯‘𝑆) ∈ ℕ ↔ 𝑆 ≠ ∅)) | |
9 | 8 | biimparc 472 | . . . . . . 7 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) ∈ ℕ) |
10 | eqid 2797 | . . . . . . . 8 ⊢ (Base‘(ℤ/nℤ‘(♯‘𝑆))) = (Base‘(ℤ/nℤ‘(♯‘𝑆))) | |
11 | 3, 10 | znhash 20225 | . . . . . . 7 ⊢ ((♯‘𝑆) ∈ ℕ → (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆)))) = (♯‘𝑆)) |
12 | 9, 11 | syl 17 | . . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆)))) = (♯‘𝑆)) |
13 | 12 | eqcomd 2803 | . . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) = (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆))))) |
14 | simpr 478 | . . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ Fin) | |
15 | 3, 10 | znfi 20226 | . . . . . . 7 ⊢ ((♯‘𝑆) ∈ ℕ → (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin) |
16 | 9, 15 | syl 17 | . . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin) |
17 | hashen 13383 | . . . . . 6 ⊢ ((𝑆 ∈ Fin ∧ (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin) → ((♯‘𝑆) = (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆)))) ↔ 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆))))) | |
18 | 14, 16, 17 | syl2anc 580 | . . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → ((♯‘𝑆) = (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆)))) ↔ 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆))))) |
19 | 13, 18 | mpbid 224 | . . . 4 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆)))) |
20 | 10 | isnumbasgrplem1 38444 | . . . 4 ⊢ (((ℤ/nℤ‘(♯‘𝑆)) ∈ Abel ∧ 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆)))) → 𝑆 ∈ (Base “ Abel)) |
21 | 7, 19, 20 | syl2anc 580 | . . 3 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel)) |
22 | 21 | adantll 706 | . 2 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel)) |
23 | 2nn0 11595 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
24 | eqid 2797 | . . . . . . . 8 ⊢ (ℤ/nℤ‘2) = (ℤ/nℤ‘2) | |
25 | 24 | zncrng 20211 | . . . . . . 7 ⊢ (2 ∈ ℕ0 → (ℤ/nℤ‘2) ∈ CRing) |
26 | crngring 18871 | . . . . . . 7 ⊢ ((ℤ/nℤ‘2) ∈ CRing → (ℤ/nℤ‘2) ∈ Ring) | |
27 | 23, 25, 26 | mp2b 10 | . . . . . 6 ⊢ (ℤ/nℤ‘2) ∈ Ring |
28 | eqid 2797 | . . . . . . 7 ⊢ ((ℤ/nℤ‘2) freeLMod 𝑆) = ((ℤ/nℤ‘2) freeLMod 𝑆) | |
29 | 28 | frlmlmod 20415 | . . . . . 6 ⊢ (((ℤ/nℤ‘2) ∈ Ring ∧ 𝑆 ∈ dom card) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod) |
30 | 27, 29 | mpan 682 | . . . . 5 ⊢ (𝑆 ∈ dom card → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod) |
31 | lmodabl 19225 | . . . . 5 ⊢ (((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel) | |
32 | 30, 31 | syl 17 | . . . 4 ⊢ (𝑆 ∈ dom card → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel) |
33 | 32 | ad2antrr 718 | . . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel) |
34 | eqid 2797 | . . . . . . 7 ⊢ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) = (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) | |
35 | 24, 28, 34 | frlmpwfi 38441 | . . . . . 6 ⊢ (𝑆 ∈ dom card → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin)) |
36 | 35 | ad2antrr 718 | . . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin)) |
37 | simpll 784 | . . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ∈ dom card) | |
38 | numinfctb 38446 | . . . . . . 7 ⊢ ((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆) | |
39 | 38 | adantlr 707 | . . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆) |
40 | infpwfien 9169 | . . . . . 6 ⊢ ((𝑆 ∈ dom card ∧ ω ≼ 𝑆) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆) | |
41 | 37, 39, 40 | syl2anc 580 | . . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆) |
42 | entr 8245 | . . . . 5 ⊢ (((Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin) ∧ (𝒫 𝑆 ∩ Fin) ≈ 𝑆) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ 𝑆) | |
43 | 36, 41, 42 | syl2anc 580 | . . . 4 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ 𝑆) |
44 | 43 | ensymd 8244 | . . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))) |
45 | 34 | isnumbasgrplem1 38444 | . . 3 ⊢ ((((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel ∧ 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))) → 𝑆 ∈ (Base “ Abel)) |
46 | 33, 44, 45 | syl2anc 580 | . 2 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel)) |
47 | 22, 46 | pm2.61dan 848 | 1 ⊢ ((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (Base “ Abel)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 ∩ cin 3766 ∅c0 4113 𝒫 cpw 4347 class class class wbr 4841 dom cdm 5310 “ cima 5313 ‘cfv 6099 (class class class)co 6876 ωcom 7297 ≈ cen 8190 ≼ cdom 8191 Fincfn 8193 cardccrd 9045 ℕcn 11310 2c2 11364 ℕ0cn0 11576 ♯chash 13366 Basecbs 16181 Abelcabl 18506 Ringcrg 18860 CRingccrg 18861 LModclmod 19178 ℤ/nℤczn 20170 freeLMod cfrlm 20412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-inf2 8786 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-pre-sup 10300 ax-addf 10301 ax-mulf 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-se 5270 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-isom 6108 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-supp 7531 df-tpos 7588 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-seqom 7780 df-1o 7797 df-2o 7798 df-oadd 7801 df-er 7980 df-ec 7982 df-qs 7986 df-map 8095 df-ixp 8147 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-fsupp 8516 df-sup 8588 df-inf 8589 df-oi 8655 df-card 9049 df-acn 9052 df-cda 9276 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-rp 12071 df-fz 12577 df-fzo 12717 df-fl 12844 df-mod 12920 df-seq 13052 df-hash 13367 df-dvds 15317 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-starv 16279 df-sca 16280 df-vsca 16281 df-ip 16282 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-hom 16288 df-cco 16289 df-0g 16414 df-prds 16420 df-pws 16422 df-imas 16480 df-qus 16481 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-mhm 17647 df-grp 17738 df-minusg 17739 df-sbg 17740 df-mulg 17854 df-subg 17901 df-nsg 17902 df-eqg 17903 df-ghm 17968 df-gim 18011 df-gic 18012 df-cmn 18507 df-abl 18508 df-mgp 18803 df-ur 18815 df-ring 18862 df-cring 18863 df-oppr 18936 df-dvdsr 18954 df-rnghom 19030 df-subrg 19093 df-lmod 19180 df-lss 19248 df-lsp 19290 df-sra 19492 df-rgmod 19493 df-lidl 19494 df-rsp 19495 df-2idl 19552 df-cnfld 20066 df-zring 20138 df-zrh 20171 df-zn 20174 df-dsmm 20398 df-frlm 20413 |
This theorem is referenced by: isnumbasabl 38449 dfacbasgrp 38451 |
Copyright terms: Public domain | W3C validator |