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Mathbox for Stefan O'Rear |
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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 13437 | . . . . . 6 ⊢ (𝑆 ∈ Fin → (♯‘𝑆) ∈ ℕ0) | |
2 | 1 | adantl 475 | . . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) ∈ ℕ0) |
3 | eqid 2825 | . . . . . 6 ⊢ (ℤ/nℤ‘(♯‘𝑆)) = (ℤ/nℤ‘(♯‘𝑆)) | |
4 | 3 | zncrng 20252 | . . . . 5 ⊢ ((♯‘𝑆) ∈ ℕ0 → (ℤ/nℤ‘(♯‘𝑆)) ∈ CRing) |
5 | crngring 18912 | . . . . 5 ⊢ ((ℤ/nℤ‘(♯‘𝑆)) ∈ CRing → (ℤ/nℤ‘(♯‘𝑆)) ∈ Ring) | |
6 | ringabl 18934 | . . . . 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 13447 | . . . . . . . 8 ⊢ (𝑆 ∈ Fin → ((♯‘𝑆) ∈ ℕ ↔ 𝑆 ≠ ∅)) | |
9 | 8 | biimparc 473 | . . . . . . 7 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) ∈ ℕ) |
10 | eqid 2825 | . . . . . . . 8 ⊢ (Base‘(ℤ/nℤ‘(♯‘𝑆))) = (Base‘(ℤ/nℤ‘(♯‘𝑆))) | |
11 | 3, 10 | znhash 20266 | . . . . . . 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 2831 | . . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) = (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆))))) |
14 | simpr 479 | . . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ Fin) | |
15 | 3, 10 | znfi 20267 | . . . . . . 7 ⊢ ((♯‘𝑆) ∈ ℕ → (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin) |
16 | 9, 15 | syl 17 | . . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin) |
17 | hashen 13427 | . . . . . 6 ⊢ ((𝑆 ∈ Fin ∧ (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin) → ((♯‘𝑆) = (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆)))) ↔ 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆))))) | |
18 | 14, 16, 17 | syl2anc 581 | . . . . 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 38514 | . . . 4 ⊢ (((ℤ/nℤ‘(♯‘𝑆)) ∈ Abel ∧ 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆)))) → 𝑆 ∈ (Base “ Abel)) |
21 | 7, 19, 20 | syl2anc 581 | . . 3 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel)) |
22 | 21 | adantll 707 | . 2 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel)) |
23 | 2nn0 11637 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
24 | eqid 2825 | . . . . . . . 8 ⊢ (ℤ/nℤ‘2) = (ℤ/nℤ‘2) | |
25 | 24 | zncrng 20252 | . . . . . . 7 ⊢ (2 ∈ ℕ0 → (ℤ/nℤ‘2) ∈ CRing) |
26 | crngring 18912 | . . . . . . 7 ⊢ ((ℤ/nℤ‘2) ∈ CRing → (ℤ/nℤ‘2) ∈ Ring) | |
27 | 23, 25, 26 | mp2b 10 | . . . . . 6 ⊢ (ℤ/nℤ‘2) ∈ Ring |
28 | eqid 2825 | . . . . . . 7 ⊢ ((ℤ/nℤ‘2) freeLMod 𝑆) = ((ℤ/nℤ‘2) freeLMod 𝑆) | |
29 | 28 | frlmlmod 20456 | . . . . . 6 ⊢ (((ℤ/nℤ‘2) ∈ Ring ∧ 𝑆 ∈ dom card) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod) |
30 | 27, 29 | mpan 683 | . . . . 5 ⊢ (𝑆 ∈ dom card → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod) |
31 | lmodabl 19266 | . . . . 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 719 | . . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel) |
34 | eqid 2825 | . . . . . . 7 ⊢ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) = (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) | |
35 | 24, 28, 34 | frlmpwfi 38511 | . . . . . 6 ⊢ (𝑆 ∈ dom card → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin)) |
36 | 35 | ad2antrr 719 | . . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin)) |
37 | simpll 785 | . . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ∈ dom card) | |
38 | numinfctb 38516 | . . . . . . 7 ⊢ ((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆) | |
39 | 38 | adantlr 708 | . . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆) |
40 | infpwfien 9198 | . . . . . 6 ⊢ ((𝑆 ∈ dom card ∧ ω ≼ 𝑆) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆) | |
41 | 37, 39, 40 | syl2anc 581 | . . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆) |
42 | entr 8274 | . . . . 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 581 | . . . 4 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ 𝑆) |
44 | 43 | ensymd 8273 | . . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))) |
45 | 34 | isnumbasgrplem1 38514 | . . 3 ⊢ ((((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel ∧ 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))) → 𝑆 ∈ (Base “ Abel)) |
46 | 33, 44, 45 | syl2anc 581 | . 2 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel)) |
47 | 22, 46 | pm2.61dan 849 | 1 ⊢ ((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (Base “ Abel)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∩ cin 3797 ∅c0 4144 𝒫 cpw 4378 class class class wbr 4873 dom cdm 5342 “ cima 5345 ‘cfv 6123 (class class class)co 6905 ωcom 7326 ≈ cen 8219 ≼ cdom 8220 Fincfn 8222 cardccrd 9074 ℕcn 11350 2c2 11406 ℕ0cn0 11618 ♯chash 13410 Basecbs 16222 Abelcabl 18547 Ringcrg 18901 CRingccrg 18902 LModclmod 19219 ℤ/nℤczn 20211 freeLMod cfrlm 20453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 ax-addf 10331 ax-mulf 10332 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-tpos 7617 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-seqom 7809 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-ec 8011 df-qs 8015 df-map 8124 df-ixp 8176 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-acn 9081 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-rp 12113 df-fz 12620 df-fzo 12761 df-fl 12888 df-mod 12964 df-seq 13096 df-hash 13411 df-dvds 15358 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-starv 16320 df-sca 16321 df-vsca 16322 df-ip 16323 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-hom 16329 df-cco 16330 df-0g 16455 df-prds 16461 df-pws 16463 df-imas 16521 df-qus 16522 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-mhm 17688 df-grp 17779 df-minusg 17780 df-sbg 17781 df-mulg 17895 df-subg 17942 df-nsg 17943 df-eqg 17944 df-ghm 18009 df-gim 18052 df-gic 18053 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-cring 18904 df-oppr 18977 df-dvdsr 18995 df-rnghom 19071 df-subrg 19134 df-lmod 19221 df-lss 19289 df-lsp 19331 df-sra 19533 df-rgmod 19534 df-lidl 19535 df-rsp 19536 df-2idl 19593 df-cnfld 20107 df-zring 20179 df-zrh 20212 df-zn 20215 df-dsmm 20439 df-frlm 20454 |
This theorem is referenced by: isnumbasabl 38519 dfacbasgrp 38521 |
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