isnumbasgrplem3 - Mathbox for Stefan O'Rear

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Theorem isnumbasgrplem3 43533

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.)

**Assertion**
Ref	Expression
isnumbasgrplem3	⊢ ((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (Base “ Abel))

**Proof of Theorem isnumbasgrplem3**
Step	Hyp	Ref	Expression
1		hashcl 14318	. . . . . 6 ⊢ (𝑆 ∈ Fin → (♯‘𝑆) ∈ ℕ₀)
2	1	adantl 481	. . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) ∈ ℕ₀)
3		eqid 2736	. . . . . 6 ⊢ (ℤ/nℤ‘(♯‘𝑆)) = (ℤ/nℤ‘(♯‘𝑆))
4	3	zncrng 21524	. . . . 5 ⊢ ((♯‘𝑆) ∈ ℕ₀ → (ℤ/nℤ‘(♯‘𝑆)) ∈ CRing)
5		crngring 20226	. . . . 5 ⊢ ((ℤ/nℤ‘(♯‘𝑆)) ∈ CRing → (ℤ/nℤ‘(♯‘𝑆)) ∈ Ring)
6		ringabl 20262	. . . . 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 14328	. . . . . . . 8 ⊢ (𝑆 ∈ Fin → ((♯‘𝑆) ∈ ℕ ↔ 𝑆 ≠ ∅))
9	8	biimparc 479	. . . . . . 7 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) ∈ ℕ)
10		eqid 2736	. . . . . . . 8 ⊢ (Base‘(ℤ/nℤ‘(♯‘𝑆))) = (Base‘(ℤ/nℤ‘(♯‘𝑆)))
11	3, 10	znhash 21538	. . . . . . 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 2742	. . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) = (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆)))))
14		simpr 484	. . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ Fin)
15	3, 10	znfi 21539	. . . . . . 7 ⊢ ((♯‘𝑆) ∈ ℕ → (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin)
16	9, 15	syl 17	. . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin)
17		hashen 14309	. . . . . 6 ⊢ ((𝑆 ∈ Fin ∧ (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin) → ((♯‘𝑆) = (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆)))) ↔ 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆)))))
18	14, 16, 17	syl2anc 585	. . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → ((♯‘𝑆) = (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆)))) ↔ 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆)))))
19	13, 18	mpbid 232	. . . 4 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆))))
20	10	isnumbasgrplem1 43529	. . . 4 ⊢ (((ℤ/nℤ‘(♯‘𝑆)) ∈ Abel ∧ 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆)))) → 𝑆 ∈ (Base “ Abel))
21	7, 19, 20	syl2anc 585	. . 3 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel))
22	21	adantll 715	. 2 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel))
23		2nn0 12454	. . . . . . 7 ⊢ 2 ∈ ℕ₀
24		eqid 2736	. . . . . . . 8 ⊢ (ℤ/nℤ‘2) = (ℤ/nℤ‘2)
25	24	zncrng 21524	. . . . . . 7 ⊢ (2 ∈ ℕ₀ → (ℤ/nℤ‘2) ∈ CRing)
26		crngring 20226	. . . . . . 7 ⊢ ((ℤ/nℤ‘2) ∈ CRing → (ℤ/nℤ‘2) ∈ Ring)
27	23, 25, 26	mp2b 10	. . . . . 6 ⊢ (ℤ/nℤ‘2) ∈ Ring
28		eqid 2736	. . . . . . 7 ⊢ ((ℤ/nℤ‘2) freeLMod 𝑆) = ((ℤ/nℤ‘2) freeLMod 𝑆)
29	28	frlmlmod 21729	. . . . . 6 ⊢ (((ℤ/nℤ‘2) ∈ Ring ∧ 𝑆 ∈ dom card) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod)
30	27, 29	mpan 691	. . . . 5 ⊢ (𝑆 ∈ dom card → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod)
31		lmodabl 20904	. . . . 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 727	. . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel)
34		eqid 2736	. . . . . . 7 ⊢ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) = (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))
35	24, 28, 34	frlmpwfi 43526	. . . . . 6 ⊢ (𝑆 ∈ dom card → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin))
36	35	ad2antrr 727	. . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin))
37		simpll 767	. . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ∈ dom card)
38		numinfctb 43531	. . . . . . 7 ⊢ ((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆)
39	38	adantlr 716	. . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆)
40		infpwfien 9984	. . . . . 6 ⊢ ((𝑆 ∈ dom card ∧ ω ≼ 𝑆) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆)
41	37, 39, 40	syl2anc 585	. . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆)
42		entr 8953	. . . . 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 585	. . . 4 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ 𝑆)
44	43	ensymd 8952	. . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)))
45	34	isnumbasgrplem1 43529	. . 3 ⊢ ((((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel ∧ 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))) → 𝑆 ∈ (Base “ Abel))
46	33, 44, 45	syl2anc 585	. 2 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel))
47	22, 46	pm2.61dan 813	1 ⊢ ((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (Base “ Abel))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∩ cin 3888 ∅c0 4273 𝒫 cpw 4541 class class class wbr 5085 dom cdm 5631 “ cima 5634 ‘cfv 6498 (class class class)co 7367 ωcom 7817 ≈ cen 8890 ≼ cdom 8891 Fincfn 8893 cardccrd 9859 ℕcn 12174 2c2 12236 ℕ₀cn0 12437 ♯chash 14292 Basecbs 17179 Abelcabl 19756 Ringcrg 20214 CRingccrg 20215 LModclmod 20855 ℤ/nℤczn 21482 freeLMod cfrlm 21726

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-seqom 8387 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-hash 14293 df-dvds 16222 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-0g 17404 df-prds 17410 df-pws 17412 df-imas 17472 df-qus 17473 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-nsg 19100 df-eqg 19101 df-ghm 19188 df-gim 19234 df-gic 19235 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-rhm 20452 df-subrng 20523 df-subrg 20547 df-lmod 20857 df-lss 20927 df-lsp 20967 df-sra 21168 df-rgmod 21169 df-lidl 21206 df-rsp 21207 df-2idl 21248 df-cnfld 21353 df-zring 21427 df-zrh 21483 df-zn 21486 df-dsmm 21712 df-frlm 21727

This theorem is referenced by: isnumbasabl 43534 dfacbasgrp 43536

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