isnumbasgrplem3 - Mathbox for Stefan O'Rear

	Mathbox for Stefan O'Rear		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > isnumbasgrplem3		Structured version Visualization version GIF version

Theorem isnumbasgrplem3 37992

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 13185	. . . . . 6 ⊢ (𝑆 ∈ Fin → (#‘𝑆) ∈ ℕ₀)
2	1	adantl 481	. . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (#‘𝑆) ∈ ℕ₀)
3		eqid 2651	. . . . . 6 ⊢ (ℤ/nℤ‘(#‘𝑆)) = (ℤ/nℤ‘(#‘𝑆))
4	3	zncrng 19941	. . . . 5 ⊢ ((#‘𝑆) ∈ ℕ₀ → (ℤ/nℤ‘(#‘𝑆)) ∈ CRing)
5		crngring 18604	. . . . 5 ⊢ ((ℤ/nℤ‘(#‘𝑆)) ∈ CRing → (ℤ/nℤ‘(#‘𝑆)) ∈ Ring)
6		ringabl 18626	. . . . 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 13195	. . . . . . . 8 ⊢ (𝑆 ∈ Fin → ((#‘𝑆) ∈ ℕ ↔ 𝑆 ≠ ∅))
9	8	biimparc 503	. . . . . . 7 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (#‘𝑆) ∈ ℕ)
10		eqid 2651	. . . . . . . 8 ⊢ (Base‘(ℤ/nℤ‘(#‘𝑆))) = (Base‘(ℤ/nℤ‘(#‘𝑆)))
11	3, 10	znhash 19955	. . . . . . 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 2657	. . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (#‘𝑆) = (#‘(Base‘(ℤ/nℤ‘(#‘𝑆)))))
14		simpr 476	. . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ Fin)
15	3, 10	znfi 19956	. . . . . . 7 ⊢ ((#‘𝑆) ∈ ℕ → (Base‘(ℤ/nℤ‘(#‘𝑆))) ∈ Fin)
16	9, 15	syl 17	. . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (Base‘(ℤ/nℤ‘(#‘𝑆))) ∈ Fin)
17		hashen 13175	. . . . . 6 ⊢ ((𝑆 ∈ Fin ∧ (Base‘(ℤ/nℤ‘(#‘𝑆))) ∈ Fin) → ((#‘𝑆) = (#‘(Base‘(ℤ/nℤ‘(#‘𝑆)))) ↔ 𝑆 ≈ (Base‘(ℤ/nℤ‘(#‘𝑆)))))
18	14, 16, 17	syl2anc 694	. . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → ((#‘𝑆) = (#‘(Base‘(ℤ/nℤ‘(#‘𝑆)))) ↔ 𝑆 ≈ (Base‘(ℤ/nℤ‘(#‘𝑆)))))
19	13, 18	mpbid 222	. . . 4 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ≈ (Base‘(ℤ/nℤ‘(#‘𝑆))))
20	10	isnumbasgrplem1 37988	. . . 4 ⊢ (((ℤ/nℤ‘(#‘𝑆)) ∈ Abel ∧ 𝑆 ≈ (Base‘(ℤ/nℤ‘(#‘𝑆)))) → 𝑆 ∈ (Base “ Abel))
21	7, 19, 20	syl2anc 694	. . 3 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel))
22	21	adantll 750	. 2 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel))
23		2nn0 11347	. . . . . . 7 ⊢ 2 ∈ ℕ₀
24		eqid 2651	. . . . . . . 8 ⊢ (ℤ/nℤ‘2) = (ℤ/nℤ‘2)
25	24	zncrng 19941	. . . . . . 7 ⊢ (2 ∈ ℕ₀ → (ℤ/nℤ‘2) ∈ CRing)
26		crngring 18604	. . . . . . 7 ⊢ ((ℤ/nℤ‘2) ∈ CRing → (ℤ/nℤ‘2) ∈ Ring)
27	23, 25, 26	mp2b 10	. . . . . 6 ⊢ (ℤ/nℤ‘2) ∈ Ring
28		eqid 2651	. . . . . . 7 ⊢ ((ℤ/nℤ‘2) freeLMod 𝑆) = ((ℤ/nℤ‘2) freeLMod 𝑆)
29	28	frlmlmod 20141	. . . . . 6 ⊢ (((ℤ/nℤ‘2) ∈ Ring ∧ 𝑆 ∈ dom card) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod)
30	27, 29	mpan 706	. . . . 5 ⊢ (𝑆 ∈ dom card → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod)
31		lmodabl 18958	. . . . 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 762	. . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel)
34		eqid 2651	. . . . . . 7 ⊢ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) = (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))
35	24, 28, 34	frlmpwfi 37985	. . . . . 6 ⊢ (𝑆 ∈ dom card → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin))
36	35	ad2antrr 762	. . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin))
37		simpll 805	. . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ∈ dom card)
38		numinfctb 37990	. . . . . . 7 ⊢ ((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆)
39	38	adantlr 751	. . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆)
40		infpwfien 8923	. . . . . 6 ⊢ ((𝑆 ∈ dom card ∧ ω ≼ 𝑆) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆)
41	37, 39, 40	syl2anc 694	. . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆)
42		entr 8049	. . . . 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 694	. . . 4 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ 𝑆)
44	43	ensymd 8048	. . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)))
45	34	isnumbasgrplem1 37988	. . 3 ⊢ ((((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel ∧ 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))) → 𝑆 ∈ (Base “ Abel))
46	33, 44, 45	syl2anc 694	. 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 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∩ cin 3606 ∅c0 3948 𝒫 cpw 4191 class class class wbr 4685 dom cdm 5143 “ cima 5146 ‘cfv 5926 (class class class)co 6690 ωcom 7107 ≈ cen 7994 ≼ cdom 7995 Fincfn 7997 cardccrd 8799 ℕcn 11058 2c2 11108 ℕ₀cn0 11330 #chash 13157 Basecbs 15904 Abelcabl 18240 Ringcrg 18593 CRingccrg 18594 LModclmod 18911 ℤ/nℤczn 19899 freeLMod cfrlm 20138

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-seqom 7588 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-ec 7789 df-qs 7793 df-map 7901 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-acn 8806 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-rp 11871 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-seq 12842 df-hash 13158 df-dvds 15028 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-0g 16149 df-prds 16155 df-pws 16157 df-imas 16215 df-qus 16216 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-grp 17472 df-minusg 17473 df-sbg 17474 df-mulg 17588 df-subg 17638 df-nsg 17639 df-eqg 17640 df-ghm 17705 df-gim 17748 df-gic 17749 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-cring 18596 df-oppr 18669 df-dvdsr 18687 df-rnghom 18763 df-subrg 18826 df-lmod 18913 df-lss 18981 df-lsp 19020 df-sra 19220 df-rgmod 19221 df-lidl 19222 df-rsp 19223 df-2idl 19280 df-cnfld 19795 df-zring 19867 df-zrh 19900 df-zn 19903 df-dsmm 20124 df-frlm 20139

This theorem is referenced by: isnumbasabl 37993 dfacbasgrp 37995

W3C validator