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 43550

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 14309	. . . . . 6 ⊢ (𝑆 ∈ Fin → (♯‘𝑆) ∈ ℕ₀)
2	1	adantl 482	. . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) ∈ ℕ₀)
3		eqid 2739	. . . . . 6 ⊢ (ℤ/nℤ‘(♯‘𝑆)) = (ℤ/nℤ‘(♯‘𝑆))
4	3	zncrng 21519	. . . . 5 ⊢ ((♯‘𝑆) ∈ ℕ₀ → (ℤ/nℤ‘(♯‘𝑆)) ∈ CRing)
5		crngring 20217	. . . . 5 ⊢ ((ℤ/nℤ‘(♯‘𝑆)) ∈ CRing → (ℤ/nℤ‘(♯‘𝑆)) ∈ Ring)
6		ringabl 20253	. . . . 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 14319	. . . . . . . 8 ⊢ (𝑆 ∈ Fin → ((♯‘𝑆) ∈ ℕ ↔ 𝑆 ≠ ∅))
9	8	biimparc 480	. . . . . . 7 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) ∈ ℕ)
10		eqid 2739	. . . . . . . 8 ⊢ (Base‘(ℤ/nℤ‘(♯‘𝑆))) = (Base‘(ℤ/nℤ‘(♯‘𝑆)))
11	3, 10	znhash 21533	. . . . . . 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 2745	. . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) = (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆)))))
14		simpr 485	. . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ Fin)
15	3, 10	znfi 21534	. . . . . . 7 ⊢ ((♯‘𝑆) ∈ ℕ → (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin)
16	9, 15	syl 17	. . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin)
17		hashen 14300	. . . . . 6 ⊢ ((𝑆 ∈ Fin ∧ (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin) → ((♯‘𝑆) = (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆)))) ↔ 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆)))))
18	14, 16, 17	syl2anc 590	. . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → ((♯‘𝑆) = (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆)))) ↔ 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆)))))
19	13, 18	mpbid 233	. . . 4 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆))))
20	10	isnumbasgrplem1 43546	. . . 4 ⊢ (((ℤ/nℤ‘(♯‘𝑆)) ∈ Abel ∧ 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆)))) → 𝑆 ∈ (Base “ Abel))
21	7, 19, 20	syl2anc 590	. . 3 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel))
22	21	adantll 720	. 2 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel))
23		2nn0 12445	. . . . . . 7 ⊢ 2 ∈ ℕ₀
24		eqid 2739	. . . . . . . 8 ⊢ (ℤ/nℤ‘2) = (ℤ/nℤ‘2)
25	24	zncrng 21519	. . . . . . 7 ⊢ (2 ∈ ℕ₀ → (ℤ/nℤ‘2) ∈ CRing)
26		crngring 20217	. . . . . . 7 ⊢ ((ℤ/nℤ‘2) ∈ CRing → (ℤ/nℤ‘2) ∈ Ring)
27	23, 25, 26	mp2b 10	. . . . . 6 ⊢ (ℤ/nℤ‘2) ∈ Ring
28		eqid 2739	. . . . . . 7 ⊢ ((ℤ/nℤ‘2) freeLMod 𝑆) = ((ℤ/nℤ‘2) freeLMod 𝑆)
29	28	frlmlmod 21724	. . . . . 6 ⊢ (((ℤ/nℤ‘2) ∈ Ring ∧ 𝑆 ∈ dom card) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod)
30	27, 29	mpan 696	. . . . 5 ⊢ (𝑆 ∈ dom card → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod)
31		lmodabl 20899	. . . . 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 732	. . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel)
34		eqid 2739	. . . . . . 7 ⊢ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) = (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))
35	24, 28, 34	frlmpwfi 43543	. . . . . 6 ⊢ (𝑆 ∈ dom card → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin))
36	35	ad2antrr 732	. . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin))
37		simpll 772	. . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ∈ dom card)
38		numinfctb 43548	. . . . . . 7 ⊢ ((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆)
39	38	adantlr 721	. . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆)
40		infpwfien 9975	. . . . . 6 ⊢ ((𝑆 ∈ dom card ∧ ω ≼ 𝑆) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆)
41	37, 39, 40	syl2anc 590	. . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆)
42		entr 8943	. . . . 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 590	. . . 4 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ 𝑆)
44	43	ensymd 8942	. . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)))
45	34	isnumbasgrplem1 43546	. . 3 ⊢ ((((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel ∧ 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))) → 𝑆 ∈ (Base “ Abel))
46	33, 44, 45	syl2anc 590	. 2 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel))
47	22, 46	pm2.61dan 818	1 ⊢ ((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (Base “ Abel))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∩ cin 3882 ∅c0 4261 𝒫 cpw 4529 class class class wbr 5072 dom cdm 5618 “ cima 5621 ‘cfv 6485 (class class class)co 7356 ωcom 7806 ≈ cen 8880 ≼ cdom 8881 Fincfn 8883 cardccrd 9850 ℕcn 12165 2c2 12227 ℕ₀cn0 12428 ♯chash 14283 Basecbs 17170 Abelcabl 19747 Ringcrg 20205 CRingccrg 20206 LModclmod 20850 ℤ/nℤczn 21477 freeLMod cfrlm 21721

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-seqom 8377 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-ec 8635 df-qs 8639 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9816 df-card 9854 df-acn 9857 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-rp 12934 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-hash 14284 df-dvds 16213 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-0g 17395 df-prds 17401 df-pws 17403 df-imas 17463 df-qus 17464 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-nsg 19091 df-eqg 19092 df-ghm 19179 df-gim 19225 df-gic 19226 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-rhm 20443 df-subrng 20518 df-subrg 20542 df-lmod 20852 df-lss 20922 df-lsp 20962 df-sra 21163 df-rgmod 21164 df-lidl 21201 df-rsp 21202 df-2idl 21243 df-cnfld 21348 df-zring 21422 df-zrh 21478 df-zn 21481 df-dsmm 21707 df-frlm 21722

This theorem is referenced by: isnumbasabl 43551 dfacbasgrp 43553

W3C validator