isnumbasgrplem3 - Mathbox for Stefan O'Rear

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Theorem isnumbasgrplem3 43262

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 14270	. . . . . 6 ⊢ (𝑆 ∈ Fin → (♯‘𝑆) ∈ ℕ₀)
2	1	adantl 481	. . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) ∈ ℕ₀)
3		eqid 2733	. . . . . 6 ⊢ (ℤ/nℤ‘(♯‘𝑆)) = (ℤ/nℤ‘(♯‘𝑆))
4	3	zncrng 21490	. . . . 5 ⊢ ((♯‘𝑆) ∈ ℕ₀ → (ℤ/nℤ‘(♯‘𝑆)) ∈ CRing)
5		crngring 20171	. . . . 5 ⊢ ((ℤ/nℤ‘(♯‘𝑆)) ∈ CRing → (ℤ/nℤ‘(♯‘𝑆)) ∈ Ring)
6		ringabl 20207	. . . . 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 14280	. . . . . . . 8 ⊢ (𝑆 ∈ Fin → ((♯‘𝑆) ∈ ℕ ↔ 𝑆 ≠ ∅))
9	8	biimparc 479	. . . . . . 7 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) ∈ ℕ)
10		eqid 2733	. . . . . . . 8 ⊢ (Base‘(ℤ/nℤ‘(♯‘𝑆))) = (Base‘(ℤ/nℤ‘(♯‘𝑆)))
11	3, 10	znhash 21504	. . . . . . 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 2739	. . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (♯‘𝑆) = (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆)))))
14		simpr 484	. . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ Fin)
15	3, 10	znfi 21505	. . . . . . 7 ⊢ ((♯‘𝑆) ∈ ℕ → (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin)
16	9, 15	syl 17	. . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin)
17		hashen 14261	. . . . . 6 ⊢ ((𝑆 ∈ Fin ∧ (Base‘(ℤ/nℤ‘(♯‘𝑆))) ∈ Fin) → ((♯‘𝑆) = (♯‘(Base‘(ℤ/nℤ‘(♯‘𝑆)))) ↔ 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆)))))
18	14, 16, 17	syl2anc 584	. . . . 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 43258	. . . 4 ⊢ (((ℤ/nℤ‘(♯‘𝑆)) ∈ Abel ∧ 𝑆 ≈ (Base‘(ℤ/nℤ‘(♯‘𝑆)))) → 𝑆 ∈ (Base “ Abel))
21	7, 19, 20	syl2anc 584	. . 3 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel))
22	21	adantll 714	. 2 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel))
23		2nn0 12409	. . . . . . 7 ⊢ 2 ∈ ℕ₀
24		eqid 2733	. . . . . . . 8 ⊢ (ℤ/nℤ‘2) = (ℤ/nℤ‘2)
25	24	zncrng 21490	. . . . . . 7 ⊢ (2 ∈ ℕ₀ → (ℤ/nℤ‘2) ∈ CRing)
26		crngring 20171	. . . . . . 7 ⊢ ((ℤ/nℤ‘2) ∈ CRing → (ℤ/nℤ‘2) ∈ Ring)
27	23, 25, 26	mp2b 10	. . . . . 6 ⊢ (ℤ/nℤ‘2) ∈ Ring
28		eqid 2733	. . . . . . 7 ⊢ ((ℤ/nℤ‘2) freeLMod 𝑆) = ((ℤ/nℤ‘2) freeLMod 𝑆)
29	28	frlmlmod 21695	. . . . . 6 ⊢ (((ℤ/nℤ‘2) ∈ Ring ∧ 𝑆 ∈ dom card) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod)
30	27, 29	mpan 690	. . . . 5 ⊢ (𝑆 ∈ dom card → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod)
31		lmodabl 20851	. . . . 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 726	. . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel)
34		eqid 2733	. . . . . . 7 ⊢ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) = (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))
35	24, 28, 34	frlmpwfi 43255	. . . . . 6 ⊢ (𝑆 ∈ dom card → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin))
36	35	ad2antrr 726	. . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin))
37		simpll 766	. . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ∈ dom card)
38		numinfctb 43260	. . . . . . 7 ⊢ ((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆)
39	38	adantlr 715	. . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆)
40		infpwfien 9964	. . . . . 6 ⊢ ((𝑆 ∈ dom card ∧ ω ≼ 𝑆) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆)
41	37, 39, 40	syl2anc 584	. . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆)
42		entr 8939	. . . . 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 584	. . . 4 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ 𝑆)
44	43	ensymd 8938	. . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)))
45	34	isnumbasgrplem1 43258	. . 3 ⊢ ((((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel ∧ 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))) → 𝑆 ∈ (Base “ Abel))
46	33, 44, 45	syl2anc 584	. 2 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel))
47	22, 46	pm2.61dan 812	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 1541 ∈ wcel 2113 ≠ wne 2929 ∩ cin 3897 ∅c0 4282 𝒫 cpw 4551 class class class wbr 5095 dom cdm 5621 “ cima 5624 ‘cfv 6489 (class class class)co 7355 ωcom 7805 ≈ cen 8876 ≼ cdom 8877 Fincfn 8879 cardccrd 9839 ℕcn 12136 2c2 12191 ℕ₀cn0 12392 ♯chash 14244 Basecbs 17127 Abelcabl 19701 Ringcrg 20159 CRingccrg 20160 LModclmod 20802 ℤ/nℤczn 21448 freeLMod cfrlm 21692

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9542 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095 ax-addf 11096 ax-mulf 11097

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-tpos 8165 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-seqom 8376 df-1o 8394 df-2o 8395 df-oadd 8398 df-er 8631 df-ec 8633 df-qs 8637 df-map 8761 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-sup 9337 df-inf 9338 df-oi 9407 df-dju 9805 df-card 9843 df-acn 9846 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-rp 12897 df-fz 13415 df-fzo 13562 df-fl 13703 df-mod 13781 df-seq 13916 df-hash 14245 df-dvds 16171 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-starv 17183 df-sca 17184 df-vsca 17185 df-ip 17186 df-tset 17187 df-ple 17188 df-ds 17190 df-unif 17191 df-hom 17192 df-cco 17193 df-0g 17352 df-prds 17358 df-pws 17360 df-imas 17420 df-qus 17421 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-mhm 18699 df-grp 18857 df-minusg 18858 df-sbg 18859 df-mulg 18989 df-subg 19044 df-nsg 19045 df-eqg 19046 df-ghm 19133 df-gim 19179 df-gic 19180 df-cmn 19702 df-abl 19703 df-mgp 20067 df-rng 20079 df-ur 20108 df-ring 20161 df-cring 20162 df-oppr 20264 df-dvdsr 20284 df-rhm 20399 df-subrng 20470 df-subrg 20494 df-lmod 20804 df-lss 20874 df-lsp 20914 df-sra 21116 df-rgmod 21117 df-lidl 21154 df-rsp 21155 df-2idl 21196 df-cnfld 21301 df-zring 21393 df-zrh 21449 df-zn 21452 df-dsmm 21678 df-frlm 21693

This theorem is referenced by: isnumbasabl 43263 dfacbasgrp 43265

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