frlmpwfi - Mathbox for Stefan O'Rear

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Theorem frlmpwfi 38162

Description: Formal linear combinations over Z/2Z are equivalent to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Proof shortened by AV, 14-Jun-2020.)

**Hypotheses**
Ref	Expression
frlmpwfi.r	⊢ 𝑅 = (ℤ/nℤ‘2)
frlmpwfi.y	⊢ 𝑌 = (𝑅 freeLMod 𝐼)
frlmpwfi.b	⊢ 𝐵 = (Base‘𝑌)

**Assertion**
Ref	Expression
frlmpwfi	⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ (𝒫 𝐼 ∩ Fin))

Proof of Theorem frlmpwfi Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frlmpwfi.r	. . . . . 6 ⊢ 𝑅 = (ℤ/nℤ‘2)
2		fvex 6354	. . . . . 6 ⊢ (ℤ/nℤ‘2) ∈ V
3	1, 2	eqeltri 2827	. . . . 5 ⊢ 𝑅 ∈ V
4		frlmpwfi.y	. . . . . 6 ⊢ 𝑌 = (𝑅 freeLMod 𝐼)
5		eqid 2752	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2752	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
7		eqid 2752	. . . . . 6 ⊢ {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)}
8	4, 5, 6, 7	frlmbas 20293	. . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = (Base‘𝑌))
9	3, 8	mpan 708	. . . 4 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = (Base‘𝑌))
10		frlmpwfi.b	. . . 4 ⊢ 𝐵 = (Base‘𝑌)
11	9, 10	syl6eqr 2804	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = 𝐵)
12		eqid 2752	. . . 4 ⊢ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅}
13		enrefg 8145	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ≈ 𝐼)
14		2nn 11369	. . . . . . . 8 ⊢ 2 ∈ ℕ
15	1, 5	znhash 20101	. . . . . . . 8 ⊢ (2 ∈ ℕ → (♯‘(Base‘𝑅)) = 2)
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ (♯‘(Base‘𝑅)) = 2
17		hash2 13377	. . . . . . 7 ⊢ (♯‘2_𝑜) = 2
18	16, 17	eqtr4i 2777	. . . . . 6 ⊢ (♯‘(Base‘𝑅)) = (♯‘2_𝑜)
19		2nn0 11493	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
20	16, 19	eqeltri 2827	. . . . . . . 8 ⊢ (♯‘(Base‘𝑅)) ∈ ℕ₀
21		fvex 6354	. . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V
22		hashclb 13333	. . . . . . . . 9 ⊢ ((Base‘𝑅) ∈ V → ((Base‘𝑅) ∈ Fin ↔ (♯‘(Base‘𝑅)) ∈ ℕ₀))
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ ((Base‘𝑅) ∈ Fin ↔ (♯‘(Base‘𝑅)) ∈ ℕ₀)
24	20, 23	mpbir 221	. . . . . . 7 ⊢ (Base‘𝑅) ∈ Fin
25		2onn 7881	. . . . . . . 8 ⊢ 2_𝑜 ∈ ω
26		nnfi 8310	. . . . . . . 8 ⊢ (2_𝑜 ∈ ω → 2_𝑜 ∈ Fin)
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ 2_𝑜 ∈ Fin
28		hashen 13321	. . . . . . 7 ⊢ (((Base‘𝑅) ∈ Fin ∧ 2_𝑜 ∈ Fin) → ((♯‘(Base‘𝑅)) = (♯‘2_𝑜) ↔ (Base‘𝑅) ≈ 2_𝑜))
29	24, 27, 28	mp2an 710	. . . . . 6 ⊢ ((♯‘(Base‘𝑅)) = (♯‘2_𝑜) ↔ (Base‘𝑅) ≈ 2_𝑜)
30	18, 29	mpbi 220	. . . . 5 ⊢ (Base‘𝑅) ≈ 2_𝑜
31	30	a1i 11	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑅) ≈ 2_𝑜)
32	1	zncrng 20087	. . . . . 6 ⊢ (2 ∈ ℕ₀ → 𝑅 ∈ CRing)
33		crngring 18750	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
34	19, 32, 33	mp2b 10	. . . . 5 ⊢ 𝑅 ∈ Ring
35	5, 6	ring0cl 18761	. . . . 5 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
36	34, 35	mp1i 13	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝑅) ∈ (Base‘𝑅))
37		2on0 7730	. . . . . 6 ⊢ 2_𝑜 ≠ ∅
38		2on 7729	. . . . . . 7 ⊢ 2_𝑜 ∈ On
39		on0eln0 5933	. . . . . . 7 ⊢ (2_𝑜 ∈ On → (∅ ∈ 2_𝑜 ↔ 2_𝑜 ≠ ∅))
40	38, 39	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ 2_𝑜 ↔ 2_𝑜 ≠ ∅)
41	37, 40	mpbir 221	. . . . 5 ⊢ ∅ ∈ 2_𝑜
42	41	a1i 11	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ 2_𝑜)
43	7, 12, 13, 31, 36, 42	mapfien2 8471	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} ≈ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅})
44	11, 43	eqbrtrrd 4820	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅})
45	12	pwfi2en 38161	. 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin))
46		entr 8165	. 2 ⊢ ((𝐵 ≈ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅} ∧ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin)) → 𝐵 ≈ (𝒫 𝐼 ∩ Fin))
47	44, 45, 46	syl2anc 696	1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ (𝒫 𝐼 ∩ Fin))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 = wceq 1624 ∈ wcel 2131 ≠ wne 2924 {crab 3046 Vcvv 3332 ∩ cin 3706 ∅c0 4050 𝒫 cpw 4294 class class class wbr 4796 Oncon0 5876 ‘cfv 6041 (class class class)co 6805 ωcom 7222 2_𝑜c2o 7715 ↑_𝑚 cmap 8015 ≈ cen 8110 Fincfn 8113 finSupp cfsupp 8432 ℕcn 11204 2c2 11254 ℕ₀cn0 11476 ♯chash 13303 Basecbs 16051 0_gc0g 16294 Ringcrg 18739 CRingccrg 18740 ℤ/nℤczn 20045 freeLMod cfrlm 20284

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-inf2 8703 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 ax-pre-sup 10198 ax-addf 10199 ax-mulf 10200

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-supp 7456 df-tpos 7513 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-2o 7722 df-oadd 7725 df-er 7903 df-ec 7905 df-qs 7909 df-map 8017 df-ixp 8067 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fsupp 8433 df-sup 8505 df-inf 8506 df-card 8947 df-cda 9174 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-div 10869 df-nn 11205 df-2 11263 df-3 11264 df-4 11265 df-5 11266 df-6 11267 df-7 11268 df-8 11269 df-9 11270 df-n0 11477 df-z 11562 df-dec 11678 df-uz 11872 df-rp 12018 df-fz 12512 df-fzo 12652 df-fl 12779 df-mod 12855 df-seq 12988 df-hash 13304 df-dvds 15175 df-struct 16053 df-ndx 16054 df-slot 16055 df-base 16057 df-sets 16058 df-ress 16059 df-plusg 16148 df-mulr 16149 df-starv 16150 df-sca 16151 df-vsca 16152 df-ip 16153 df-tset 16154 df-ple 16155 df-ds 16158 df-unif 16159 df-hom 16160 df-cco 16161 df-0g 16296 df-prds 16302 df-pws 16304 df-imas 16362 df-qus 16363 df-mgm 17435 df-sgrp 17477 df-mnd 17488 df-mhm 17528 df-grp 17618 df-minusg 17619 df-sbg 17620 df-mulg 17734 df-subg 17784 df-nsg 17785 df-eqg 17786 df-ghm 17851 df-cmn 18387 df-abl 18388 df-mgp 18682 df-ur 18694 df-ring 18741 df-cring 18742 df-oppr 18815 df-dvdsr 18833 df-rnghom 18909 df-subrg 18972 df-lmod 19059 df-lss 19127 df-lsp 19166 df-sra 19366 df-rgmod 19367 df-lidl 19368 df-rsp 19369 df-2idl 19426 df-cnfld 19941 df-zring 20013 df-zrh 20046 df-zn 20049 df-dsmm 20270 df-frlm 20285

This theorem is referenced by: isnumbasgrplem3 38169

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