frlmpwfi - Mathbox for Stefan O'Rear

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Theorem frlmpwfi 38189

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 6364	. . . . . 6 ⊢ (ℤ/nℤ‘2) ∈ V
3	1, 2	eqeltri 2836	. . . . 5 ⊢ 𝑅 ∈ V
4		frlmpwfi.y	. . . . . 6 ⊢ 𝑌 = (𝑅 freeLMod 𝐼)
5		eqid 2761	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2761	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
7		eqid 2761	. . . . . 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 20322	. . . . 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 2813	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = 𝐵)
12		eqid 2761	. . . 4 ⊢ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅}
13		enrefg 8156	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ≈ 𝐼)
14		2nn 11398	. . . . . . . 8 ⊢ 2 ∈ ℕ
15	1, 5	znhash 20130	. . . . . . . 8 ⊢ (2 ∈ ℕ → (♯‘(Base‘𝑅)) = 2)
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ (♯‘(Base‘𝑅)) = 2
17		hash2 13406	. . . . . . 7 ⊢ (♯‘2_𝑜) = 2
18	16, 17	eqtr4i 2786	. . . . . 6 ⊢ (♯‘(Base‘𝑅)) = (♯‘2_𝑜)
19		2nn0 11522	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
20	16, 19	eqeltri 2836	. . . . . . . 8 ⊢ (♯‘(Base‘𝑅)) ∈ ℕ₀
21		fvex 6364	. . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V
22		hashclb 13362	. . . . . . . . 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 7892	. . . . . . . 8 ⊢ 2_𝑜 ∈ ω
26		nnfi 8321	. . . . . . . 8 ⊢ (2_𝑜 ∈ ω → 2_𝑜 ∈ Fin)
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ 2_𝑜 ∈ Fin
28		hashen 13350	. . . . . . 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 20116	. . . . . 6 ⊢ (2 ∈ ℕ₀ → 𝑅 ∈ CRing)
33		crngring 18779	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
34	19, 32, 33	mp2b 10	. . . . 5 ⊢ 𝑅 ∈ Ring
35	5, 6	ring0cl 18790	. . . . 5 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
36	34, 35	mp1i 13	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝑅) ∈ (Base‘𝑅))
37		2on0 7741	. . . . . 6 ⊢ 2_𝑜 ≠ ∅
38		2on 7740	. . . . . . 7 ⊢ 2_𝑜 ∈ On
39		on0eln0 5942	. . . . . . 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 8482	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} ≈ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅})
44	11, 43	eqbrtrrd 4829	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅})
45	12	pwfi2en 38188	. 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin))
46		entr 8176	. 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 1632 ∈ wcel 2140 ≠ wne 2933 {crab 3055 Vcvv 3341 ∩ cin 3715 ∅c0 4059 𝒫 cpw 4303 class class class wbr 4805 Oncon0 5885 ‘cfv 6050 (class class class)co 6815 ωcom 7232 2_𝑜c2o 7725 ↑_𝑚 cmap 8026 ≈ cen 8121 Fincfn 8124 finSupp cfsupp 8443 ℕcn 11233 2c2 11283 ℕ₀cn0 11505 ♯chash 13332 Basecbs 16080 0_gc0g 16323 Ringcrg 18768 CRingccrg 18769 ℤ/nℤczn 20074 freeLMod cfrlm 20313

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-rep 4924 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-inf2 8714 ax-cnex 10205 ax-resscn 10206 ax-1cn 10207 ax-icn 10208 ax-addcl 10209 ax-addrcl 10210 ax-mulcl 10211 ax-mulrcl 10212 ax-mulcom 10213 ax-addass 10214 ax-mulass 10215 ax-distr 10216 ax-i2m1 10217 ax-1ne0 10218 ax-1rid 10219 ax-rnegex 10220 ax-rrecex 10221 ax-cnre 10222 ax-pre-lttri 10223 ax-pre-lttrn 10224 ax-pre-ltadd 10225 ax-pre-mulgt0 10226 ax-pre-sup 10227 ax-addf 10228 ax-mulf 10229

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-int 4629 df-iun 4675 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-riota 6776 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-om 7233 df-1st 7335 df-2nd 7336 df-supp 7466 df-tpos 7523 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-1o 7731 df-2o 7732 df-oadd 7735 df-er 7914 df-ec 7916 df-qs 7920 df-map 8028 df-ixp 8078 df-en 8125 df-dom 8126 df-sdom 8127 df-fin 8128 df-fsupp 8444 df-sup 8516 df-inf 8517 df-card 8976 df-cda 9203 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293 df-sub 10481 df-neg 10482 df-div 10898 df-nn 11234 df-2 11292 df-3 11293 df-4 11294 df-5 11295 df-6 11296 df-7 11297 df-8 11298 df-9 11299 df-n0 11506 df-z 11591 df-dec 11707 df-uz 11901 df-rp 12047 df-fz 12541 df-fzo 12681 df-fl 12808 df-mod 12884 df-seq 13017 df-hash 13333 df-dvds 15204 df-struct 16082 df-ndx 16083 df-slot 16084 df-base 16086 df-sets 16087 df-ress 16088 df-plusg 16177 df-mulr 16178 df-starv 16179 df-sca 16180 df-vsca 16181 df-ip 16182 df-tset 16183 df-ple 16184 df-ds 16187 df-unif 16188 df-hom 16189 df-cco 16190 df-0g 16325 df-prds 16331 df-pws 16333 df-imas 16391 df-qus 16392 df-mgm 17464 df-sgrp 17506 df-mnd 17517 df-mhm 17557 df-grp 17647 df-minusg 17648 df-sbg 17649 df-mulg 17763 df-subg 17813 df-nsg 17814 df-eqg 17815 df-ghm 17880 df-cmn 18416 df-abl 18417 df-mgp 18711 df-ur 18723 df-ring 18770 df-cring 18771 df-oppr 18844 df-dvdsr 18862 df-rnghom 18938 df-subrg 19001 df-lmod 19088 df-lss 19156 df-lsp 19195 df-sra 19395 df-rgmod 19396 df-lidl 19397 df-rsp 19398 df-2idl 19455 df-cnfld 19970 df-zring 20042 df-zrh 20075 df-zn 20078 df-dsmm 20299 df-frlm 20314

This theorem is referenced by: isnumbasgrplem3 38196

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