frlmpwfi - Mathbox for Stefan O'Rear

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Theorem frlmpwfi 43055

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	1	fvexi 6934	. . . . 5 ⊢ 𝑅 ∈ V
3		frlmpwfi.y	. . . . . 6 ⊢ 𝑌 = (𝑅 freeLMod 𝐼)
4		eqid 2740	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2740	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
6		eqid 2740	. . . . . 6 ⊢ {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)}
7	3, 4, 5, 6	frlmbas 21798	. . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = (Base‘𝑌))
8	2, 7	mpan 689	. . . 4 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = (Base‘𝑌))
9		frlmpwfi.b	. . . 4 ⊢ 𝐵 = (Base‘𝑌)
10	8, 9	eqtr4di 2798	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = 𝐵)
11		eqid 2740	. . . 4 ⊢ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅}
12		enrefg 9044	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ≈ 𝐼)
13		2nn 12366	. . . . . . . 8 ⊢ 2 ∈ ℕ
14	1, 4	znhash 21600	. . . . . . . 8 ⊢ (2 ∈ ℕ → (♯‘(Base‘𝑅)) = 2)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (♯‘(Base‘𝑅)) = 2
16		hash2 14454	. . . . . . 7 ⊢ (♯‘2_o) = 2
17	15, 16	eqtr4i 2771	. . . . . 6 ⊢ (♯‘(Base‘𝑅)) = (♯‘2_o)
18		2nn0 12570	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
19	15, 18	eqeltri 2840	. . . . . . . 8 ⊢ (♯‘(Base‘𝑅)) ∈ ℕ₀
20		fvex 6933	. . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V
21		hashclb 14407	. . . . . . . . 9 ⊢ ((Base‘𝑅) ∈ V → ((Base‘𝑅) ∈ Fin ↔ (♯‘(Base‘𝑅)) ∈ ℕ₀))
22	20, 21	ax-mp 5	. . . . . . . 8 ⊢ ((Base‘𝑅) ∈ Fin ↔ (♯‘(Base‘𝑅)) ∈ ℕ₀)
23	19, 22	mpbir 231	. . . . . . 7 ⊢ (Base‘𝑅) ∈ Fin
24		2onn 8698	. . . . . . . 8 ⊢ 2_o ∈ ω
25		nnfi 9233	. . . . . . . 8 ⊢ (2_o ∈ ω → 2_o ∈ Fin)
26	24, 25	ax-mp 5	. . . . . . 7 ⊢ 2_o ∈ Fin
27		hashen 14396	. . . . . . 7 ⊢ (((Base‘𝑅) ∈ Fin ∧ 2_o ∈ Fin) → ((♯‘(Base‘𝑅)) = (♯‘2_o) ↔ (Base‘𝑅) ≈ 2_o))
28	23, 26, 27	mp2an 691	. . . . . 6 ⊢ ((♯‘(Base‘𝑅)) = (♯‘2_o) ↔ (Base‘𝑅) ≈ 2_o)
29	17, 28	mpbi 230	. . . . 5 ⊢ (Base‘𝑅) ≈ 2_o
30	29	a1i 11	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑅) ≈ 2_o)
31	1	zncrng 21586	. . . . . 6 ⊢ (2 ∈ ℕ₀ → 𝑅 ∈ CRing)
32		crngring 20272	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
33	18, 31, 32	mp2b 10	. . . . 5 ⊢ 𝑅 ∈ Ring
34	4, 5	ring0cl 20290	. . . . 5 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
35	33, 34	mp1i 13	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝑅) ∈ (Base‘𝑅))
36		2on0 8538	. . . . . 6 ⊢ 2_o ≠ ∅
37		2on 8536	. . . . . . 7 ⊢ 2_o ∈ On
38		on0eln0 6451	. . . . . . 7 ⊢ (2_o ∈ On → (∅ ∈ 2_o ↔ 2_o ≠ ∅))
39	37, 38	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ 2_o ↔ 2_o ≠ ∅)
40	36, 39	mpbir 231	. . . . 5 ⊢ ∅ ∈ 2_o
41	40	a1i 11	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ 2_o)
42	6, 11, 12, 30, 35, 41	mapfien2 9478	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} ≈ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅})
43	10, 42	eqbrtrrd 5190	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅})
44	11	pwfi2en 43054	. 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin))
45		entr 9066	. 2 ⊢ ((𝐵 ≈ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅} ∧ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin)) → 𝐵 ≈ (𝒫 𝐼 ∩ Fin))
46	43, 44, 45	syl2anc 583	1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ (𝒫 𝐼 ∩ Fin))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 {crab 3443 Vcvv 3488 ∩ cin 3975 ∅c0 4352 𝒫 cpw 4622 class class class wbr 5166 Oncon0 6395 ‘cfv 6573 (class class class)co 7448 ωcom 7903 2_oc2o 8516 ↑_m cmap 8884 ≈ cen 9000 Fincfn 9003 finSupp cfsupp 9431 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ♯chash 14379 Basecbs 17258 0_gc0g 17499 Ringcrg 20260 CRingccrg 20261 ℤ/nℤczn 21536 freeLMod cfrlm 21789

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-ec 8765 df-qs 8769 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-inf 9512 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-hash 14380 df-dvds 16303 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-0g 17501 df-prds 17507 df-pws 17509 df-imas 17568 df-qus 17569 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-nsg 19164 df-eqg 19165 df-ghm 19253 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-lmod 20882 df-lss 20953 df-lsp 20993 df-sra 21195 df-rgmod 21196 df-lidl 21241 df-rsp 21242 df-2idl 21283 df-cnfld 21388 df-zring 21481 df-zrh 21537 df-zn 21540 df-dsmm 21775 df-frlm 21790

This theorem is referenced by: isnumbasgrplem3 43062

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