frlmpwfi - Mathbox for Stefan O'Rear

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Theorem frlmpwfi 43080

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 6854	. . . . 5 ⊢ 𝑅 ∈ V
3		frlmpwfi.y	. . . . . 6 ⊢ 𝑌 = (𝑅 freeLMod 𝐼)
4		eqid 2729	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2729	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
6		eqid 2729	. . . . . 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 21697	. . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = (Base‘𝑌))
8	2, 7	mpan 690	. . . 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 2782	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = 𝐵)
11		eqid 2729	. . . 4 ⊢ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅}
12		enrefg 8932	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ≈ 𝐼)
13		2nn 12235	. . . . . . . 8 ⊢ 2 ∈ ℕ
14	1, 4	znhash 21500	. . . . . . . 8 ⊢ (2 ∈ ℕ → (♯‘(Base‘𝑅)) = 2)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (♯‘(Base‘𝑅)) = 2
16		hash2 14346	. . . . . . 7 ⊢ (♯‘2_o) = 2
17	15, 16	eqtr4i 2755	. . . . . 6 ⊢ (♯‘(Base‘𝑅)) = (♯‘2_o)
18		2nn0 12435	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
19	15, 18	eqeltri 2824	. . . . . . . 8 ⊢ (♯‘(Base‘𝑅)) ∈ ℕ₀
20		fvex 6853	. . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V
21		hashclb 14299	. . . . . . . . 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 8583	. . . . . . . 8 ⊢ 2_o ∈ ω
25		nnfi 9108	. . . . . . . 8 ⊢ (2_o ∈ ω → 2_o ∈ Fin)
26	24, 25	ax-mp 5	. . . . . . 7 ⊢ 2_o ∈ Fin
27		hashen 14288	. . . . . . 7 ⊢ (((Base‘𝑅) ∈ Fin ∧ 2_o ∈ Fin) → ((♯‘(Base‘𝑅)) = (♯‘2_o) ↔ (Base‘𝑅) ≈ 2_o))
28	23, 26, 27	mp2an 692	. . . . . 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 21486	. . . . . 6 ⊢ (2 ∈ ℕ₀ → 𝑅 ∈ CRing)
32		crngring 20165	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
33	18, 31, 32	mp2b 10	. . . . 5 ⊢ 𝑅 ∈ Ring
34	4, 5	ring0cl 20187	. . . . 5 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
35	33, 34	mp1i 13	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝑅) ∈ (Base‘𝑅))
36		2on0 8425	. . . . . 6 ⊢ 2_o ≠ ∅
37		2on 8424	. . . . . . 7 ⊢ 2_o ∈ On
38		on0eln0 6377	. . . . . . 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 9336	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} ≈ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅})
43	10, 42	eqbrtrrd 5126	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅})
44	11	pwfi2en 43079	. 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin))
45		entr 8954	. 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 584	1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ (𝒫 𝐼 ∩ Fin))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3402 Vcvv 3444 ∩ cin 3910 ∅c0 4292 𝒫 cpw 4559 class class class wbr 5102 Oncon0 6320 ‘cfv 6499 (class class class)co 7369 ωcom 7822 2_oc2o 8405 ↑_m cmap 8776 ≈ cen 8892 Fincfn 8895 finSupp cfsupp 9288 ℕcn 12162 2c2 12217 ℕ₀cn0 12418 ♯chash 14271 Basecbs 17155 0_gc0g 17378 Ringcrg 20153 CRingccrg 20154 ℤ/nℤczn 21444 freeLMod cfrlm 21688

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-ec 8650 df-qs 8654 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-sup 9369 df-inf 9370 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-rp 12928 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-hash 14272 df-dvds 16199 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-0g 17380 df-prds 17386 df-pws 17388 df-imas 17447 df-qus 17448 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-grp 18850 df-minusg 18851 df-sbg 18852 df-mulg 18982 df-subg 19037 df-nsg 19038 df-eqg 19039 df-ghm 19127 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-cring 20156 df-oppr 20257 df-dvdsr 20277 df-rhm 20392 df-subrng 20466 df-subrg 20490 df-lmod 20800 df-lss 20870 df-lsp 20910 df-sra 21112 df-rgmod 21113 df-lidl 21150 df-rsp 21151 df-2idl 21192 df-cnfld 21297 df-zring 21389 df-zrh 21445 df-zn 21448 df-dsmm 21674 df-frlm 21689

This theorem is referenced by: isnumbasgrplem3 43087

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