frlmpwfi - Mathbox for Stefan O'Rear

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Theorem frlmpwfi 37668

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 6201	. . . . . 6 ⊢ (ℤ/nℤ‘2) ∈ V
3	1, 2	eqeltri 2697	. . . . 5 ⊢ 𝑅 ∈ V
4		frlmpwfi.y	. . . . . 6 ⊢ 𝑌 = (𝑅 freeLMod 𝐼)
5		eqid 2622	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2622	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
7		eqid 2622	. . . . . 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 20099	. . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = (Base‘𝑌))
9	3, 8	mpan 706	. . . 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 2674	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = 𝐵)
12		eqid 2622	. . . 4 ⊢ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅}
13		enrefg 7987	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ≈ 𝐼)
14		2nn 11185	. . . . . . . 8 ⊢ 2 ∈ ℕ
15	1, 5	znhash 19907	. . . . . . . 8 ⊢ (2 ∈ ℕ → (#‘(Base‘𝑅)) = 2)
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ (#‘(Base‘𝑅)) = 2
17		hash2 13193	. . . . . . 7 ⊢ (#‘2_𝑜) = 2
18	16, 17	eqtr4i 2647	. . . . . 6 ⊢ (#‘(Base‘𝑅)) = (#‘2_𝑜)
19		2nn0 11309	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
20	16, 19	eqeltri 2697	. . . . . . . 8 ⊢ (#‘(Base‘𝑅)) ∈ ℕ₀
21		fvex 6201	. . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V
22		hashclb 13149	. . . . . . . . 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 7720	. . . . . . . 8 ⊢ 2_𝑜 ∈ ω
26		nnfi 8153	. . . . . . . 8 ⊢ (2_𝑜 ∈ ω → 2_𝑜 ∈ Fin)
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ 2_𝑜 ∈ Fin
28		hashen 13135	. . . . . . 7 ⊢ (((Base‘𝑅) ∈ Fin ∧ 2_𝑜 ∈ Fin) → ((#‘(Base‘𝑅)) = (#‘2_𝑜) ↔ (Base‘𝑅) ≈ 2_𝑜))
29	24, 27, 28	mp2an 708	. . . . . 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 19893	. . . . . 6 ⊢ (2 ∈ ℕ₀ → 𝑅 ∈ CRing)
33		crngring 18558	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
34	19, 32, 33	mp2b 10	. . . . 5 ⊢ 𝑅 ∈ Ring
35	5, 6	ring0cl 18569	. . . . 5 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
36	34, 35	mp1i 13	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝑅) ∈ (Base‘𝑅))
37		2on0 7569	. . . . . 6 ⊢ 2_𝑜 ≠ ∅
38		2on 7568	. . . . . . 7 ⊢ 2_𝑜 ∈ On
39		on0eln0 5780	. . . . . . 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 8314	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} ≈ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅})
44	11, 43	eqbrtrrd 4677	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅})
45	12	pwfi2en 37667	. 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin))
46		entr 8008	. 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 693	1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ (𝒫 𝐼 ∩ Fin))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {crab 2916 Vcvv 3200 ∩ cin 3573 ∅c0 3915 𝒫 cpw 4158 class class class wbr 4653 Oncon0 5723 ‘cfv 5888 (class class class)co 6650 ωcom 7065 2_𝑜c2o 7554 ↑_𝑚 cmap 7857 ≈ cen 7952 Fincfn 7955 finSupp cfsupp 8275 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 #chash 13117 Basecbs 15857 0_gc0g 16100 Ringcrg 18547 CRingccrg 18548 ℤ/nℤczn 19851 freeLMod cfrlm 20090

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-inf 8349 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-hash 13118 df-dvds 14984 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-0g 16102 df-prds 16108 df-pws 16110 df-imas 16168 df-qus 16169 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-nsg 17592 df-eqg 17593 df-ghm 17658 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-rnghom 18715 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-sra 19172 df-rgmod 19173 df-lidl 19174 df-rsp 19175 df-2idl 19232 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-zn 19855 df-dsmm 20076 df-frlm 20091

This theorem is referenced by: isnumbasgrplem3 37675

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