frlmpwfi - Mathbox for Stefan O'Rear

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Theorem frlmpwfi 41548

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 6887	. . . . 5 ⊢ 𝑅 ∈ V
3		frlmpwfi.y	. . . . . 6 ⊢ 𝑌 = (𝑅 freeLMod 𝐼)
4		eqid 2731	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2731	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
6		eqid 2731	. . . . . 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 21236	. . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = (Base‘𝑌))
8	2, 7	mpan 688	. . . 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 2789	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = 𝐵)
11		eqid 2731	. . . 4 ⊢ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅}
12		enrefg 8958	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ≈ 𝐼)
13		2nn 12262	. . . . . . . 8 ⊢ 2 ∈ ℕ
14	1, 4	znhash 21040	. . . . . . . 8 ⊢ (2 ∈ ℕ → (♯‘(Base‘𝑅)) = 2)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (♯‘(Base‘𝑅)) = 2
16		hash2 14342	. . . . . . 7 ⊢ (♯‘2_o) = 2
17	15, 16	eqtr4i 2762	. . . . . 6 ⊢ (♯‘(Base‘𝑅)) = (♯‘2_o)
18		2nn0 12466	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
19	15, 18	eqeltri 2828	. . . . . . . 8 ⊢ (♯‘(Base‘𝑅)) ∈ ℕ₀
20		fvex 6886	. . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V
21		hashclb 14295	. . . . . . . . 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 230	. . . . . . 7 ⊢ (Base‘𝑅) ∈ Fin
24		2onn 8619	. . . . . . . 8 ⊢ 2_o ∈ ω
25		nnfi 9145	. . . . . . . 8 ⊢ (2_o ∈ ω → 2_o ∈ Fin)
26	24, 25	ax-mp 5	. . . . . . 7 ⊢ 2_o ∈ Fin
27		hashen 14284	. . . . . . 7 ⊢ (((Base‘𝑅) ∈ Fin ∧ 2_o ∈ Fin) → ((♯‘(Base‘𝑅)) = (♯‘2_o) ↔ (Base‘𝑅) ≈ 2_o))
28	23, 26, 27	mp2an 690	. . . . . 6 ⊢ ((♯‘(Base‘𝑅)) = (♯‘2_o) ↔ (Base‘𝑅) ≈ 2_o)
29	17, 28	mpbi 229	. . . . 5 ⊢ (Base‘𝑅) ≈ 2_o
30	29	a1i 11	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑅) ≈ 2_o)
31	1	zncrng 21026	. . . . . 6 ⊢ (2 ∈ ℕ₀ → 𝑅 ∈ CRing)
32		crngring 20021	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
33	18, 31, 32	mp2b 10	. . . . 5 ⊢ 𝑅 ∈ Ring
34	4, 5	ring0cl 20036	. . . . 5 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
35	33, 34	mp1i 13	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝑅) ∈ (Base‘𝑅))
36		2on0 8459	. . . . . 6 ⊢ 2_o ≠ ∅
37		2on 8457	. . . . . . 7 ⊢ 2_o ∈ On
38		on0eln0 6404	. . . . . . 7 ⊢ (2_o ∈ On → (∅ ∈ 2_o ↔ 2_o ≠ ∅))
39	37, 38	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ 2_o ↔ 2_o ≠ ∅)
40	36, 39	mpbir 230	. . . . 5 ⊢ ∅ ∈ 2_o
41	40	a1i 11	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ 2_o)
42	6, 11, 12, 30, 35, 41	mapfien2 9381	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} ≈ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅})
43	10, 42	eqbrtrrd 5160	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅})
44	11	pwfi2en 41547	. 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin))
45		entr 8980	. 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 205 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 {crab 3428 Vcvv 3469 ∩ cin 3938 ∅c0 4313 𝒫 cpw 4591 class class class wbr 5136 Oncon0 6348 ‘cfv 6527 (class class class)co 7388 ωcom 7833 2_oc2o 8437 ↑_m cmap 8798 ≈ cen 8914 Fincfn 8917 finSupp cfsupp 9339 ℕcn 12189 2c2 12244 ℕ₀cn0 12449 ♯chash 14267 Basecbs 17121 0_gc0g 17362 Ringcrg 20009 CRingccrg 20010 ℤ/nℤczn 20978 freeLMod cfrlm 21227

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5273 ax-sep 5287 ax-nul 5294 ax-pow 5351 ax-pr 5415 ax-un 7703 ax-cnex 11143 ax-resscn 11144 ax-1cn 11145 ax-icn 11146 ax-addcl 11147 ax-addrcl 11148 ax-mulcl 11149 ax-mulrcl 11150 ax-mulcom 11151 ax-addass 11152 ax-mulass 11153 ax-distr 11154 ax-i2m1 11155 ax-1ne0 11156 ax-1rid 11157 ax-rnegex 11158 ax-rrecex 11159 ax-cnre 11160 ax-pre-lttri 11161 ax-pre-lttrn 11162 ax-pre-ltadd 11163 ax-pre-mulgt0 11164 ax-pre-sup 11165 ax-addf 11166 ax-mulf 11167

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3374 df-reu 3375 df-rab 3429 df-v 3471 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4314 df-if 4518 df-pw 4593 df-sn 4618 df-pr 4620 df-tp 4622 df-op 4624 df-uni 4897 df-int 4939 df-iun 4987 df-br 5137 df-opab 5199 df-mpt 5220 df-tr 5254 df-id 5562 df-eprel 5568 df-po 5576 df-so 5577 df-fr 5619 df-we 5621 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6284 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7344 df-ov 7391 df-oprab 7392 df-mpo 7393 df-om 7834 df-1st 7952 df-2nd 7953 df-supp 8124 df-tpos 8188 df-frecs 8243 df-wrecs 8274 df-recs 8348 df-rdg 8387 df-1o 8443 df-2o 8444 df-oadd 8447 df-er 8681 df-ec 8683 df-qs 8687 df-map 8800 df-ixp 8870 df-en 8918 df-dom 8919 df-sdom 8920 df-fin 8921 df-fsupp 9340 df-sup 9414 df-inf 9415 df-dju 9873 df-card 9911 df-pnf 11227 df-mnf 11228 df-xr 11229 df-ltxr 11230 df-le 11231 df-sub 11423 df-neg 11424 df-div 11849 df-nn 12190 df-2 12252 df-3 12253 df-4 12254 df-5 12255 df-6 12256 df-7 12257 df-8 12258 df-9 12259 df-n0 12450 df-z 12536 df-dec 12655 df-uz 12800 df-rp 12952 df-fz 13462 df-fzo 13605 df-fl 13734 df-mod 13812 df-seq 13944 df-hash 14268 df-dvds 16175 df-struct 17057 df-sets 17074 df-slot 17092 df-ndx 17104 df-base 17122 df-ress 17151 df-plusg 17187 df-mulr 17188 df-starv 17189 df-sca 17190 df-vsca 17191 df-ip 17192 df-tset 17193 df-ple 17194 df-ds 17196 df-unif 17197 df-hom 17198 df-cco 17199 df-0g 17364 df-prds 17370 df-pws 17372 df-imas 17431 df-qus 17432 df-mgm 18538 df-sgrp 18587 df-mnd 18598 df-mhm 18642 df-grp 18792 df-minusg 18793 df-sbg 18794 df-mulg 18918 df-subg 18970 df-nsg 18971 df-eqg 18972 df-ghm 19051 df-cmn 19609 df-abl 19610 df-mgp 19942 df-ur 19959 df-ring 20011 df-cring 20012 df-oppr 20097 df-dvdsr 20118 df-rnghom 20196 df-subrg 20303 df-lmod 20415 df-lss 20485 df-lsp 20525 df-sra 20727 df-rgmod 20728 df-lidl 20729 df-rsp 20730 df-2idl 20796 df-cnfld 20872 df-zring 20945 df-zrh 20979 df-zn 20982 df-dsmm 21213 df-frlm 21228

This theorem is referenced by: isnumbasgrplem3 41555

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