frlmpwfi - Mathbox for Stefan O'Rear

	Mathbox for Stefan O'Rear		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > frlmpwfi		Structured version Visualization version GIF version

Theorem frlmpwfi 43377

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 6847	. . . . 5 ⊢ 𝑅 ∈ V
3		frlmpwfi.y	. . . . . 6 ⊢ 𝑌 = (𝑅 freeLMod 𝐼)
4		eqid 2735	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2735	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
6		eqid 2735	. . . . . 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 21712	. . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = (Base‘𝑌))
8	2, 7	mpan 691	. . . 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 2788	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = 𝐵)
11		eqid 2735	. . . 4 ⊢ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅}
12		enrefg 8923	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ≈ 𝐼)
13		2nn 12220	. . . . . . . 8 ⊢ 2 ∈ ℕ
14	1, 4	znhash 21515	. . . . . . . 8 ⊢ (2 ∈ ℕ → (♯‘(Base‘𝑅)) = 2)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (♯‘(Base‘𝑅)) = 2
16		hash2 14330	. . . . . . 7 ⊢ (♯‘2_o) = 2
17	15, 16	eqtr4i 2761	. . . . . 6 ⊢ (♯‘(Base‘𝑅)) = (♯‘2_o)
18		2nn0 12420	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
19	15, 18	eqeltri 2831	. . . . . . . 8 ⊢ (♯‘(Base‘𝑅)) ∈ ℕ₀
20		fvex 6846	. . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V
21		hashclb 14283	. . . . . . . . 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 8570	. . . . . . . 8 ⊢ 2_o ∈ ω
25		nnfi 9094	. . . . . . . 8 ⊢ (2_o ∈ ω → 2_o ∈ Fin)
26	24, 25	ax-mp 5	. . . . . . 7 ⊢ 2_o ∈ Fin
27		hashen 14272	. . . . . . 7 ⊢ (((Base‘𝑅) ∈ Fin ∧ 2_o ∈ Fin) → ((♯‘(Base‘𝑅)) = (♯‘2_o) ↔ (Base‘𝑅) ≈ 2_o))
28	23, 26, 27	mp2an 693	. . . . . 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 21501	. . . . . 6 ⊢ (2 ∈ ℕ₀ → 𝑅 ∈ CRing)
32		crngring 20182	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
33	18, 31, 32	mp2b 10	. . . . 5 ⊢ 𝑅 ∈ Ring
34	4, 5	ring0cl 20204	. . . . 5 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
35	33, 34	mp1i 13	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝑅) ∈ (Base‘𝑅))
36		2on0 8411	. . . . . 6 ⊢ 2_o ≠ ∅
37		2on 8410	. . . . . . 7 ⊢ 2_o ∈ On
38		on0eln0 6373	. . . . . . 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 9314	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} ≈ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅})
43	10, 42	eqbrtrrd 5121	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅})
44	11	pwfi2en 43376	. 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin))
45		entr 8945	. 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 585	1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ (𝒫 𝐼 ∩ Fin))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1542 ∈ wcel 2114 ≠ wne 2931 {crab 3398 Vcvv 3439 ∩ cin 3899 ∅c0 4284 𝒫 cpw 4553 class class class wbr 5097 Oncon0 6316 ‘cfv 6491 (class class class)co 7358 ωcom 7808 2_oc2o 8391 ↑_m cmap 8765 ≈ cen 8882 Fincfn 8885 finSupp cfsupp 9266 ℕcn 12147 2c2 12202 ℕ₀cn0 12403 ♯chash 14255 Basecbs 17138 0_gc0g 17361 Ringcrg 20170 CRingccrg 20171 ℤ/nℤczn 21459 freeLMod cfrlm 21703

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-mulf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-ec 8637 df-qs 8641 df-map 8767 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-sup 9347 df-inf 9348 df-dju 9815 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-rp 12908 df-fz 13426 df-fzo 13573 df-fl 13714 df-mod 13792 df-seq 13927 df-hash 14256 df-dvds 16182 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-0g 17363 df-prds 17369 df-pws 17371 df-imas 17431 df-qus 17432 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-nsg 19056 df-eqg 19057 df-ghm 19144 df-cmn 19713 df-abl 19714 df-mgp 20078 df-rng 20090 df-ur 20119 df-ring 20172 df-cring 20173 df-oppr 20275 df-dvdsr 20295 df-rhm 20410 df-subrng 20481 df-subrg 20505 df-lmod 20815 df-lss 20885 df-lsp 20925 df-sra 21127 df-rgmod 21128 df-lidl 21165 df-rsp 21166 df-2idl 21207 df-cnfld 21312 df-zring 21404 df-zrh 21460 df-zn 21463 df-dsmm 21689 df-frlm 21704

This theorem is referenced by: isnumbasgrplem3 43384

W3C validator