frlmpwfi - Mathbox for Stefan O'Rear

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Theorem frlmpwfi 42143

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 6905	. . . . 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 21530	. . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = (Base‘𝑌))
8	2, 7	mpan 687	. . . 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 8984	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ≈ 𝐼)
13		2nn 12290	. . . . . . . 8 ⊢ 2 ∈ ℕ
14	1, 4	znhash 21334	. . . . . . . 8 ⊢ (2 ∈ ℕ → (♯‘(Base‘𝑅)) = 2)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (♯‘(Base‘𝑅)) = 2
16		hash2 14370	. . . . . . 7 ⊢ (♯‘2_o) = 2
17	15, 16	eqtr4i 2762	. . . . . 6 ⊢ (♯‘(Base‘𝑅)) = (♯‘2_o)
18		2nn0 12494	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
19	15, 18	eqeltri 2828	. . . . . . . 8 ⊢ (♯‘(Base‘𝑅)) ∈ ℕ₀
20		fvex 6904	. . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V
21		hashclb 14323	. . . . . . . . 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 8645	. . . . . . . 8 ⊢ 2_o ∈ ω
25		nnfi 9171	. . . . . . . 8 ⊢ (2_o ∈ ω → 2_o ∈ Fin)
26	24, 25	ax-mp 5	. . . . . . 7 ⊢ 2_o ∈ Fin
27		hashen 14312	. . . . . . 7 ⊢ (((Base‘𝑅) ∈ Fin ∧ 2_o ∈ Fin) → ((♯‘(Base‘𝑅)) = (♯‘2_o) ↔ (Base‘𝑅) ≈ 2_o))
28	23, 26, 27	mp2an 689	. . . . . 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 21320	. . . . . 6 ⊢ (2 ∈ ℕ₀ → 𝑅 ∈ CRing)
32		crngring 20140	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
33	18, 31, 32	mp2b 10	. . . . 5 ⊢ 𝑅 ∈ Ring
34	4, 5	ring0cl 20156	. . . . 5 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
35	33, 34	mp1i 13	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝑅) ∈ (Base‘𝑅))
36		2on0 8486	. . . . . 6 ⊢ 2_o ≠ ∅
37		2on 8484	. . . . . . 7 ⊢ 2_o ∈ On
38		on0eln0 6420	. . . . . . 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 9408	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_m 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} ≈ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅})
43	10, 42	eqbrtrrd 5172	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅})
44	11	pwfi2en 42142	. 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2_o ↑_m 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin))
45		entr 9006	. 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 205 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 {crab 3431 Vcvv 3473 ∩ cin 3947 ∅c0 4322 𝒫 cpw 4602 class class class wbr 5148 Oncon0 6364 ‘cfv 6543 (class class class)co 7412 ωcom 7859 2_oc2o 8464 ↑_m cmap 8824 ≈ cen 8940 Fincfn 8943 finSupp cfsupp 9365 ℕcn 12217 2c2 12272 ℕ₀cn0 12477 ♯chash 14295 Basecbs 17149 0_gc0g 17390 Ringcrg 20128 CRingccrg 20129 ℤ/nℤczn 21272 freeLMod cfrlm 21521

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-er 8707 df-ec 8709 df-qs 8713 df-map 8826 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-sup 9441 df-inf 9442 df-dju 9900 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-hash 14296 df-dvds 16203 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-0g 17392 df-prds 17398 df-pws 17400 df-imas 17459 df-qus 17460 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-nsg 19041 df-eqg 19042 df-ghm 19129 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-rhm 20364 df-subrng 20435 df-subrg 20460 df-lmod 20617 df-lss 20688 df-lsp 20728 df-sra 20931 df-rgmod 20932 df-lidl 20933 df-rsp 20934 df-2idl 21007 df-cnfld 21146 df-zring 21219 df-zrh 21273 df-zn 21276 df-dsmm 21507 df-frlm 21522

This theorem is referenced by: isnumbasgrplem3 42150

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