frlmpwfi - Mathbox for Stefan O'Rear

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Theorem frlmpwfi 36686

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 6113	. . . . . 6 ⊢ (ℤ/nℤ‘2) ∈ V
3	1, 2	eqeltri 2684	. . . . 5 ⊢ 𝑅 ∈ V
4		frlmpwfi.y	. . . . . 6 ⊢ 𝑌 = (𝑅 freeLMod 𝐼)
5		eqid 2610	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2610	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
7		eqid 2610	. . . . . 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 19918	. . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = (Base‘𝑌))
9	3, 8	mpan 702	. . . 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 2662	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} = 𝐵)
12		eqid 2610	. . . 4 ⊢ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅}
13		enrefg 7873	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ≈ 𝐼)
14		2nn 11062	. . . . . . . 8 ⊢ 2 ∈ ℕ
15	1, 5	znhash 19726	. . . . . . . 8 ⊢ (2 ∈ ℕ → (#‘(Base‘𝑅)) = 2)
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ (#‘(Base‘𝑅)) = 2
17		hash2 13054	. . . . . . 7 ⊢ (#‘2_𝑜) = 2
18	16, 17	eqtr4i 2635	. . . . . 6 ⊢ (#‘(Base‘𝑅)) = (#‘2_𝑜)
19		2nn0 11186	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
20	16, 19	eqeltri 2684	. . . . . . . 8 ⊢ (#‘(Base‘𝑅)) ∈ ℕ₀
21		fvex 6113	. . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V
22		hashclb 13011	. . . . . . . . 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 220	. . . . . . 7 ⊢ (Base‘𝑅) ∈ Fin
25		2onn 7607	. . . . . . . 8 ⊢ 2_𝑜 ∈ ω
26		nnfi 8038	. . . . . . . 8 ⊢ (2_𝑜 ∈ ω → 2_𝑜 ∈ Fin)
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ 2_𝑜 ∈ Fin
28		hashen 12997	. . . . . . 7 ⊢ (((Base‘𝑅) ∈ Fin ∧ 2_𝑜 ∈ Fin) → ((#‘(Base‘𝑅)) = (#‘2_𝑜) ↔ (Base‘𝑅) ≈ 2_𝑜))
29	24, 27, 28	mp2an 704	. . . . . 6 ⊢ ((#‘(Base‘𝑅)) = (#‘2_𝑜) ↔ (Base‘𝑅) ≈ 2_𝑜)
30	18, 29	mpbi 219	. . . . 5 ⊢ (Base‘𝑅) ≈ 2_𝑜
31	30	a1i 11	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑅) ≈ 2_𝑜)
32	1	zncrng 19712	. . . . . 6 ⊢ (2 ∈ ℕ₀ → 𝑅 ∈ CRing)
33		crngring 18381	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
34	19, 32, 33	mp2b 10	. . . . 5 ⊢ 𝑅 ∈ Ring
35	5, 6	ring0cl 18392	. . . . 5 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
36	34, 35	mp1i 13	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝑅) ∈ (Base‘𝑅))
37		2on0 7456	. . . . . 6 ⊢ 2_𝑜 ≠ ∅
38		2on 7455	. . . . . . 7 ⊢ 2_𝑜 ∈ On
39		on0eln0 5697	. . . . . . 7 ⊢ (2_𝑜 ∈ On → (∅ ∈ 2_𝑜 ↔ 2_𝑜 ≠ ∅))
40	38, 39	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ 2_𝑜 ↔ 2_𝑜 ≠ ∅)
41	37, 40	mpbir 220	. . . . 5 ⊢ ∅ ∈ 2_𝑜
42	41	a1i 11	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ 2_𝑜)
43	7, 12, 13, 31, 36, 42	mapfien2 8197	. . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑_𝑚 𝐼) ∣ 𝑥 finSupp (0_g‘𝑅)} ≈ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅})
44	11, 43	eqbrtrrd 4607	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅})
45	12	pwfi2en 36685	. 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2_𝑜 ↑_𝑚 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin))
46		entr 7894	. 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 691	1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ (𝒫 𝐼 ∩ Fin))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ∩ cin 3539 ∅c0 3874 𝒫 cpw 4108 class class class wbr 4583 Oncon0 5640 ‘cfv 5804 (class class class)co 6549 ωcom 6957 2_𝑜c2o 7441 ↑_𝑚 cmap 7744 ≈ cen 7838 Fincfn 7841 finSupp cfsupp 8158 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 #chash 12979 Basecbs 15695 0_gc0g 15923 Ringcrg 18370 CRingccrg 18371 ℤ/nℤczn 19670 freeLMod cfrlm 19909

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-inf 8232 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-hash 12980 df-dvds 14822 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-0g 15925 df-prds 15931 df-pws 15933 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-nsg 17415 df-eqg 17416 df-ghm 17481 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-rnghom 18538 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-2idl 19053 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-zn 19674 df-dsmm 19895 df-frlm 19910

This theorem is referenced by: isnumbasgrplem3 36694

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