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Mirrors > Home > MPE Home > Th. List > Mathboxes > frlmpwfi | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
frlmpwfi.r | ⊢ 𝑅 = (ℤ/nℤ‘2) |
frlmpwfi.y | ⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
frlmpwfi.b | ⊢ 𝐵 = (Base‘𝑌) |
Ref | Expression |
---|---|
frlmpwfi | ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ (𝒫 𝐼 ∩ Fin)) |
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 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | eqid 2610 | . . . . . 6 ⊢ {𝑥 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} = {𝑥 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} | |
8 | 4, 5, 6, 7 | frlmbas 19918 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑥 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} = (Base‘𝑌)) |
9 | 3, 8 | mpan 702 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑥 finSupp (0g‘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 (0g‘𝑅)} = 𝐵) |
12 | eqid 2610 | . . . 4 ⊢ {𝑥 ∈ (2𝑜 ↑𝑚 𝐼) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (2𝑜 ↑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 ⊢ (#‘2𝑜) = 2 | |
18 | 16, 17 | eqtr4i 2635 | . . . . . 6 ⊢ (#‘(Base‘𝑅)) = (#‘2𝑜) |
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 ⊢ 2𝑜 ∈ ω | |
26 | nnfi 8038 | . . . . . . . 8 ⊢ (2𝑜 ∈ ω → 2𝑜 ∈ Fin) | |
27 | 25, 26 | ax-mp 5 | . . . . . . 7 ⊢ 2𝑜 ∈ Fin |
28 | hashen 12997 | . . . . . . 7 ⊢ (((Base‘𝑅) ∈ Fin ∧ 2𝑜 ∈ Fin) → ((#‘(Base‘𝑅)) = (#‘2𝑜) ↔ (Base‘𝑅) ≈ 2𝑜)) | |
29 | 24, 27, 28 | mp2an 704 | . . . . . 6 ⊢ ((#‘(Base‘𝑅)) = (#‘2𝑜) ↔ (Base‘𝑅) ≈ 2𝑜) |
30 | 18, 29 | mpbi 219 | . . . . 5 ⊢ (Base‘𝑅) ≈ 2𝑜 |
31 | 30 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑅) ≈ 2𝑜) |
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 → (0g‘𝑅) ∈ (Base‘𝑅)) |
36 | 34, 35 | mp1i 13 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝑅) ∈ (Base‘𝑅)) |
37 | 2on0 7456 | . . . . . 6 ⊢ 2𝑜 ≠ ∅ | |
38 | 2on 7455 | . . . . . . 7 ⊢ 2𝑜 ∈ On | |
39 | on0eln0 5697 | . . . . . . 7 ⊢ (2𝑜 ∈ On → (∅ ∈ 2𝑜 ↔ 2𝑜 ≠ ∅)) | |
40 | 38, 39 | ax-mp 5 | . . . . . 6 ⊢ (∅ ∈ 2𝑜 ↔ 2𝑜 ≠ ∅) |
41 | 37, 40 | mpbir 220 | . . . . 5 ⊢ ∅ ∈ 2𝑜 |
42 | 41 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ 2𝑜) |
43 | 7, 12, 13, 31, 36, 42 | mapfien2 8197 | . . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} ≈ {𝑥 ∈ (2𝑜 ↑𝑚 𝐼) ∣ 𝑥 finSupp ∅}) |
44 | 11, 43 | eqbrtrrd 4607 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2𝑜 ↑𝑚 𝐼) ∣ 𝑥 finSupp ∅}) |
45 | 12 | pwfi2en 36685 | . 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2𝑜 ↑𝑚 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin)) |
46 | entr 7894 | . 2 ⊢ ((𝐵 ≈ {𝑥 ∈ (2𝑜 ↑𝑚 𝐼) ∣ 𝑥 finSupp ∅} ∧ {𝑥 ∈ (2𝑜 ↑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 2𝑜c2o 7441 ↑𝑚 cmap 7744 ≈ cen 7838 Fincfn 7841 finSupp cfsupp 8158 ℕcn 10897 2c2 10947 ℕ0cn0 11169 #chash 12979 Basecbs 15695 0gc0g 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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