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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 | 1 | fvexi 6919 | . . . . 5 ⊢ 𝑅 ∈ V | 
| 3 | frlmpwfi.y | . . . . . 6 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
| 4 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 5 | eqid 2736 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 6 | eqid 2736 | . . . . . 6 ⊢ {𝑥 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} = {𝑥 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} | |
| 7 | 3, 4, 5, 6 | frlmbas 21776 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑥 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} = (Base‘𝑌)) | 
| 8 | 2, 7 | mpan 690 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} = (Base‘𝑌)) | 
| 9 | frlmpwfi.b | . . . 4 ⊢ 𝐵 = (Base‘𝑌) | |
| 10 | 8, 9 | eqtr4di 2794 | . . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} = 𝐵) | 
| 11 | eqid 2736 | . . . 4 ⊢ {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅} | |
| 12 | enrefg 9025 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ≈ 𝐼) | |
| 13 | 2nn 12340 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
| 14 | 1, 4 | znhash 21578 | . . . . . . . 8 ⊢ (2 ∈ ℕ → (♯‘(Base‘𝑅)) = 2) | 
| 15 | 13, 14 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘(Base‘𝑅)) = 2 | 
| 16 | hash2 14445 | . . . . . . 7 ⊢ (♯‘2o) = 2 | |
| 17 | 15, 16 | eqtr4i 2767 | . . . . . 6 ⊢ (♯‘(Base‘𝑅)) = (♯‘2o) | 
| 18 | 2nn0 12545 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
| 19 | 15, 18 | eqeltri 2836 | . . . . . . . 8 ⊢ (♯‘(Base‘𝑅)) ∈ ℕ0 | 
| 20 | fvex 6918 | . . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V | |
| 21 | hashclb 14398 | . . . . . . . . 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 8681 | . . . . . . . 8 ⊢ 2o ∈ ω | |
| 25 | nnfi 9208 | . . . . . . . 8 ⊢ (2o ∈ ω → 2o ∈ Fin) | |
| 26 | 24, 25 | ax-mp 5 | . . . . . . 7 ⊢ 2o ∈ Fin | 
| 27 | hashen 14387 | . . . . . . 7 ⊢ (((Base‘𝑅) ∈ Fin ∧ 2o ∈ Fin) → ((♯‘(Base‘𝑅)) = (♯‘2o) ↔ (Base‘𝑅) ≈ 2o)) | |
| 28 | 23, 26, 27 | mp2an 692 | . . . . . 6 ⊢ ((♯‘(Base‘𝑅)) = (♯‘2o) ↔ (Base‘𝑅) ≈ 2o) | 
| 29 | 17, 28 | mpbi 230 | . . . . 5 ⊢ (Base‘𝑅) ≈ 2o | 
| 30 | 29 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑅) ≈ 2o) | 
| 31 | 1 | zncrng 21564 | . . . . . 6 ⊢ (2 ∈ ℕ0 → 𝑅 ∈ CRing) | 
| 32 | crngring 20243 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 33 | 18, 31, 32 | mp2b 10 | . . . . 5 ⊢ 𝑅 ∈ Ring | 
| 34 | 4, 5 | ring0cl 20265 | . . . . 5 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅)) | 
| 35 | 33, 34 | mp1i 13 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝑅) ∈ (Base‘𝑅)) | 
| 36 | 2on0 8523 | . . . . . 6 ⊢ 2o ≠ ∅ | |
| 37 | 2on 8521 | . . . . . . 7 ⊢ 2o ∈ On | |
| 38 | on0eln0 6439 | . . . . . . 7 ⊢ (2o ∈ On → (∅ ∈ 2o ↔ 2o ≠ ∅)) | |
| 39 | 37, 38 | ax-mp 5 | . . . . . 6 ⊢ (∅ ∈ 2o ↔ 2o ≠ ∅) | 
| 40 | 36, 39 | mpbir 231 | . . . . 5 ⊢ ∅ ∈ 2o | 
| 41 | 40 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ 2o) | 
| 42 | 6, 11, 12, 30, 35, 41 | mapfien2 9450 | . . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} ≈ {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅}) | 
| 43 | 10, 42 | eqbrtrrd 5166 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅}) | 
| 44 | 11 | pwfi2en 43114 | . 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin)) | 
| 45 | entr 9047 | . 2 ⊢ ((𝐵 ≈ {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅} ∧ {𝑥 ∈ (2o ↑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 206 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 {crab 3435 Vcvv 3479 ∩ cin 3949 ∅c0 4332 𝒫 cpw 4599 class class class wbr 5142 Oncon0 6383 ‘cfv 6560 (class class class)co 7432 ωcom 7888 2oc2o 8501 ↑m cmap 8867 ≈ cen 8983 Fincfn 8986 finSupp cfsupp 9402 ℕcn 12267 2c2 12322 ℕ0cn0 12528 ♯chash 14370 Basecbs 17248 0gc0g 17485 Ringcrg 20231 CRingccrg 20232 ℤ/nℤczn 21514 freeLMod cfrlm 21767 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-inf 9484 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-hash 14371 df-dvds 16292 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17487 df-prds 17493 df-pws 17495 df-imas 17554 df-qus 17555 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-nsg 19143 df-eqg 19144 df-ghm 19232 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-rhm 20473 df-subrng 20547 df-subrg 20571 df-lmod 20861 df-lss 20931 df-lsp 20971 df-sra 21173 df-rgmod 21174 df-lidl 21219 df-rsp 21220 df-2idl 21261 df-cnfld 21366 df-zring 21459 df-zrh 21515 df-zn 21518 df-dsmm 21753 df-frlm 21768 | 
| This theorem is referenced by: isnumbasgrplem3 43122 | 
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