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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 6683 | . . . . 5 ⊢ 𝑅 ∈ V |
3 | frlmpwfi.y | . . . . . 6 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
4 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | eqid 2821 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | eqid 2821 | . . . . . 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 20898 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑥 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} = (Base‘𝑌)) |
8 | 2, 7 | mpan 688 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} = (Base‘𝑌)) |
9 | frlmpwfi.b | . . . 4 ⊢ 𝐵 = (Base‘𝑌) | |
10 | 8, 9 | syl6eqr 2874 | . . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} = 𝐵) |
11 | eqid 2821 | . . . 4 ⊢ {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅} | |
12 | enrefg 8540 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ≈ 𝐼) | |
13 | 2nn 11709 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
14 | 1, 4 | znhash 20704 | . . . . . . . 8 ⊢ (2 ∈ ℕ → (♯‘(Base‘𝑅)) = 2) |
15 | 13, 14 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘(Base‘𝑅)) = 2 |
16 | hash2 13765 | . . . . . . 7 ⊢ (♯‘2o) = 2 | |
17 | 15, 16 | eqtr4i 2847 | . . . . . 6 ⊢ (♯‘(Base‘𝑅)) = (♯‘2o) |
18 | 2nn0 11913 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
19 | 15, 18 | eqeltri 2909 | . . . . . . . 8 ⊢ (♯‘(Base‘𝑅)) ∈ ℕ0 |
20 | fvex 6682 | . . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V | |
21 | hashclb 13718 | . . . . . . . . 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 233 | . . . . . . 7 ⊢ (Base‘𝑅) ∈ Fin |
24 | 2onn 8265 | . . . . . . . 8 ⊢ 2o ∈ ω | |
25 | nnfi 8710 | . . . . . . . 8 ⊢ (2o ∈ ω → 2o ∈ Fin) | |
26 | 24, 25 | ax-mp 5 | . . . . . . 7 ⊢ 2o ∈ Fin |
27 | hashen 13706 | . . . . . . 7 ⊢ (((Base‘𝑅) ∈ Fin ∧ 2o ∈ Fin) → ((♯‘(Base‘𝑅)) = (♯‘2o) ↔ (Base‘𝑅) ≈ 2o)) | |
28 | 23, 26, 27 | mp2an 690 | . . . . . 6 ⊢ ((♯‘(Base‘𝑅)) = (♯‘2o) ↔ (Base‘𝑅) ≈ 2o) |
29 | 17, 28 | mpbi 232 | . . . . 5 ⊢ (Base‘𝑅) ≈ 2o |
30 | 29 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑅) ≈ 2o) |
31 | 1 | zncrng 20690 | . . . . . 6 ⊢ (2 ∈ ℕ0 → 𝑅 ∈ CRing) |
32 | crngring 19307 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
33 | 18, 31, 32 | mp2b 10 | . . . . 5 ⊢ 𝑅 ∈ Ring |
34 | 4, 5 | ring0cl 19318 | . . . . 5 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅)) |
35 | 33, 34 | mp1i 13 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝑅) ∈ (Base‘𝑅)) |
36 | 2on0 8112 | . . . . . 6 ⊢ 2o ≠ ∅ | |
37 | 2on 8110 | . . . . . . 7 ⊢ 2o ∈ On | |
38 | on0eln0 6245 | . . . . . . 7 ⊢ (2o ∈ On → (∅ ∈ 2o ↔ 2o ≠ ∅)) | |
39 | 37, 38 | ax-mp 5 | . . . . . 6 ⊢ (∅ ∈ 2o ↔ 2o ≠ ∅) |
40 | 36, 39 | mpbir 233 | . . . . 5 ⊢ ∅ ∈ 2o |
41 | 40 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → ∅ ∈ 2o) |
42 | 6, 11, 12, 30, 35, 41 | mapfien2 8871 | . . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} ≈ {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅}) |
43 | 10, 42 | eqbrtrrd 5089 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅}) |
44 | 11 | pwfi2en 39695 | . 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin)) |
45 | entr 8560 | . 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 586 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ (𝒫 𝐼 ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {crab 3142 Vcvv 3494 ∩ cin 3934 ∅c0 4290 𝒫 cpw 4538 class class class wbr 5065 Oncon0 6190 ‘cfv 6354 (class class class)co 7155 ωcom 7579 2oc2o 8095 ↑m cmap 8405 ≈ cen 8505 Fincfn 8508 finSupp cfsupp 8832 ℕcn 11637 2c2 11691 ℕ0cn0 11896 ♯chash 13689 Basecbs 16482 0gc0g 16712 Ringcrg 19296 CRingccrg 19297 ℤ/nℤczn 20649 freeLMod cfrlm 20889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-tpos 7891 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-ec 8290 df-qs 8294 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-sup 8905 df-inf 8906 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-hash 13690 df-dvds 15607 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-0g 16714 df-prds 16720 df-pws 16722 df-imas 16780 df-qus 16781 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-nsg 18276 df-eqg 18277 df-ghm 18355 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-cring 19299 df-oppr 19372 df-dvdsr 19390 df-rnghom 19466 df-subrg 19532 df-lmod 19635 df-lss 19703 df-lsp 19743 df-sra 19943 df-rgmod 19944 df-lidl 19945 df-rsp 19946 df-2idl 20004 df-cnfld 20545 df-zring 20617 df-zrh 20650 df-zn 20653 df-dsmm 20875 df-frlm 20890 |
This theorem is referenced by: isnumbasgrplem3 39703 |
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