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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 6678 | . . . . 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 20893 | . . . . 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 8535 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ≈ 𝐼) | |
13 | 2nn 11704 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
14 | 1, 4 | znhash 20699 | . . . . . . . 8 ⊢ (2 ∈ ℕ → (♯‘(Base‘𝑅)) = 2) |
15 | 13, 14 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘(Base‘𝑅)) = 2 |
16 | hash2 13760 | . . . . . . 7 ⊢ (♯‘2o) = 2 | |
17 | 15, 16 | eqtr4i 2847 | . . . . . 6 ⊢ (♯‘(Base‘𝑅)) = (♯‘2o) |
18 | 2nn0 11908 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
19 | 15, 18 | eqeltri 2909 | . . . . . . . 8 ⊢ (♯‘(Base‘𝑅)) ∈ ℕ0 |
20 | fvex 6677 | . . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V | |
21 | hashclb 13713 | . . . . . . . . 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 8260 | . . . . . . . 8 ⊢ 2o ∈ ω | |
25 | nnfi 8705 | . . . . . . . 8 ⊢ (2o ∈ ω → 2o ∈ Fin) | |
26 | 24, 25 | ax-mp 5 | . . . . . . 7 ⊢ 2o ∈ Fin |
27 | hashen 13701 | . . . . . . 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 20685 | . . . . . 6 ⊢ (2 ∈ ℕ0 → 𝑅 ∈ CRing) |
32 | crngring 19302 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
33 | 18, 31, 32 | mp2b 10 | . . . . 5 ⊢ 𝑅 ∈ Ring |
34 | 4, 5 | ring0cl 19313 | . . . . 5 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅)) |
35 | 33, 34 | mp1i 13 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝑅) ∈ (Base‘𝑅)) |
36 | 2on0 8107 | . . . . . 6 ⊢ 2o ≠ ∅ | |
37 | 2on 8105 | . . . . . . 7 ⊢ 2o ∈ On | |
38 | on0eln0 6240 | . . . . . . 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 8866 | . . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} ≈ {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅}) |
43 | 10, 42 | eqbrtrrd 5082 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅}) |
44 | 11 | pwfi2en 39690 | . 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin)) |
45 | entr 8555 | . 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 5058 Oncon0 6185 ‘cfv 6349 (class class class)co 7150 ωcom 7574 2oc2o 8090 ↑m cmap 8400 ≈ cen 8500 Fincfn 8503 finSupp cfsupp 8827 ℕcn 11632 2c2 11686 ℕ0cn0 11891 ♯chash 13684 Basecbs 16477 0gc0g 16707 Ringcrg 19291 CRingccrg 19292 ℤ/nℤczn 20644 freeLMod cfrlm 20884 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-inf 8901 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-hash 13685 df-dvds 15602 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-0g 16709 df-prds 16715 df-pws 16717 df-imas 16775 df-qus 16776 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-nsg 18271 df-eqg 18272 df-ghm 18350 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-rnghom 19461 df-subrg 19527 df-lmod 19630 df-lss 19698 df-lsp 19738 df-sra 19938 df-rgmod 19939 df-lidl 19940 df-rsp 19941 df-2idl 19999 df-cnfld 20540 df-zring 20612 df-zrh 20645 df-zn 20648 df-dsmm 20870 df-frlm 20885 |
This theorem is referenced by: isnumbasgrplem3 39698 |
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