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Mathbox for Stefan O'Rear |
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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 6921 | . . . . 5 ⊢ 𝑅 ∈ V |
3 | frlmpwfi.y | . . . . . 6 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
4 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | eqid 2735 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | eqid 2735 | . . . . . 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 21793 | . . . . 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 2793 | . . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} = 𝐵) |
11 | eqid 2735 | . . . 4 ⊢ {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅} | |
12 | enrefg 9023 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ≈ 𝐼) | |
13 | 2nn 12337 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
14 | 1, 4 | znhash 21595 | . . . . . . . 8 ⊢ (2 ∈ ℕ → (♯‘(Base‘𝑅)) = 2) |
15 | 13, 14 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘(Base‘𝑅)) = 2 |
16 | hash2 14441 | . . . . . . 7 ⊢ (♯‘2o) = 2 | |
17 | 15, 16 | eqtr4i 2766 | . . . . . 6 ⊢ (♯‘(Base‘𝑅)) = (♯‘2o) |
18 | 2nn0 12541 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
19 | 15, 18 | eqeltri 2835 | . . . . . . . 8 ⊢ (♯‘(Base‘𝑅)) ∈ ℕ0 |
20 | fvex 6920 | . . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V | |
21 | hashclb 14394 | . . . . . . . . 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 8679 | . . . . . . . 8 ⊢ 2o ∈ ω | |
25 | nnfi 9206 | . . . . . . . 8 ⊢ (2o ∈ ω → 2o ∈ Fin) | |
26 | 24, 25 | ax-mp 5 | . . . . . . 7 ⊢ 2o ∈ Fin |
27 | hashen 14383 | . . . . . . 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 21581 | . . . . . 6 ⊢ (2 ∈ ℕ0 → 𝑅 ∈ CRing) |
32 | crngring 20263 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
33 | 18, 31, 32 | mp2b 10 | . . . . 5 ⊢ 𝑅 ∈ Ring |
34 | 4, 5 | ring0cl 20281 | . . . . 5 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅)) |
35 | 33, 34 | mp1i 13 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝑅) ∈ (Base‘𝑅)) |
36 | 2on0 8521 | . . . . . 6 ⊢ 2o ≠ ∅ | |
37 | 2on 8519 | . . . . . . 7 ⊢ 2o ∈ On | |
38 | on0eln0 6442 | . . . . . . 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 9447 | . . 3 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑥 finSupp (0g‘𝑅)} ≈ {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅}) |
43 | 10, 42 | eqbrtrrd 5172 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅}) |
44 | 11 | pwfi2en 43086 | . 2 ⊢ (𝐼 ∈ 𝑉 → {𝑥 ∈ (2o ↑m 𝐼) ∣ 𝑥 finSupp ∅} ≈ (𝒫 𝐼 ∩ Fin)) |
45 | entr 9045 | . 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 1537 ∈ wcel 2106 ≠ wne 2938 {crab 3433 Vcvv 3478 ∩ cin 3962 ∅c0 4339 𝒫 cpw 4605 class class class wbr 5148 Oncon0 6386 ‘cfv 6563 (class class class)co 7431 ωcom 7887 2oc2o 8499 ↑m cmap 8865 ≈ cen 8981 Fincfn 8984 finSupp cfsupp 9399 ℕcn 12264 2c2 12319 ℕ0cn0 12524 ♯chash 14366 Basecbs 17245 0gc0g 17486 Ringcrg 20251 CRingccrg 20252 ℤ/nℤczn 21531 freeLMod cfrlm 21784 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-inf 9481 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-hash 14367 df-dvds 16288 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 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 17488 df-prds 17494 df-pws 17496 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-nsg 19155 df-eqg 19156 df-ghm 19244 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-zn 21535 df-dsmm 21770 df-frlm 21785 |
This theorem is referenced by: isnumbasgrplem3 43094 |
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