![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > psrnegcl | Structured version Visualization version GIF version |
Description: The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
psrgrp.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
psrgrp.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
psrgrp.r | ⊢ (𝜑 → 𝑅 ∈ Grp) |
psrnegcl.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
psrnegcl.i | ⊢ 𝑁 = (invg‘𝑅) |
psrnegcl.b | ⊢ 𝐵 = (Base‘𝑆) |
psrnegcl.z | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
psrnegcl | ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | psrnegcl.i | . . . . . 6 ⊢ 𝑁 = (invg‘𝑅) | |
3 | psrgrp.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
4 | 1, 2, 3 | grpinvf1o 17883 | . . . . 5 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅)) |
5 | f1of 6393 | . . . . 5 ⊢ (𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅) → 𝑁:(Base‘𝑅)⟶(Base‘𝑅)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁:(Base‘𝑅)⟶(Base‘𝑅)) |
7 | psrgrp.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
8 | psrnegcl.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
9 | psrnegcl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
10 | psrnegcl.z | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 7, 1, 8, 9, 10 | psrelbas 19787 | . . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) |
12 | fco 6310 | . . . 4 ⊢ ((𝑁:(Base‘𝑅)⟶(Base‘𝑅) ∧ 𝑋:𝐷⟶(Base‘𝑅)) → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) | |
13 | 6, 11, 12 | syl2anc 579 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) |
14 | fvex 6461 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
15 | ovex 6956 | . . . . 5 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V | |
16 | 8, 15 | rabex2 5053 | . . . 4 ⊢ 𝐷 ∈ V |
17 | 14, 16 | elmap 8171 | . . 3 ⊢ ((𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑𝑚 𝐷) ↔ (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) |
18 | 13, 17 | sylibr 226 | . 2 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑𝑚 𝐷)) |
19 | psrgrp.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
20 | 7, 1, 8, 9, 19 | psrbas 19786 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑𝑚 𝐷)) |
21 | 18, 20 | eleqtrrd 2862 | 1 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 {crab 3094 ◡ccnv 5356 “ cima 5360 ∘ ccom 5361 ⟶wf 6133 –1-1-onto→wf1o 6136 ‘cfv 6137 (class class class)co 6924 ↑𝑚 cmap 8142 Fincfn 8243 ℕcn 11379 ℕ0cn0 11647 Basecbs 16266 Grpcgrp 17820 invgcminusg 17821 mPwSer cmps 19759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-plusg 16362 df-mulr 16363 df-sca 16365 df-vsca 16366 df-tset 16368 df-0g 16499 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-grp 17823 df-minusg 17824 df-psr 19764 |
This theorem is referenced by: psrlinv 19805 psrgrp 19806 psrneg 19808 |
Copyright terms: Public domain | W3C validator |