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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 2728 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | psrnegcl.i | . . . . . 6 ⊢ 𝑁 = (invg‘𝑅) | |
3 | psrgrp.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
4 | 1, 2, 3 | grpinvf1o 18979 | . . . . 5 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅)) |
5 | f1of 6844 | . . . . 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 21893 | . . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) |
12 | fco 6752 | . . . 4 ⊢ ((𝑁:(Base‘𝑅)⟶(Base‘𝑅) ∧ 𝑋:𝐷⟶(Base‘𝑅)) → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) | |
13 | 6, 11, 12 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) |
14 | fvex 6915 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
15 | ovex 7459 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
16 | 8, 15 | rabex2 5340 | . . . 4 ⊢ 𝐷 ∈ V |
17 | 14, 16 | elmap 8898 | . . 3 ⊢ ((𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑m 𝐷) ↔ (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) |
18 | 13, 17 | sylibr 233 | . 2 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑m 𝐷)) |
19 | psrgrp.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
20 | 7, 1, 8, 9, 19 | psrbas 21892 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m 𝐷)) |
21 | 18, 20 | eleqtrrd 2832 | 1 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3430 ◡ccnv 5681 “ cima 5685 ∘ ccom 5686 ⟶wf 6549 –1-1-onto→wf1o 6552 ‘cfv 6553 (class class class)co 7426 ↑m cmap 8853 Fincfn 8972 ℕcn 12252 ℕ0cn0 12512 Basecbs 17189 Grpcgrp 18904 invgcminusg 18905 mPwSer cmps 21851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-struct 17125 df-slot 17160 df-ndx 17172 df-base 17190 df-plusg 17255 df-mulr 17256 df-sca 17258 df-vsca 17259 df-tset 17261 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-minusg 18908 df-psr 21856 |
This theorem is referenced by: psrlinv 21914 psrgrpOLD 21916 psrneg 21918 |
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