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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 2737 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 2 | psrnegcl.i | . . . . . 6 ⊢ 𝑁 = (invg‘𝑅) | |
| 3 | psrgrp.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
| 4 | 1, 2, 3 | grpinvf1o 19027 | . . . . 5 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅)) | 
| 5 | f1of 6848 | . . . . 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 21954 | . . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) | 
| 12 | fco 6760 | . . . 4 ⊢ ((𝑁:(Base‘𝑅)⟶(Base‘𝑅) ∧ 𝑋:𝐷⟶(Base‘𝑅)) → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) | |
| 13 | 6, 11, 12 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) | 
| 14 | fvex 6919 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
| 15 | ovex 7464 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
| 16 | 8, 15 | rabex2 5341 | . . . 4 ⊢ 𝐷 ∈ V | 
| 17 | 14, 16 | elmap 8911 | . . 3 ⊢ ((𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑m 𝐷) ↔ (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) | 
| 18 | 13, 17 | sylibr 234 | . 2 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑m 𝐷)) | 
| 19 | psrgrp.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
| 20 | 7, 1, 8, 9, 19 | psrbas 21953 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m 𝐷)) | 
| 21 | 18, 20 | eleqtrrd 2844 | 1 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 ◡ccnv 5684 “ cima 5688 ∘ ccom 5689 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Fincfn 8985 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 Grpcgrp 18951 invgcminusg 18952 mPwSer cmps 21924 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-psr 21929 | 
| This theorem is referenced by: psrlinv 21975 psrgrpOLD 21977 psrneg 21979 | 
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