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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 2733 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | psrnegcl.i | . . . . . 6 ⊢ 𝑁 = (invg‘𝑅) | |
3 | psrgrp.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
4 | 1, 2, 3 | grpinvf1o 18822 | . . . . 5 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅)) |
5 | f1of 6785 | . . . . 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 21363 | . . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) |
12 | fco 6693 | . . . 4 ⊢ ((𝑁:(Base‘𝑅)⟶(Base‘𝑅) ∧ 𝑋:𝐷⟶(Base‘𝑅)) → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) | |
13 | 6, 11, 12 | syl2anc 585 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) |
14 | fvex 6856 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
15 | ovex 7391 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
16 | 8, 15 | rabex2 5292 | . . . 4 ⊢ 𝐷 ∈ V |
17 | 14, 16 | elmap 8812 | . . 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 21362 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m 𝐷)) |
21 | 18, 20 | eleqtrrd 2837 | 1 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3406 ◡ccnv 5633 “ cima 5637 ∘ ccom 5638 ⟶wf 6493 –1-1-onto→wf1o 6496 ‘cfv 6497 (class class class)co 7358 ↑m cmap 8768 Fincfn 8886 ℕcn 12158 ℕ0cn0 12418 Basecbs 17088 Grpcgrp 18753 invgcminusg 18754 mPwSer cmps 21322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-tset 17157 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-psr 21327 |
This theorem is referenced by: psrlinv 21381 psrgrpOLD 21383 psrneg 21385 |
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