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Mirrors > Home > MPE Home > Th. List > psrneg | Structured version Visualization version GIF version |
Description: The negative function of 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) |
psrneg.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
psrneg.i | ⊢ 𝑁 = (invg‘𝑅) |
psrneg.b | ⊢ 𝐵 = (Base‘𝑆) |
psrneg.m | ⊢ 𝑀 = (invg‘𝑆) |
psrneg.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
psrneg | ⊢ (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrgrp.s | . . . 4 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
2 | psrgrp.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
3 | psrgrp.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
4 | psrneg.d | . . . 4 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
5 | psrneg.i | . . . 4 ⊢ 𝑁 = (invg‘𝑅) | |
6 | psrneg.b | . . . 4 ⊢ 𝐵 = (Base‘𝑆) | |
7 | psrneg.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | eqid 2736 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
9 | eqid 2736 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | psrlinv 20876 | . . 3 ⊢ (𝜑 → ((𝑁 ∘ 𝑋)(+g‘𝑆)𝑋) = (𝐷 × {(0g‘𝑅)})) |
11 | eqid 2736 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
12 | 1, 2, 3, 4, 8, 11 | psr0 20878 | . . 3 ⊢ (𝜑 → (0g‘𝑆) = (𝐷 × {(0g‘𝑅)})) |
13 | 10, 12 | eqtr4d 2774 | . 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋)(+g‘𝑆)𝑋) = (0g‘𝑆)) |
14 | 1, 2, 3 | psrgrp 20877 | . . 3 ⊢ (𝜑 → 𝑆 ∈ Grp) |
15 | 1, 2, 3, 4, 5, 6, 7 | psrnegcl 20875 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵) |
16 | psrneg.m | . . . 4 ⊢ 𝑀 = (invg‘𝑆) | |
17 | 6, 9, 11, 16 | grpinvid2 18373 | . . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝑁 ∘ 𝑋) ∈ 𝐵) → ((𝑀‘𝑋) = (𝑁 ∘ 𝑋) ↔ ((𝑁 ∘ 𝑋)(+g‘𝑆)𝑋) = (0g‘𝑆))) |
18 | 14, 7, 15, 17 | syl3anc 1373 | . 2 ⊢ (𝜑 → ((𝑀‘𝑋) = (𝑁 ∘ 𝑋) ↔ ((𝑁 ∘ 𝑋)(+g‘𝑆)𝑋) = (0g‘𝑆))) |
19 | 13, 18 | mpbird 260 | 1 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 {crab 3055 {csn 4527 × cxp 5534 ◡ccnv 5535 “ cima 5539 ∘ ccom 5540 ‘cfv 6358 (class class class)co 7191 ↑m cmap 8486 Fincfn 8604 ℕcn 11795 ℕ0cn0 12055 Basecbs 16666 +gcplusg 16749 0gc0g 16898 Grpcgrp 18319 invgcminusg 18320 mPwSer cmps 20817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-tset 16768 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-minusg 18323 df-psr 20822 |
This theorem is referenced by: mplsubglem 20915 mplneg 20924 |
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