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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 2728 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
9 | eqid 2728 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | psrlinv 21898 | . . 3 ⊢ (𝜑 → ((𝑁 ∘ 𝑋)(+g‘𝑆)𝑋) = (𝐷 × {(0g‘𝑅)})) |
11 | eqid 2728 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
12 | 1, 2, 3, 4, 8, 11 | psr0 21901 | . . 3 ⊢ (𝜑 → (0g‘𝑆) = (𝐷 × {(0g‘𝑅)})) |
13 | 10, 12 | eqtr4d 2771 | . 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋)(+g‘𝑆)𝑋) = (0g‘𝑆)) |
14 | 1, 2, 3 | psrgrp 21899 | . . 3 ⊢ (𝜑 → 𝑆 ∈ Grp) |
15 | 1, 2, 3, 4, 5, 6, 7 | psrnegcl 21897 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵) |
16 | psrneg.m | . . . 4 ⊢ 𝑀 = (invg‘𝑆) | |
17 | 6, 9, 11, 16 | grpinvid2 18949 | . . 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 1369 | . 2 ⊢ (𝜑 → ((𝑀‘𝑋) = (𝑁 ∘ 𝑋) ↔ ((𝑁 ∘ 𝑋)(+g‘𝑆)𝑋) = (0g‘𝑆))) |
19 | 13, 18 | mpbird 257 | 1 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 {crab 3429 {csn 4629 × cxp 5676 ◡ccnv 5677 “ cima 5681 ∘ ccom 5682 ‘cfv 6548 (class class class)co 7420 ↑m cmap 8845 Fincfn 8964 ℕcn 12243 ℕ0cn0 12503 Basecbs 17180 +gcplusg 17233 0gc0g 17421 Grpcgrp 18890 invgcminusg 18891 mPwSer cmps 21837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-struct 17116 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-0g 17423 df-prds 17429 df-pws 17431 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-psr 21842 |
This theorem is referenced by: mplsubglem 21941 mplneg 21952 |
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