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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 2726 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
9 | eqid 2726 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | psrlinv 21854 | . . 3 ⊢ (𝜑 → ((𝑁 ∘ 𝑋)(+g‘𝑆)𝑋) = (𝐷 × {(0g‘𝑅)})) |
11 | eqid 2726 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
12 | 1, 2, 3, 4, 8, 11 | psr0 21857 | . . 3 ⊢ (𝜑 → (0g‘𝑆) = (𝐷 × {(0g‘𝑅)})) |
13 | 10, 12 | eqtr4d 2769 | . 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋)(+g‘𝑆)𝑋) = (0g‘𝑆)) |
14 | 1, 2, 3 | psrgrp 21855 | . . 3 ⊢ (𝜑 → 𝑆 ∈ Grp) |
15 | 1, 2, 3, 4, 5, 6, 7 | psrnegcl 21853 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵) |
16 | psrneg.m | . . . 4 ⊢ 𝑀 = (invg‘𝑆) | |
17 | 6, 9, 11, 16 | grpinvid2 18920 | . . 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 1368 | . 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 1533 ∈ wcel 2098 {crab 3426 {csn 4623 × cxp 5667 ◡ccnv 5668 “ cima 5672 ∘ ccom 5673 ‘cfv 6536 (class class class)co 7404 ↑m cmap 8819 Fincfn 8938 ℕcn 12213 ℕ0cn0 12473 Basecbs 17151 +gcplusg 17204 0gc0g 17392 Grpcgrp 18861 invgcminusg 18862 mPwSer cmps 21794 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-prds 17400 df-pws 17402 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-psr 21799 |
This theorem is referenced by: mplsubglem 21896 mplneg 21907 |
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