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Mirrors > Home > MPE Home > Th. List > psrlinv | 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) |
psrlinv.o | ⊢ 0 = (0g‘𝑅) |
psrlinv.p | ⊢ + = (+g‘𝑆) |
Ref | Expression |
---|---|
psrlinv | ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrnegcl.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
2 | ovex 7459 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
3 | 1, 2 | rabex2 5340 | . . . 4 ⊢ 𝐷 ∈ V |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
5 | fvexd 6917 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑁‘(𝑋‘𝑥)) ∈ V) | |
6 | psrgrp.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
7 | eqid 2728 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | psrnegcl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
9 | psrnegcl.z | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
10 | 6, 7, 1, 8, 9 | psrelbas 21886 | . . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) |
11 | 10 | ffvelcdmda 7099 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋‘𝑥) ∈ (Base‘𝑅)) |
12 | 10 | feqmptd 6972 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐷 ↦ (𝑋‘𝑥))) |
13 | psrnegcl.i | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑅) | |
14 | psrgrp.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
15 | 7, 13, 14 | grpinvf1o 18972 | . . . . . 6 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅)) |
16 | f1of 6844 | . . . . . 6 ⊢ (𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅) → 𝑁:(Base‘𝑅)⟶(Base‘𝑅)) | |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁:(Base‘𝑅)⟶(Base‘𝑅)) |
18 | 17 | feqmptd 6972 | . . . 4 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ (Base‘𝑅) ↦ (𝑁‘𝑦))) |
19 | fveq2 6902 | . . . 4 ⊢ (𝑦 = (𝑋‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑋‘𝑥))) | |
20 | 11, 12, 18, 19 | fmptco 7144 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐷 ↦ (𝑁‘(𝑋‘𝑥)))) |
21 | 4, 5, 11, 20, 12 | offval2 7711 | . 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) ∘f (+g‘𝑅)𝑋) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥)))) |
22 | eqid 2728 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
23 | psrlinv.p | . . 3 ⊢ + = (+g‘𝑆) | |
24 | psrgrp.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
25 | 6, 24, 14, 1, 13, 8, 9 | psrnegcl 21904 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵) |
26 | 6, 8, 22, 23, 25, 9 | psradd 21889 | . 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = ((𝑁 ∘ 𝑋) ∘f (+g‘𝑅)𝑋)) |
27 | fconstmpt 5744 | . . 3 ⊢ (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ 0 ) | |
28 | psrlinv.o | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
29 | 7, 22, 28, 13 | grplinv 18953 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑥) ∈ (Base‘𝑅)) → ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥)) = 0 ) |
30 | 14, 11, 29 | syl2an2r 683 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥)) = 0 ) |
31 | 30 | mpteq2dva 5252 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥))) = (𝑥 ∈ 𝐷 ↦ 0 )) |
32 | 27, 31 | eqtr4id 2787 | . 2 ⊢ (𝜑 → (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥)))) |
33 | 21, 26, 32 | 3eqtr4d 2778 | 1 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3430 Vcvv 3473 {csn 4632 ↦ cmpt 5235 × cxp 5680 ◡ccnv 5681 “ cima 5685 ∘ ccom 5686 ⟶wf 6549 –1-1-onto→wf1o 6552 ‘cfv 6553 (class class class)co 7426 ∘f cof 7689 ↑m cmap 8851 Fincfn 8970 ℕcn 12250 ℕ0cn0 12510 Basecbs 17187 +gcplusg 17240 0gc0g 17428 Grpcgrp 18897 invgcminusg 18898 mPwSer cmps 21844 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-tset 17259 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-psr 21849 |
This theorem is referenced by: psrgrpOLD 21907 psrneg 21909 |
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