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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 7464 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
| 3 | 1, 2 | rabex2 5341 | . . . 4 ⊢ 𝐷 ∈ V |
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
| 5 | fvexd 6921 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑁‘(𝑋‘𝑥)) ∈ V) | |
| 6 | psrgrp.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
| 7 | eqid 2737 | . . . . 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 21954 | . . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) |
| 11 | 10 | ffvelcdmda 7104 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋‘𝑥) ∈ (Base‘𝑅)) |
| 12 | 10 | feqmptd 6977 | . . . 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 19027 | . . . . . 6 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅)) |
| 16 | f1of 6848 | . . . . . 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 6977 | . . . 4 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ (Base‘𝑅) ↦ (𝑁‘𝑦))) |
| 19 | fveq2 6906 | . . . 4 ⊢ (𝑦 = (𝑋‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑋‘𝑥))) | |
| 20 | 11, 12, 18, 19 | fmptco 7149 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐷 ↦ (𝑁‘(𝑋‘𝑥)))) |
| 21 | 4, 5, 11, 20, 12 | offval2 7717 | . 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) ∘f (+g‘𝑅)𝑋) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥)))) |
| 22 | eqid 2737 | . . 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 21974 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵) |
| 26 | 6, 8, 22, 23, 25, 9 | psradd 21957 | . 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = ((𝑁 ∘ 𝑋) ∘f (+g‘𝑅)𝑋)) |
| 27 | fconstmpt 5747 | . . 3 ⊢ (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ 0 ) | |
| 28 | psrlinv.o | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
| 29 | 7, 22, 28, 13 | grplinv 19007 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑥) ∈ (Base‘𝑅)) → ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥)) = 0 ) |
| 30 | 14, 11, 29 | syl2an2r 685 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥)) = 0 ) |
| 31 | 30 | mpteq2dva 5242 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥))) = (𝑥 ∈ 𝐷 ↦ 0 )) |
| 32 | 27, 31 | eqtr4id 2796 | . 2 ⊢ (𝜑 → (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥)))) |
| 33 | 21, 26, 32 | 3eqtr4d 2787 | 1 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 })) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 {csn 4626 ↦ cmpt 5225 × cxp 5683 ◡ccnv 5684 “ cima 5688 ∘ ccom 5689 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 ↑m cmap 8866 Fincfn 8985 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Grpcgrp 18951 invgcminusg 18952 mPwSer cmps 21924 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-psr 21929 |
| This theorem is referenced by: psrgrpOLD 21977 psrneg 21979 |
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