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Mirrors > Home > MPE Home > Th. List > psr0cl | Structured version Visualization version GIF version |
Description: The zero element 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) |
psr0cl.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
psr0cl.o | ⊢ 0 = (0g‘𝑅) |
psr0cl.b | ⊢ 𝐵 = (Base‘𝑆) |
Ref | Expression |
---|---|
psr0cl | ⊢ (𝜑 → (𝐷 × { 0 }) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrgrp.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
2 | eqid 2736 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | psr0cl.o | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | grpidcl 18349 | . . . 4 ⊢ (𝑅 ∈ Grp → 0 ∈ (Base‘𝑅)) |
5 | fconst6g 6586 | . . . 4 ⊢ ( 0 ∈ (Base‘𝑅) → (𝐷 × { 0 }):𝐷⟶(Base‘𝑅)) | |
6 | 1, 4, 5 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐷 × { 0 }):𝐷⟶(Base‘𝑅)) |
7 | fvex 6708 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
8 | psr0cl.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
9 | ovex 7224 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
10 | 8, 9 | rabex2 5212 | . . . 4 ⊢ 𝐷 ∈ V |
11 | 7, 10 | elmap 8530 | . . 3 ⊢ ((𝐷 × { 0 }) ∈ ((Base‘𝑅) ↑m 𝐷) ↔ (𝐷 × { 0 }):𝐷⟶(Base‘𝑅)) |
12 | 6, 11 | sylibr 237 | . 2 ⊢ (𝜑 → (𝐷 × { 0 }) ∈ ((Base‘𝑅) ↑m 𝐷)) |
13 | psrgrp.s | . . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
14 | psr0cl.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
15 | psrgrp.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
16 | 13, 2, 8, 14, 15 | psrbas 20857 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m 𝐷)) |
17 | 12, 16 | eleqtrrd 2834 | 1 ⊢ (𝜑 → (𝐷 × { 0 }) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 {crab 3055 {csn 4527 × cxp 5534 ◡ccnv 5535 “ cima 5539 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ↑m cmap 8486 Fincfn 8604 ℕcn 11795 ℕ0cn0 12055 Basecbs 16666 0gc0g 16898 Grpcgrp 18319 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-psr 20822 |
This theorem is referenced by: psr0lid 20874 psrgrp 20877 psr0 20878 mplsubglem 20915 |
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