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Mirrors > Home > MPE Home > Th. List > psr1cl | Structured version Visualization version GIF version |
Description: The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
psrring.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
psrring.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
psrring.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
psr1cl.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
psr1cl.z | ⊢ 0 = (0g‘𝑅) |
psr1cl.o | ⊢ 1 = (1r‘𝑅) |
psr1cl.u | ⊢ 𝑈 = (𝑥 ∈ 𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 )) |
psr1cl.b | ⊢ 𝐵 = (Base‘𝑆) |
Ref | Expression |
---|---|
psr1cl | ⊢ (𝜑 → 𝑈 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrring.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | eqid 2733 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | psr1cl.o | . . . . . . . 8 ⊢ 1 = (1r‘𝑅) | |
4 | 2, 3 | ringidcl 19994 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
5 | psr1cl.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
6 | 2, 5 | ring0cl 19995 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
7 | 4, 6 | ifcld 4533 | . . . . . 6 ⊢ (𝑅 ∈ Ring → if(𝑥 = (𝐼 × {0}), 1 , 0 ) ∈ (Base‘𝑅)) |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → if(𝑥 = (𝐼 × {0}), 1 , 0 ) ∈ (Base‘𝑅)) |
9 | 8 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → if(𝑥 = (𝐼 × {0}), 1 , 0 ) ∈ (Base‘𝑅)) |
10 | psr1cl.u | . . . 4 ⊢ 𝑈 = (𝑥 ∈ 𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 )) | |
11 | 9, 10 | fmptd 7063 | . . 3 ⊢ (𝜑 → 𝑈:𝐷⟶(Base‘𝑅)) |
12 | fvex 6856 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
13 | psr1cl.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
14 | ovex 7391 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
15 | 13, 14 | rabex2 5292 | . . . 4 ⊢ 𝐷 ∈ V |
16 | 12, 15 | elmap 8812 | . . 3 ⊢ (𝑈 ∈ ((Base‘𝑅) ↑m 𝐷) ↔ 𝑈:𝐷⟶(Base‘𝑅)) |
17 | 11, 16 | sylibr 233 | . 2 ⊢ (𝜑 → 𝑈 ∈ ((Base‘𝑅) ↑m 𝐷)) |
18 | psrring.s | . . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
19 | psr1cl.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
20 | psrring.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
21 | 18, 2, 13, 19, 20 | psrbas 21362 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m 𝐷)) |
22 | 17, 21 | eleqtrrd 2837 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3406 ifcif 4487 {csn 4587 ↦ cmpt 5189 × cxp 5632 ◡ccnv 5633 “ cima 5637 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 ↑m cmap 8768 Fincfn 8886 0cc0 11056 ℕcn 12158 ℕ0cn0 12418 Basecbs 17088 0gc0g 17326 1rcur 19918 Ringcrg 19969 mPwSer cmps 21322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-tset 17157 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-mgp 19902 df-ur 19919 df-ring 19971 df-psr 21327 |
This theorem is referenced by: psrlidm 21388 psrridm 21389 psrring 21396 psr1 21397 |
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