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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 2732 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | psr1cl.o | . . . . . . . 8 ⊢ 1 = (1r‘𝑅) | |
4 | 2, 3 | ringidcl 20076 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
5 | psr1cl.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
6 | 2, 5 | ring0cl 20077 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
7 | 4, 6 | ifcld 4573 | . . . . . 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 481 | . . . 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 7110 | . . 3 ⊢ (𝜑 → 𝑈:𝐷⟶(Base‘𝑅)) |
12 | fvex 6901 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
13 | psr1cl.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
14 | ovex 7438 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
15 | 13, 14 | rabex2 5333 | . . . 4 ⊢ 𝐷 ∈ V |
16 | 12, 15 | elmap 8861 | . . 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 21488 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m 𝐷)) |
22 | 17, 21 | eleqtrrd 2836 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {crab 3432 ifcif 4527 {csn 4627 ↦ cmpt 5230 × cxp 5673 ◡ccnv 5674 “ cima 5678 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑m cmap 8816 Fincfn 8935 0cc0 11106 ℕcn 12208 ℕ0cn0 12468 Basecbs 17140 0gc0g 17381 1rcur 19998 Ringcrg 20049 mPwSer cmps 21448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-tset 17212 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-mgp 19982 df-ur 19999 df-ring 20051 df-psr 21453 |
This theorem is referenced by: psrlidm 21514 psrridm 21515 psrring 21522 psr1 21523 |
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