![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2726 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | psr1cl.o | . . . . . . . 8 ⊢ 1 = (1r‘𝑅) | |
4 | 2, 3 | ringidcl 20163 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
5 | psr1cl.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
6 | 2, 5 | ring0cl 20164 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
7 | 4, 6 | ifcld 4569 | . . . . . 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 480 | . . . 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 7108 | . . 3 ⊢ (𝜑 → 𝑈:𝐷⟶(Base‘𝑅)) |
12 | fvex 6897 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
13 | psr1cl.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
14 | ovex 7437 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
15 | 13, 14 | rabex2 5327 | . . . 4 ⊢ 𝐷 ∈ V |
16 | 12, 15 | elmap 8864 | . . 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 21834 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m 𝐷)) |
22 | 17, 21 | eleqtrrd 2830 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3426 ifcif 4523 {csn 4623 ↦ cmpt 5224 × cxp 5667 ◡ccnv 5668 “ cima 5672 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 ↑m cmap 8819 Fincfn 8938 0cc0 11109 ℕcn 12213 ℕ0cn0 12473 Basecbs 17151 0gc0g 17392 1rcur 20084 Ringcrg 20136 mPwSer cmps 21794 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-tset 17223 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-mgp 20038 df-ur 20085 df-ring 20138 df-psr 21799 |
This theorem is referenced by: psrlidm 21861 psrridm 21862 psrring 21869 psr1 21870 |
Copyright terms: Public domain | W3C validator |