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Mirrors > Home > MPE Home > Th. List > psr0lid | Structured version Visualization version GIF version |
Description: The zero element of the ring of power series is a left identity. (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) |
psr0lid.p | ⊢ + = (+g‘𝑆) |
psr0lid.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
psr0lid | ⊢ (𝜑 → ((𝐷 × { 0 }) + 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrgrp.s | . . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
2 | psr0cl.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
3 | eqid 2773 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | psr0lid.p | . . 3 ⊢ + = (+g‘𝑆) | |
5 | psrgrp.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
6 | psrgrp.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
7 | psr0cl.d | . . . 4 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
8 | psr0cl.o | . . . 4 ⊢ 0 = (0g‘𝑅) | |
9 | 1, 5, 6, 7, 8, 2 | psr0cl 19901 | . . 3 ⊢ (𝜑 → (𝐷 × { 0 }) ∈ 𝐵) |
10 | psr0lid.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 1, 2, 3, 4, 9, 10 | psradd 19889 | . 2 ⊢ (𝜑 → ((𝐷 × { 0 }) + 𝑋) = ((𝐷 × { 0 }) ∘𝑓 (+g‘𝑅)𝑋)) |
12 | ovex 7007 | . . . . 5 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V | |
13 | 7, 12 | rabex2 5090 | . . . 4 ⊢ 𝐷 ∈ V |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
15 | eqid 2773 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
16 | 1, 15, 7, 2, 10 | psrelbas 19886 | . . 3 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) |
17 | 8 | fvexi 6511 | . . . 4 ⊢ 0 ∈ V |
18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
19 | 15, 3, 8 | grplid 17934 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ( 0 (+g‘𝑅)𝑥) = 𝑥) |
20 | 6, 19 | sylan 572 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ( 0 (+g‘𝑅)𝑥) = 𝑥) |
21 | 14, 16, 18, 20 | caofid0l 7254 | . 2 ⊢ (𝜑 → ((𝐷 × { 0 }) ∘𝑓 (+g‘𝑅)𝑋) = 𝑋) |
22 | 11, 21 | eqtrd 2809 | 1 ⊢ (𝜑 → ((𝐷 × { 0 }) + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 {crab 3087 Vcvv 3410 {csn 4436 × cxp 5402 ◡ccnv 5403 “ cima 5407 ‘cfv 6186 (class class class)co 6975 ∘𝑓 cof 7224 ↑𝑚 cmap 8205 Fincfn 8305 ℕcn 11438 ℕ0cn0 11706 Basecbs 16338 +gcplusg 16420 0gc0g 16568 Grpcgrp 17904 mPwSer cmps 19858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-of 7226 df-om 7396 df-1st 7500 df-2nd 7501 df-supp 7633 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-oadd 7908 df-er 8088 df-map 8207 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-fsupp 8628 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-5 11505 df-6 11506 df-7 11507 df-8 11508 df-9 11509 df-n0 11707 df-z 11793 df-uz 12058 df-fz 12708 df-struct 16340 df-ndx 16341 df-slot 16342 df-base 16344 df-plusg 16433 df-mulr 16434 df-sca 16436 df-vsca 16437 df-tset 16439 df-0g 16570 df-mgm 17723 df-sgrp 17765 df-mnd 17776 df-grp 17907 df-psr 19863 |
This theorem is referenced by: psrgrp 19905 psr0 19906 |
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