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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 2728 | . . 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 21902 | . . 3 ⊢ (𝜑 → (𝐷 × { 0 }) ∈ 𝐵) |
10 | psr0lid.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 1, 2, 3, 4, 9, 10 | psradd 21889 | . 2 ⊢ (𝜑 → ((𝐷 × { 0 }) + 𝑋) = ((𝐷 × { 0 }) ∘f (+g‘𝑅)𝑋)) |
12 | ovex 7459 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
13 | 7, 12 | rabex2 5340 | . . . 4 ⊢ 𝐷 ∈ V |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
15 | eqid 2728 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
16 | 1, 15, 7, 2, 10 | psrelbas 21886 | . . 3 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) |
17 | 8 | fvexi 6916 | . . . 4 ⊢ 0 ∈ V |
18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
19 | 15, 3, 8 | grplid 18931 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ( 0 (+g‘𝑅)𝑥) = 𝑥) |
20 | 6, 19 | sylan 578 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ( 0 (+g‘𝑅)𝑥) = 𝑥) |
21 | 14, 16, 18, 20 | caofid0l 7722 | . 2 ⊢ (𝜑 → ((𝐷 × { 0 }) ∘f (+g‘𝑅)𝑋) = 𝑋) |
22 | 11, 21 | eqtrd 2768 | 1 ⊢ (𝜑 → ((𝐷 × { 0 }) + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3430 Vcvv 3473 {csn 4632 × cxp 5680 ◡ccnv 5681 “ cima 5685 ‘cfv 6553 (class class class)co 7426 ∘f cof 7689 ↑m cmap 8851 Fincfn 8970 ℕcn 12250 ℕ0cn0 12510 Basecbs 17187 +gcplusg 17240 0gc0g 17428 Grpcgrp 18897 mPwSer cmps 21844 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-tset 17259 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-psr 21849 |
This theorem is referenced by: psrgrpOLD 21907 psr0 21908 |
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