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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 2821 | . . 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 20174 | . . 3 ⊢ (𝜑 → (𝐷 × { 0 }) ∈ 𝐵) |
10 | psr0lid.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 1, 2, 3, 4, 9, 10 | psradd 20162 | . 2 ⊢ (𝜑 → ((𝐷 × { 0 }) + 𝑋) = ((𝐷 × { 0 }) ∘f (+g‘𝑅)𝑋)) |
12 | ovex 7189 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
13 | 7, 12 | rabex2 5237 | . . . 4 ⊢ 𝐷 ∈ V |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
15 | eqid 2821 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
16 | 1, 15, 7, 2, 10 | psrelbas 20159 | . . 3 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) |
17 | 8 | fvexi 6684 | . . . 4 ⊢ 0 ∈ V |
18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
19 | 15, 3, 8 | grplid 18133 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ( 0 (+g‘𝑅)𝑥) = 𝑥) |
20 | 6, 19 | sylan 582 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ( 0 (+g‘𝑅)𝑥) = 𝑥) |
21 | 14, 16, 18, 20 | caofid0l 7437 | . 2 ⊢ (𝜑 → ((𝐷 × { 0 }) ∘f (+g‘𝑅)𝑋) = 𝑋) |
22 | 11, 21 | eqtrd 2856 | 1 ⊢ (𝜑 → ((𝐷 × { 0 }) + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3494 {csn 4567 × cxp 5553 ◡ccnv 5554 “ cima 5558 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 ↑m cmap 8406 Fincfn 8509 ℕcn 11638 ℕ0cn0 11898 Basecbs 16483 +gcplusg 16565 0gc0g 16713 Grpcgrp 18103 mPwSer cmps 20131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-tset 16584 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-psr 20136 |
This theorem is referenced by: psrgrp 20178 psr0 20179 |
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