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Mirrors > Home > MPE Home > Th. List > psraddcl | Structured version Visualization version GIF version |
Description: Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.) Generalize to magmas. (Revised by SN, 12-Apr-2025.) |
Ref | Expression |
---|---|
psraddcl.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
psraddcl.b | ⊢ 𝐵 = (Base‘𝑆) |
psraddcl.p | ⊢ + = (+g‘𝑆) |
psraddcl.r | ⊢ (𝜑 → 𝑅 ∈ Mgm) |
psraddcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
psraddcl.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
psraddcl | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psraddcl.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Mgm) | |
2 | eqid 2728 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2728 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | 2, 3 | mgmcl 18612 | . . . . . 6 ⊢ ((𝑅 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
5 | 4 | 3expb 1117 | . . . . 5 ⊢ ((𝑅 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
6 | 1, 5 | sylan 578 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
7 | psraddcl.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
8 | eqid 2728 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
9 | psraddcl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
10 | psraddcl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 7, 2, 8, 9, 10 | psrelbas 21893 | . . . 4 ⊢ (𝜑 → 𝑋:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
12 | psraddcl.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
13 | 7, 2, 8, 9, 12 | psrelbas 21893 | . . . 4 ⊢ (𝜑 → 𝑌:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
14 | ovex 7459 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
15 | 14 | rabex 5338 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) |
17 | inidm 4221 | . . . 4 ⊢ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∩ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
18 | 6, 11, 13, 16, 16, 17 | off 7710 | . . 3 ⊢ (𝜑 → (𝑋 ∘f (+g‘𝑅)𝑌):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
19 | fvex 6915 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
20 | 19, 15 | elmap 8898 | . . 3 ⊢ ((𝑋 ∘f (+g‘𝑅)𝑌) ∈ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ↔ (𝑋 ∘f (+g‘𝑅)𝑌):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
21 | 18, 20 | sylibr 233 | . 2 ⊢ (𝜑 → (𝑋 ∘f (+g‘𝑅)𝑌) ∈ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
22 | psraddcl.p | . . 3 ⊢ + = (+g‘𝑆) | |
23 | 7, 9, 3, 22, 10, 12 | psradd 21896 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑋 ∘f (+g‘𝑅)𝑌)) |
24 | reldmpsr 21861 | . . . . . 6 ⊢ Rel dom mPwSer | |
25 | 24, 7, 9 | elbasov 17196 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
26 | 10, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
27 | 26 | simpld 493 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
28 | 7, 2, 8, 9, 27 | psrbas 21892 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
29 | 21, 23, 28 | 3eltr4d 2844 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3430 Vcvv 3473 ◡ccnv 5681 “ cima 5685 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ∘f cof 7690 ↑m cmap 8853 Fincfn 8972 ℕcn 12252 ℕ0cn0 12512 Basecbs 17189 +gcplusg 17242 Mgmcmgm 18607 mPwSer cmps 21851 |
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 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
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-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 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-struct 17125 df-slot 17160 df-ndx 17172 df-base 17190 df-plusg 17255 df-mulr 17256 df-sca 17258 df-vsca 17259 df-tset 17261 df-mgm 18609 df-psr 21856 |
This theorem is referenced by: psrgrpOLD 21916 psrlmod 21919 psrdi 21924 psrdir 21925 mplsubglem 21958 psdadd 22105 |
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