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Mirrors > Home > ILE Home > Th. List > psraddcl | 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 2193 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2193 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | 2, 3 | mgmcl 12942 | . . . . . 6 ⊢ ((𝑅 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
5 | 4 | 3expb 1206 | . . . . 5 ⊢ ((𝑅 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
6 | 1, 5 | sylan 283 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
7 | psraddcl.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
8 | eqid 2193 | . . . . 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 14160 | . . . 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 14160 | . . . 4 ⊢ (𝜑 → 𝑌:{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
14 | fnmap 6709 | . . . . . 6 ⊢ ↑𝑚 Fn (V × V) | |
15 | nn0ex 9246 | . . . . . 6 ⊢ ℕ0 ∈ V | |
16 | reldmpsr 14151 | . . . . . . . . 9 ⊢ Rel dom mPwSer | |
17 | fnpsr 14153 | . . . . . . . . . 10 ⊢ mPwSer Fn (V × V) | |
18 | fnrel 5352 | . . . . . . . . . 10 ⊢ ( mPwSer Fn (V × V) → Rel mPwSer ) | |
19 | 17, 18 | ax-mp 5 | . . . . . . . . 9 ⊢ Rel mPwSer |
20 | 16, 19, 7, 9 | relelbasov 12680 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
21 | 10, 20 | syl 14 | . . . . . . 7 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
22 | 21 | simpld 112 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
23 | fnovex 5951 | . . . . . 6 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℕ0 ∈ V ∧ 𝐼 ∈ V) → (ℕ0 ↑𝑚 𝐼) ∈ V) | |
24 | 14, 15, 22, 23 | mp3an12i 1352 | . . . . 5 ⊢ (𝜑 → (ℕ0 ↑𝑚 𝐼) ∈ V) |
25 | rabexg 4172 | . . . . 5 ⊢ ((ℕ0 ↑𝑚 𝐼) ∈ V → {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) | |
26 | 24, 25 | syl 14 | . . . 4 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) |
27 | inidm 3368 | . . . 4 ⊢ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∩ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
28 | 6, 11, 13, 26, 26, 27 | off 6143 | . . 3 ⊢ (𝜑 → (𝑋 ∘𝑓 (+g‘𝑅)𝑌):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
29 | basfn 12676 | . . . . 5 ⊢ Base Fn V | |
30 | 1 | elexd 2773 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) |
31 | funfvex 5571 | . . . . . 6 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V) | |
32 | 31 | funfni 5354 | . . . . 5 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V) |
33 | 29, 30, 32 | sylancr 414 | . . . 4 ⊢ (𝜑 → (Base‘𝑅) ∈ V) |
34 | 33, 26 | elmapd 6716 | . . 3 ⊢ (𝜑 → ((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))) |
35 | 28, 34 | mpbird 167 | . 2 ⊢ (𝜑 → (𝑋 ∘𝑓 (+g‘𝑅)𝑌) ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
36 | psraddcl.p | . . 3 ⊢ + = (+g‘𝑆) | |
37 | 7, 9, 3, 36, 10, 12 | psradd 14163 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑋 ∘𝑓 (+g‘𝑅)𝑌)) |
38 | 7, 2, 8, 9, 22, 1 | psrbasg 14159 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
39 | 35, 37, 38 | 3eltr4d 2277 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 {crab 2476 Vcvv 2760 × cxp 4657 ◡ccnv 4658 “ cima 4662 Rel wrel 4664 Fn wfn 5249 ⟶wf 5250 ‘cfv 5254 (class class class)co 5918 ∘𝑓 cof 6128 ↑𝑚 cmap 6702 Fincfn 6794 ℕcn 8982 ℕ0cn0 9240 Basecbs 12618 +gcplusg 12695 Mgmcmgm 12937 mPwSer cmps 14149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-addcom 7972 ax-addass 7974 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-0id 7980 ax-rnegex 7981 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-tp 3626 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-of 6130 df-1st 6193 df-2nd 6194 df-map 6704 df-ixp 6753 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-5 9044 df-6 9045 df-7 9046 df-8 9047 df-9 9048 df-n0 9241 df-z 9318 df-uz 9593 df-fz 10075 df-struct 12620 df-ndx 12621 df-slot 12622 df-base 12624 df-plusg 12708 df-mulr 12709 df-sca 12711 df-vsca 12712 df-tset 12714 df-rest 12852 df-topn 12853 df-topgen 12871 df-pt 12872 df-mgm 12939 df-psr 14150 |
This theorem is referenced by: (None) |
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