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| Mirrors > Home > MPE Home > Th. List > psrmulcllem | Structured version Visualization version GIF version | ||
| Description: Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.) |
| Ref | Expression |
|---|---|
| psrmulcl.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
| psrmulcl.b | ⊢ 𝐵 = (Base‘𝑆) |
| psrmulcl.t | ⊢ · = (.r‘𝑆) |
| psrmulcl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
| psrmulcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| psrmulcl.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| psrmulcl.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
| Ref | Expression |
|---|---|
| psrmulcllem | ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | psrmulcl.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
| 2 | psrmulcl.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
| 3 | psrmulcl.s | . . . . . 6 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
| 4 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 5 | psrmulcl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑆) | |
| 6 | psrmulcl.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 7 | 3, 4, 1, 5, 6 | psrelbas 21954 | . . . . 5 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) |
| 8 | psrmulcl.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 9 | 3, 4, 1, 5, 8 | psrelbas 21954 | . . . . 5 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅)) |
| 10 | 1, 2, 7, 9 | rhmpsrlem2 21961 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥))))) ∈ (Base‘𝑅)) |
| 11 | 10 | fmpttd 7135 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥)))))):𝐷⟶(Base‘𝑅)) |
| 12 | fvex 6919 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
| 13 | ovex 7464 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
| 14 | 1, 13 | rabex2 5341 | . . . 4 ⊢ 𝐷 ∈ V |
| 15 | 12, 14 | elmap 8911 | . . 3 ⊢ ((𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥)))))) ∈ ((Base‘𝑅) ↑m 𝐷) ↔ (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥)))))):𝐷⟶(Base‘𝑅)) |
| 16 | 11, 15 | sylibr 234 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥)))))) ∈ ((Base‘𝑅) ↑m 𝐷)) |
| 17 | eqid 2737 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 18 | psrmulcl.t | . . 3 ⊢ · = (.r‘𝑆) | |
| 19 | 3, 5, 17, 18, 1, 6, 8 | psrmulfval 21963 | . 2 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥))))))) |
| 20 | reldmpsr 21934 | . . . . . 6 ⊢ Rel dom mPwSer | |
| 21 | 20, 3, 5 | elbasov 17254 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
| 22 | 6, 21 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
| 23 | 22 | simpld 494 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
| 24 | 3, 4, 1, 5, 23 | psrbas 21953 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m 𝐷)) |
| 25 | 16, 19, 24 | 3eltr4d 2856 | 1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 class class class wbr 5143 ↦ cmpt 5225 ◡ccnv 5684 “ cima 5688 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 ∘r cofr 7696 ↑m cmap 8866 Fincfn 8985 ≤ cle 11296 − cmin 11492 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 .rcmulr 17298 Σg cgsu 17485 Ringcrg 20230 mPwSer cmps 21924 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-ur 20179 df-ring 20232 df-psr 21929 |
| This theorem is referenced by: psrmulcl 21966 |
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