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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 2725 | . . . . . 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 21896 | . . . . 5 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) |
8 | psrmulcl.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | 3, 4, 1, 5, 8 | psrelbas 21896 | . . . . 5 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅)) |
10 | 1, 2, 7, 9 | rhmpsrlem2 21903 | . . . 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 7124 | . . 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 6909 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
13 | ovex 7452 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
14 | 1, 13 | rabex2 5337 | . . . 4 ⊢ 𝐷 ∈ V |
15 | 12, 14 | elmap 8890 | . . 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 233 | . 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 2725 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
18 | psrmulcl.t | . . 3 ⊢ · = (.r‘𝑆) | |
19 | 3, 5, 17, 18, 1, 6, 8 | psrmulfval 21905 | . 2 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥))))))) |
20 | reldmpsr 21864 | . . . . . 6 ⊢ Rel dom mPwSer | |
21 | 20, 3, 5 | elbasov 17190 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
22 | 6, 21 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
23 | 22 | simpld 493 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
24 | 3, 4, 1, 5, 23 | psrbas 21895 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m 𝐷)) |
25 | 16, 19, 24 | 3eltr4d 2840 | 1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3418 Vcvv 3461 class class class wbr 5149 ↦ cmpt 5232 ◡ccnv 5677 “ cima 5681 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ∘f cof 7683 ∘r cofr 7684 ↑m cmap 8845 Fincfn 8964 ≤ cle 11281 − cmin 11476 ℕcn 12245 ℕ0cn0 12505 Basecbs 17183 .rcmulr 17237 Σg cgsu 17425 Ringcrg 20185 mPwSer cmps 21854 |
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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-ofr 7686 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-tset 17255 df-0g 17426 df-gsum 17427 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-cntz 19280 df-cmn 19749 df-abl 19750 df-mgp 20087 df-ur 20134 df-ring 20187 df-psr 21859 |
This theorem is referenced by: psrmulcl 21908 |
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