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Mirrors > Home > MPE Home > Th. List > plyaddlem | Structured version Visualization version GIF version |
Description: Lemma for plyadd 25576. (Contributed by Mario Carneiro, 21-Jul-2014.) |
Ref | Expression |
---|---|
plyadd.1 | ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) |
plyadd.2 | ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) |
plyadd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
plyadd.m | ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
plyadd.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
plyadd.a | ⊢ (𝜑 → 𝐴 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) |
plyadd.b | ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) |
plyadd.a2 | ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) |
plyadd.b2 | ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0}) |
plyadd.f | ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))) |
plyadd.g | ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))) |
Ref | Expression |
---|---|
plyaddlem | ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plyadd.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) | |
2 | plyadd.2 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) | |
3 | plyadd.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
4 | plyadd.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
5 | plyadd.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) | |
6 | plybss 25553 | . . . . . . . . . 10 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
7 | 1, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
8 | 0cnd 11147 | . . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℂ) | |
9 | 8 | snssd 4769 | . . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ ℂ) |
10 | 7, 9 | unssd 4146 | . . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ) |
11 | cnex 11131 | . . . . . . . 8 ⊢ ℂ ∈ V | |
12 | ssexg 5280 | . . . . . . . 8 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V) | |
13 | 10, 11, 12 | sylancl 586 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∪ {0}) ∈ V) |
14 | nn0ex 12418 | . . . . . . 7 ⊢ ℕ0 ∈ V | |
15 | elmapg 8777 | . . . . . . 7 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝐴 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ↔ 𝐴:ℕ0⟶(𝑆 ∪ {0}))) | |
16 | 13, 14, 15 | sylancl 586 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ↔ 𝐴:ℕ0⟶(𝑆 ∪ {0}))) |
17 | 5, 16 | mpbid 231 | . . . . 5 ⊢ (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0})) |
18 | 17, 10 | fssd 6686 | . . . 4 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
19 | plyadd.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) | |
20 | elmapg 8777 | . . . . . . 7 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝐵 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ↔ 𝐵:ℕ0⟶(𝑆 ∪ {0}))) | |
21 | 13, 14, 20 | sylancl 586 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ↔ 𝐵:ℕ0⟶(𝑆 ∪ {0}))) |
22 | 19, 21 | mpbid 231 | . . . . 5 ⊢ (𝜑 → 𝐵:ℕ0⟶(𝑆 ∪ {0})) |
23 | 22, 10 | fssd 6686 | . . . 4 ⊢ (𝜑 → 𝐵:ℕ0⟶ℂ) |
24 | plyadd.a2 | . . . 4 ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) | |
25 | plyadd.b2 | . . . 4 ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0}) | |
26 | plyadd.f | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))) | |
27 | plyadd.g | . . . 4 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))) | |
28 | 1, 2, 3, 4, 18, 23, 24, 25, 26, 27 | plyaddlem1 25572 | . . 3 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘f + 𝐵)‘𝑘) · (𝑧↑𝑘)))) |
29 | 4, 3 | ifcld 4532 | . . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0) |
30 | eqid 2736 | . . . . . . 7 ⊢ (𝑆 ∪ {0}) = (𝑆 ∪ {0}) | |
31 | plyadd.3 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
32 | 7, 30, 31 | un0addcl 12445 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∪ {0}) ∧ 𝑦 ∈ (𝑆 ∪ {0}))) → (𝑥 + 𝑦) ∈ (𝑆 ∪ {0})) |
33 | 14 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℕ0 ∈ V) |
34 | inidm 4178 | . . . . . 6 ⊢ (ℕ0 ∩ ℕ0) = ℕ0 | |
35 | 32, 17, 22, 33, 33, 34 | off 7634 | . . . . 5 ⊢ (𝜑 → (𝐴 ∘f + 𝐵):ℕ0⟶(𝑆 ∪ {0})) |
36 | elfznn0 13533 | . . . . 5 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0) | |
37 | ffvelcdm 7032 | . . . . 5 ⊢ (((𝐴 ∘f + 𝐵):ℕ0⟶(𝑆 ∪ {0}) ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘f + 𝐵)‘𝑘) ∈ (𝑆 ∪ {0})) | |
38 | 35, 36, 37 | syl2an 596 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) → ((𝐴 ∘f + 𝐵)‘𝑘) ∈ (𝑆 ∪ {0})) |
39 | 10, 29, 38 | elplyd 25561 | . . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘f + 𝐵)‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘(𝑆 ∪ {0}))) |
40 | 28, 39 | eqeltrd 2838 | . 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (Poly‘(𝑆 ∪ {0}))) |
41 | plyun0 25556 | . 2 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆) | |
42 | 40, 41 | eleqtrdi 2848 | 1 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ∪ cun 3908 ⊆ wss 3910 ifcif 4486 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 “ cima 5636 ⟶wf 6492 ‘cfv 6496 (class class class)co 7356 ∘f cof 7614 ↑m cmap 8764 ℂcc 11048 0cc0 11050 1c1 11051 + caddc 11053 · cmul 11055 ≤ cle 11189 ℕ0cn0 12412 ℤ≥cuz 12762 ...cfz 13423 ↑cexp 13966 Σcsu 15569 Polycply 25543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-inf2 9576 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-map 8766 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-sup 9377 df-oi 9445 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-n0 12413 df-z 12499 df-uz 12763 df-rp 12915 df-fz 13424 df-fzo 13567 df-seq 13906 df-exp 13967 df-hash 14230 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-clim 15369 df-sum 15570 df-ply 25547 |
This theorem is referenced by: plyadd 25576 |
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