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Mirrors > Home > MPE Home > Th. List > plyaddlem | Structured version Visualization version GIF version |
Description: Lemma for plyadd 24193. (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 24170 | . . . . . . . . . 10 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
7 | 1, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
8 | 0cnd 10239 | . . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℂ) | |
9 | 8 | snssd 4476 | . . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ ℂ) |
10 | 7, 9 | unssd 3940 | . . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ) |
11 | cnex 10223 | . . . . . . . 8 ⊢ ℂ ∈ V | |
12 | ssexg 4939 | . . . . . . . 8 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V) | |
13 | 10, 11, 12 | sylancl 574 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∪ {0}) ∈ V) |
14 | nn0ex 11505 | . . . . . . 7 ⊢ ℕ0 ∈ V | |
15 | elmapg 8026 | . . . . . . 7 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝐴:ℕ0⟶(𝑆 ∪ {0}))) | |
16 | 13, 14, 15 | sylancl 574 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝐴:ℕ0⟶(𝑆 ∪ {0}))) |
17 | 5, 16 | mpbid 222 | . . . . 5 ⊢ (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0})) |
18 | 17, 10 | fssd 6198 | . . . 4 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
19 | plyadd.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) | |
20 | elmapg 8026 | . . . . . . 7 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝐵:ℕ0⟶(𝑆 ∪ {0}))) | |
21 | 13, 14, 20 | sylancl 574 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝐵:ℕ0⟶(𝑆 ∪ {0}))) |
22 | 19, 21 | mpbid 222 | . . . . 5 ⊢ (𝜑 → 𝐵:ℕ0⟶(𝑆 ∪ {0})) |
23 | 22, 10 | fssd 6198 | . . . 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 24189 | . . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘)))) |
29 | 4, 3 | ifcld 4271 | . . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0) |
30 | eqid 2771 | . . . . . . 7 ⊢ (𝑆 ∪ {0}) = (𝑆 ∪ {0}) | |
31 | plyadd.3 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
32 | 7, 30, 31 | un0addcl 11533 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∪ {0}) ∧ 𝑦 ∈ (𝑆 ∪ {0}))) → (𝑥 + 𝑦) ∈ (𝑆 ∪ {0})) |
33 | 14 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℕ0 ∈ V) |
34 | inidm 3971 | . . . . . 6 ⊢ (ℕ0 ∩ ℕ0) = ℕ0 | |
35 | 32, 17, 22, 33, 33, 34 | off 7063 | . . . . 5 ⊢ (𝜑 → (𝐴 ∘𝑓 + 𝐵):ℕ0⟶(𝑆 ∪ {0})) |
36 | elfznn0 12640 | . . . . 5 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0) | |
37 | ffvelrn 6502 | . . . . 5 ⊢ (((𝐴 ∘𝑓 + 𝐵):ℕ0⟶(𝑆 ∪ {0}) ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘𝑓 + 𝐵)‘𝑘) ∈ (𝑆 ∪ {0})) | |
38 | 35, 36, 37 | syl2an 583 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) → ((𝐴 ∘𝑓 + 𝐵)‘𝑘) ∈ (𝑆 ∪ {0})) |
39 | 10, 29, 38 | elplyd 24178 | . . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘(𝑆 ∪ {0}))) |
40 | 28, 39 | eqeltrd 2850 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ (Poly‘(𝑆 ∪ {0}))) |
41 | plyun0 24173 | . 2 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆) | |
42 | 40, 41 | syl6eleq 2860 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ (Poly‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ∪ cun 3721 ⊆ wss 3723 ifcif 4226 {csn 4317 class class class wbr 4787 ↦ cmpt 4864 “ cima 5253 ⟶wf 6026 ‘cfv 6030 (class class class)co 6796 ∘𝑓 cof 7046 ↑𝑚 cmap 8013 ℂcc 10140 0cc0 10142 1c1 10143 + caddc 10145 · cmul 10147 ≤ cle 10281 ℕ0cn0 11499 ℤ≥cuz 11893 ...cfz 12533 ↑cexp 13067 Σcsu 14624 Polycply 24160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-of 7048 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7900 df-map 8015 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-sup 8508 df-oi 8575 df-card 8969 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-n0 11500 df-z 11585 df-uz 11894 df-rp 12036 df-fz 12534 df-fzo 12674 df-seq 13009 df-exp 13068 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-sum 14625 df-ply 24164 |
This theorem is referenced by: plyadd 24193 |
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