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| Description: Lemma for plyadd 26257. (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 26234 | . . . . . . . . . 10 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
| 7 | 1, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | 
| 8 | 0cnd 11255 | . . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℂ) | |
| 9 | 8 | snssd 4808 | . . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ ℂ) | 
| 10 | 7, 9 | unssd 4191 | . . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ) | 
| 11 | cnex 11237 | . . . . . . . 8 ⊢ ℂ ∈ V | |
| 12 | ssexg 5322 | . . . . . . . 8 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V) | |
| 13 | 10, 11, 12 | sylancl 586 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∪ {0}) ∈ V) | 
| 14 | nn0ex 12534 | . . . . . . 7 ⊢ ℕ0 ∈ V | |
| 15 | elmapg 8880 | . . . . . . 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 232 | . . . . 5 ⊢ (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0})) | 
| 18 | 17, 10 | fssd 6752 | . . . 4 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | 
| 19 | plyadd.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) | |
| 20 | elmapg 8880 | . . . . . . 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 232 | . . . . 5 ⊢ (𝜑 → 𝐵:ℕ0⟶(𝑆 ∪ {0})) | 
| 23 | 22, 10 | fssd 6752 | . . . 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 26253 | . . 3 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘f + 𝐵)‘𝑘) · (𝑧↑𝑘)))) | 
| 29 | 4, 3 | ifcld 4571 | . . . 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 12561 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∪ {0}) ∧ 𝑦 ∈ (𝑆 ∪ {0}))) → (𝑥 + 𝑦) ∈ (𝑆 ∪ {0})) | 
| 33 | 14 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℕ0 ∈ V) | 
| 34 | inidm 4226 | . . . . . 6 ⊢ (ℕ0 ∩ ℕ0) = ℕ0 | |
| 35 | 32, 17, 22, 33, 33, 34 | off 7716 | . . . . 5 ⊢ (𝜑 → (𝐴 ∘f + 𝐵):ℕ0⟶(𝑆 ∪ {0})) | 
| 36 | elfznn0 13661 | . . . . 5 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0) | |
| 37 | ffvelcdm 7100 | . . . . 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 26242 | . . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘f + 𝐵)‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘(𝑆 ∪ {0}))) | 
| 40 | 28, 39 | eqeltrd 2840 | . 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (Poly‘(𝑆 ∪ {0}))) | 
| 41 | plyun0 26237 | . 2 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆) | |
| 42 | 40, 41 | eleqtrdi 2850 | 1 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∪ cun 3948 ⊆ wss 3950 ifcif 4524 {csn 4625 class class class wbr 5142 ↦ cmpt 5224 “ cima 5687 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 ↑m cmap 8867 ℂcc 11154 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 ≤ cle 11297 ℕ0cn0 12528 ℤ≥cuz 12879 ...cfz 13548 ↑cexp 14103 Σcsu 15723 Polycply 26224 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-sum 15724 df-ply 26228 | 
| This theorem is referenced by: plyadd 26257 | 
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