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Mirrors > Home > MPE Home > Th. List > plyaddlem | Structured version Visualization version GIF version |
Description: Lemma for plyadd 23777. (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 23754 | . . . . . . . . . 10 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
7 | 1, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
8 | 0cnd 9912 | . . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℂ) | |
9 | 8 | snssd 4281 | . . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ ℂ) |
10 | 7, 9 | unssd 3751 | . . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ) |
11 | cnex 9896 | . . . . . . . 8 ⊢ ℂ ∈ V | |
12 | ssexg 4732 | . . . . . . . 8 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V) | |
13 | 10, 11, 12 | sylancl 693 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∪ {0}) ∈ V) |
14 | nn0ex 11175 | . . . . . . 7 ⊢ ℕ0 ∈ V | |
15 | elmapg 7757 | . . . . . . 7 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝐴:ℕ0⟶(𝑆 ∪ {0}))) | |
16 | 13, 14, 15 | sylancl 693 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝐴:ℕ0⟶(𝑆 ∪ {0}))) |
17 | 5, 16 | mpbid 221 | . . . . 5 ⊢ (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0})) |
18 | 17, 10 | fssd 5970 | . . . 4 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
19 | plyadd.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) | |
20 | elmapg 7757 | . . . . . . 7 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝐵:ℕ0⟶(𝑆 ∪ {0}))) | |
21 | 13, 14, 20 | sylancl 693 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝐵:ℕ0⟶(𝑆 ∪ {0}))) |
22 | 19, 21 | mpbid 221 | . . . . 5 ⊢ (𝜑 → 𝐵:ℕ0⟶(𝑆 ∪ {0})) |
23 | 22, 10 | fssd 5970 | . . . 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 23773 | . . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘)))) |
29 | 4, 3 | ifcld 4081 | . . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0) |
30 | eqid 2610 | . . . . . . 7 ⊢ (𝑆 ∪ {0}) = (𝑆 ∪ {0}) | |
31 | plyadd.3 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
32 | 7, 30, 31 | un0addcl 11203 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∪ {0}) ∧ 𝑦 ∈ (𝑆 ∪ {0}))) → (𝑥 + 𝑦) ∈ (𝑆 ∪ {0})) |
33 | 14 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℕ0 ∈ V) |
34 | inidm 3784 | . . . . . 6 ⊢ (ℕ0 ∩ ℕ0) = ℕ0 | |
35 | 32, 17, 22, 33, 33, 34 | off 6810 | . . . . 5 ⊢ (𝜑 → (𝐴 ∘𝑓 + 𝐵):ℕ0⟶(𝑆 ∪ {0})) |
36 | elfznn0 12302 | . . . . 5 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0) | |
37 | ffvelrn 6265 | . . . . 5 ⊢ (((𝐴 ∘𝑓 + 𝐵):ℕ0⟶(𝑆 ∪ {0}) ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘𝑓 + 𝐵)‘𝑘) ∈ (𝑆 ∪ {0})) | |
38 | 35, 36, 37 | syl2an 493 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) → ((𝐴 ∘𝑓 + 𝐵)‘𝑘) ∈ (𝑆 ∪ {0})) |
39 | 10, 29, 38 | elplyd 23762 | . . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘(𝑆 ∪ {0}))) |
40 | 28, 39 | eqeltrd 2688 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ (Poly‘(𝑆 ∪ {0}))) |
41 | plyun0 23757 | . 2 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆) | |
42 | 40, 41 | syl6eleq 2698 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ (Poly‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 ifcif 4036 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 ↑𝑚 cmap 7744 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ≤ cle 9954 ℕ0cn0 11169 ℤ≥cuz 11563 ...cfz 12197 ↑cexp 12722 Σcsu 14264 Polycply 23744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-ply 23748 |
This theorem is referenced by: plyadd 23777 |
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