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Mirrors > Home > MPE Home > Th. List > plyaddlem | Structured version Visualization version GIF version |
Description: Lemma for plyadd 24525. (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 24502 | . . . . . . . . . 10 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
7 | 1, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
8 | 0cnd 10430 | . . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℂ) | |
9 | 8 | snssd 4612 | . . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ ℂ) |
10 | 7, 9 | unssd 4044 | . . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ) |
11 | cnex 10414 | . . . . . . . 8 ⊢ ℂ ∈ V | |
12 | ssexg 5079 | . . . . . . . 8 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V) | |
13 | 10, 11, 12 | sylancl 578 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∪ {0}) ∈ V) |
14 | nn0ex 11712 | . . . . . . 7 ⊢ ℕ0 ∈ V | |
15 | elmapg 8217 | . . . . . . 7 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝐴:ℕ0⟶(𝑆 ∪ {0}))) | |
16 | 13, 14, 15 | sylancl 578 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝐴:ℕ0⟶(𝑆 ∪ {0}))) |
17 | 5, 16 | mpbid 224 | . . . . 5 ⊢ (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0})) |
18 | 17, 10 | fssd 6355 | . . . 4 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
19 | plyadd.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) | |
20 | elmapg 8217 | . . . . . . 7 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝐵:ℕ0⟶(𝑆 ∪ {0}))) | |
21 | 13, 14, 20 | sylancl 578 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝐵:ℕ0⟶(𝑆 ∪ {0}))) |
22 | 19, 21 | mpbid 224 | . . . . 5 ⊢ (𝜑 → 𝐵:ℕ0⟶(𝑆 ∪ {0})) |
23 | 22, 10 | fssd 6355 | . . . 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 24521 | . . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘)))) |
29 | 4, 3 | ifcld 4389 | . . . 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 11740 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∪ {0}) ∧ 𝑦 ∈ (𝑆 ∪ {0}))) → (𝑥 + 𝑦) ∈ (𝑆 ∪ {0})) |
33 | 14 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℕ0 ∈ V) |
34 | inidm 4076 | . . . . . 6 ⊢ (ℕ0 ∩ ℕ0) = ℕ0 | |
35 | 32, 17, 22, 33, 33, 34 | off 7240 | . . . . 5 ⊢ (𝜑 → (𝐴 ∘𝑓 + 𝐵):ℕ0⟶(𝑆 ∪ {0})) |
36 | elfznn0 12814 | . . . . 5 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0) | |
37 | ffvelrn 6672 | . . . . 5 ⊢ (((𝐴 ∘𝑓 + 𝐵):ℕ0⟶(𝑆 ∪ {0}) ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘𝑓 + 𝐵)‘𝑘) ∈ (𝑆 ∪ {0})) | |
38 | 35, 36, 37 | syl2an 587 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) → ((𝐴 ∘𝑓 + 𝐵)‘𝑘) ∈ (𝑆 ∪ {0})) |
39 | 10, 29, 38 | elplyd 24510 | . . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘(𝑆 ∪ {0}))) |
40 | 28, 39 | eqeltrd 2859 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ (Poly‘(𝑆 ∪ {0}))) |
41 | plyun0 24505 | . 2 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆) | |
42 | 40, 41 | syl6eleq 2869 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ (Poly‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 Vcvv 3408 ∪ cun 3820 ⊆ wss 3822 ifcif 4344 {csn 4435 class class class wbr 4925 ↦ cmpt 5004 “ cima 5406 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 ∘𝑓 cof 7223 ↑𝑚 cmap 8204 ℂcc 10331 0cc0 10333 1c1 10334 + caddc 10336 · cmul 10338 ≤ cle 10473 ℕ0cn0 11705 ℤ≥cuz 12056 ...cfz 12706 ↑cexp 13242 Σcsu 14901 Polycply 24492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-sup 8699 df-oi 8767 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-n0 11706 df-z 11792 df-uz 12057 df-rp 12203 df-fz 12707 df-fzo 12848 df-seq 13183 df-exp 13243 df-hash 13504 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-clim 14704 df-sum 14902 df-ply 24496 |
This theorem is referenced by: plyadd 24525 |
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