plyaddlem - Intuitionistic Logic Explorer

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Theorem plyaddlem 15444

Description: Lemma for plyadd 15446. (Contributed by Mario Carneiro, 21-Jul-2014.)

**Hypotheses**
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))))

**Assertion**
Ref	Expression
plyaddlem	⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) ∈ (Poly‘𝑆))

Distinct variable groups: 𝑥,𝑘,𝑦,𝑧,𝐵 𝑥,𝐹,𝑦,𝑧 𝑆,𝑘,𝑥,𝑦,𝑧 𝑥,𝐴,𝑦,𝑧 𝑥,𝐺,𝑦,𝑧 𝜑,𝑘,𝑥,𝑦,𝑧 𝑘,𝑀,𝑧 𝑘,𝑁,𝑧 Allowed substitution hints: 𝐴(𝑘) 𝐹(𝑘) 𝐺(𝑘) 𝑀(𝑥,𝑦) 𝑁(𝑥,𝑦)

**Proof of Theorem plyaddlem**
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 15428	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
7	1, 6	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
8		0cnd 8155	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℂ)
9	8	snssd 3813	. . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ ℂ)
10	7, 9	unssd 3380	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
11		cnex 8139	. . . . . . . 8 ⊢ ℂ ∈ V
12		ssexg 4223	. . . . . . . 8 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V)
13	10, 11, 12	sylancl 413	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∪ {0}) ∈ V)
14		nn0ex 9391	. . . . . . 7 ⊢ ℕ₀ ∈ V
15		elmapg 6821	. . . . . . 7 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ₀ ∈ V) → (𝐴 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀) ↔ 𝐴:ℕ₀⟶(𝑆 ∪ {0})))
16	13, 14, 15	sylancl 413	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀) ↔ 𝐴:ℕ₀⟶(𝑆 ∪ {0})))
17	5, 16	mpbid 147	. . . . 5 ⊢ (𝜑 → 𝐴:ℕ₀⟶(𝑆 ∪ {0}))
18	17, 10	fssd 5489	. . . 4 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
19		plyadd.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))
20		elmapg 6821	. . . . . . 7 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ₀ ∈ V) → (𝐵 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀) ↔ 𝐵:ℕ₀⟶(𝑆 ∪ {0})))
21	13, 14, 20	sylancl 413	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀) ↔ 𝐵:ℕ₀⟶(𝑆 ∪ {0})))
22	19, 21	mpbid 147	. . . . 5 ⊢ (𝜑 → 𝐵:ℕ₀⟶(𝑆 ∪ {0}))
23	22, 10	fssd 5489	. . . 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 15442	. . 3 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘_𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘))))
29	3	nn0zd 9583	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
30	4	nn0zd 9583	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
31		zdcle 9539	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑀 ≤ 𝑁)
32	29, 30, 31	syl2anc 411	. . . . 5 ⊢ (𝜑 → DECID 𝑀 ≤ 𝑁)
33	4, 3, 32	ifcldcd 3640	. . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ₀)
34		eqid 2229	. . . . . . 7 ⊢ (𝑆 ∪ {0}) = (𝑆 ∪ {0})
35		plyadd.3	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
36	7, 34, 35	un0addcl 9418	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∪ {0}) ∧ 𝑦 ∈ (𝑆 ∪ {0}))) → (𝑥 + 𝑦) ∈ (𝑆 ∪ {0}))
37	14	a1i 9	. . . . . 6 ⊢ (𝜑 → ℕ₀ ∈ V)
38		inidm 3413	. . . . . 6 ⊢ (ℕ₀ ∩ ℕ₀) = ℕ₀
39	36, 17, 22, 37, 37, 38	off 6240	. . . . 5 ⊢ (𝜑 → (𝐴 ∘_𝑓 + 𝐵):ℕ₀⟶(𝑆 ∪ {0}))
40		elfznn0 10327	. . . . 5 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ₀)
41		ffvelcdm 5773	. . . . 5 ⊢ (((𝐴 ∘_𝑓 + 𝐵):ℕ₀⟶(𝑆 ∪ {0}) ∧ 𝑘 ∈ ℕ₀) → ((𝐴 ∘_𝑓 + 𝐵)‘𝑘) ∈ (𝑆 ∪ {0}))
42	39, 40, 41	syl2an 289	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) → ((𝐴 ∘_𝑓 + 𝐵)‘𝑘) ∈ (𝑆 ∪ {0}))
43	10, 33, 42	elplyd 15436	. . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘_𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘(𝑆 ∪ {0})))
44	28, 43	eqeltrd 2306	. 2 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) ∈ (Poly‘(𝑆 ∪ {0})))
45		plyun0 15431	. 2 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
46	44, 45	eleqtrdi 2322	1 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) ∈ (Poly‘𝑆))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 839 = wceq 1395 ∈ wcel 2200 Vcvv 2799 ∪ cun 3195 ⊆ wss 3197 ifcif 3602 {csn 3666 class class class wbr 4083 ↦ cmpt 4145 “ cima 4723 ⟶wf 5317 ‘cfv 5321 (class class class)co 6010 ∘_𝑓 cof 6225 ↑_𝑚 cmap 6808 ℂcc 8013 0cc0 8015 1c1 8016 + caddc 8018 · cmul 8020 ≤ cle 8198 ℕ₀cn0 9385 ℤcz 9462 ℤ_≥cuz 9738 ...cfz 10221 ↑cexp 10777 Σcsu 11885 Polycply 15423

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4259 ax-pr 4294 ax-un 4525 ax-setind 4630 ax-iinf 4681 ax-cnex 8106 ax-resscn 8107 ax-1cn 8108 ax-1re 8109 ax-icn 8110 ax-addcl 8111 ax-addrcl 8112 ax-mulcl 8113 ax-mulrcl 8114 ax-addcom 8115 ax-mulcom 8116 ax-addass 8117 ax-mulass 8118 ax-distr 8119 ax-i2m1 8120 ax-0lt1 8121 ax-1rid 8122 ax-0id 8123 ax-rnegex 8124 ax-precex 8125 ax-cnre 8126 ax-pre-ltirr 8127 ax-pre-ltwlin 8128 ax-pre-lttrn 8129 ax-pre-apti 8130 ax-pre-ltadd 8131 ax-pre-mulgt0 8132 ax-pre-mulext 8133 ax-arch 8134 ax-caucvg 8135

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4385 df-po 4388 df-iso 4389 df-iord 4458 df-on 4460 df-ilim 4461 df-suc 4463 df-iom 4684 df-xp 4726 df-rel 4727 df-cnv 4728 df-co 4729 df-dm 4730 df-rn 4731 df-res 4732 df-ima 4733 df-iota 5281 df-fun 5323 df-fn 5324 df-f 5325 df-f1 5326 df-fo 5327 df-f1o 5328 df-fv 5329 df-isom 5330 df-riota 5963 df-ov 6013 df-oprab 6014 df-mpo 6015 df-of 6227 df-1st 6295 df-2nd 6296 df-recs 6462 df-irdg 6527 df-frec 6548 df-1o 6573 df-oadd 6577 df-er 6693 df-map 6810 df-en 6901 df-dom 6902 df-fin 6903 df-sup 7167 df-pnf 8199 df-mnf 8200 df-xr 8201 df-ltxr 8202 df-le 8203 df-sub 8335 df-neg 8336 df-reap 8738 df-ap 8745 df-div 8836 df-inn 9127 df-2 9185 df-3 9186 df-4 9187 df-n0 9386 df-z 9463 df-uz 9739 df-q 9832 df-rp 9867 df-fz 10222 df-fzo 10356 df-seqfrec 10687 df-exp 10778 df-ihash 11015 df-cj 11374 df-re 11375 df-im 11376 df-rsqrt 11530 df-abs 11531 df-clim 11811 df-sumdc 11886 df-ply 15425

This theorem is referenced by: plyadd 15446

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