plyaddlem - Intuitionistic Logic Explorer

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Theorem plyaddlem 15388

Description: Lemma for plyadd 15390. (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 15372	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
7	1, 6	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
8		0cnd 8107	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℂ)
9	8	snssd 3792	. . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ ℂ)
10	7, 9	unssd 3360	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
11		cnex 8091	. . . . . . . 8 ⊢ ℂ ∈ V
12		ssexg 4202	. . . . . . . 8 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V)
13	10, 11, 12	sylancl 413	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∪ {0}) ∈ V)
14		nn0ex 9343	. . . . . . 7 ⊢ ℕ₀ ∈ V
15		elmapg 6778	. . . . . . 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 5462	. . . 4 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
19		plyadd.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))
20		elmapg 6778	. . . . . . 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 5462	. . . 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 15386	. . 3 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘_𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘))))
29	3	nn0zd 9535	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
30	4	nn0zd 9535	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
31		zdcle 9491	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑀 ≤ 𝑁)
32	29, 30, 31	syl2anc 411	. . . . 5 ⊢ (𝜑 → DECID 𝑀 ≤ 𝑁)
33	4, 3, 32	ifcldcd 3620	. . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ₀)
34		eqid 2209	. . . . . . 7 ⊢ (𝑆 ∪ {0}) = (𝑆 ∪ {0})
35		plyadd.3	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
36	7, 34, 35	un0addcl 9370	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∪ {0}) ∧ 𝑦 ∈ (𝑆 ∪ {0}))) → (𝑥 + 𝑦) ∈ (𝑆 ∪ {0}))
37	14	a1i 9	. . . . . 6 ⊢ (𝜑 → ℕ₀ ∈ V)
38		inidm 3393	. . . . . 6 ⊢ (ℕ₀ ∩ ℕ₀) = ℕ₀
39	36, 17, 22, 37, 37, 38	off 6201	. . . . 5 ⊢ (𝜑 → (𝐴 ∘_𝑓 + 𝐵):ℕ₀⟶(𝑆 ∪ {0}))
40		elfznn0 10278	. . . . 5 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ₀)
41		ffvelcdm 5741	. . . . 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 15380	. . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘_𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘(𝑆 ∪ {0})))
44	28, 43	eqeltrd 2286	. 2 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) ∈ (Poly‘(𝑆 ∪ {0})))
45		plyun0 15375	. 2 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
46	44, 45	eleqtrdi 2302	1 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) ∈ (Poly‘𝑆))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 838 = wceq 1375 ∈ wcel 2180 Vcvv 2779 ∪ cun 3175 ⊆ wss 3177 ifcif 3582 {csn 3646 class class class wbr 4062 ↦ cmpt 4124 “ cima 4699 ⟶wf 5290 ‘cfv 5294 (class class class)co 5974 ∘_𝑓 cof 6186 ↑_𝑚 cmap 6765 ℂcc 7965 0cc0 7967 1c1 7968 + caddc 7970 · cmul 7972 ≤ cle 8150 ℕ₀cn0 9337 ℤcz 9414 ℤ_≥cuz 9690 ...cfz 10172 ↑cexp 10727 Σcsu 11830 Polycply 15367

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 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-nul 4189 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-mulrcl 8066 ax-addcom 8067 ax-mulcom 8068 ax-addass 8069 ax-mulass 8070 ax-distr 8071 ax-i2m1 8072 ax-0lt1 8073 ax-1rid 8074 ax-0id 8075 ax-rnegex 8076 ax-precex 8077 ax-cnre 8078 ax-pre-ltirr 8079 ax-pre-ltwlin 8080 ax-pre-lttrn 8081 ax-pre-apti 8082 ax-pre-ltadd 8083 ax-pre-mulgt0 8084 ax-pre-mulext 8085 ax-arch 8086 ax-caucvg 8087

This theorem depends on definitions: df-bi 117 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rmo 2496 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-if 3583 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-tr 4162 df-id 4361 df-po 4364 df-iso 4365 df-iord 4434 df-on 4436 df-ilim 4437 df-suc 4439 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-isom 5303 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-of 6188 df-1st 6256 df-2nd 6257 df-recs 6421 df-irdg 6486 df-frec 6507 df-1o 6532 df-oadd 6536 df-er 6650 df-map 6767 df-en 6858 df-dom 6859 df-fin 6860 df-sup 7119 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-sub 8287 df-neg 8288 df-reap 8690 df-ap 8697 df-div 8788 df-inn 9079 df-2 9137 df-3 9138 df-4 9139 df-n0 9338 df-z 9415 df-uz 9691 df-q 9783 df-rp 9818 df-fz 10173 df-fzo 10307 df-seqfrec 10637 df-exp 10728 df-ihash 10965 df-cj 11319 df-re 11320 df-im 11321 df-rsqrt 11475 df-abs 11476 df-clim 11756 df-sumdc 11831 df-ply 15369

This theorem is referenced by: plyadd 15390

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