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Theorem plyaddlem 15663

Description: Lemma for plyadd 15665. (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 15647	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
7	1, 6	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
8		0cnd 8272	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℂ)
9	8	snssd 3841	. . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ ℂ)
10	7, 9	unssd 3397	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
11		cnex 8256	. . . . . . . 8 ⊢ ℂ ∈ V
12		ssexg 4251	. . . . . . . 8 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V)
13	10, 11, 12	sylancl 413	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∪ {0}) ∈ V)
14		nn0ex 9507	. . . . . . 7 ⊢ ℕ₀ ∈ V
15		elmapg 6897	. . . . . . 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 5524	. . . 4 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
19		plyadd.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))
20		elmapg 6897	. . . . . . 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 5524	. . . 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 15661	. . 3 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘_𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘))))
29	3	nn0zd 9704	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
30	4	nn0zd 9704	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
31		zdcle 9659	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑀 ≤ 𝑁)
32	29, 30, 31	syl2anc 411	. . . . 5 ⊢ (𝜑 → DECID 𝑀 ≤ 𝑁)
33	4, 3, 32	ifcldcd 3662	. . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ₀)
34		eqid 2234	. . . . . . 7 ⊢ (𝑆 ∪ {0}) = (𝑆 ∪ {0})
35		plyadd.3	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
36	7, 34, 35	un0addcl 9534	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∪ {0}) ∧ 𝑦 ∈ (𝑆 ∪ {0}))) → (𝑥 + 𝑦) ∈ (𝑆 ∪ {0}))
37	14	a1i 9	. . . . . 6 ⊢ (𝜑 → ℕ₀ ∈ V)
38		inidm 3432	. . . . . 6 ⊢ (ℕ₀ ∩ ℕ₀) = ℕ₀
39	36, 17, 22, 37, 37, 38	off 6281	. . . . 5 ⊢ (𝜑 → (𝐴 ∘_𝑓 + 𝐵):ℕ₀⟶(𝑆 ∪ {0}))
40		elfznn0 10455	. . . . 5 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ₀)
41		ffvelcdm 5812	. . . . 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 15655	. . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘_𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘(𝑆 ∪ {0})))
44	28, 43	eqeltrd 2311	. 2 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) ∈ (Poly‘(𝑆 ∪ {0})))
45		plyun0 15650	. 2 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
46	44, 45	eleqtrdi 2327	1 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) ∈ (Poly‘𝑆))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 842 = wceq 1398 ∈ wcel 2205 Vcvv 2815 ∪ cun 3211 ⊆ wss 3213 ifcif 3622 {csn 3691 class class class wbr 4111 ↦ cmpt 4173 “ cima 4754 ⟶wf 5350 ‘cfv 5354 (class class class)co 6052 ∘_𝑓 cof 6266 ↑_𝑚 cmap 6884 ℂcc 8130 0cc0 8132 1c1 8133 + caddc 8135 · cmul 8137 ≤ cle 8314 ℕ₀cn0 9501 ℤcz 9582 ℤ_≥cuz 9859 ...cfz 10348 ↑cexp 10907 Σcsu 12046 Polycply 15642

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4227 ax-sep 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-iinf 4712 ax-cnex 8223 ax-resscn 8224 ax-1cn 8225 ax-1re 8226 ax-icn 8227 ax-addcl 8228 ax-addrcl 8229 ax-mulcl 8230 ax-mulrcl 8231 ax-addcom 8232 ax-mulcom 8233 ax-addass 8234 ax-mulass 8235 ax-distr 8236 ax-i2m1 8237 ax-0lt1 8238 ax-1rid 8239 ax-0id 8240 ax-rnegex 8241 ax-precex 8242 ax-cnre 8243 ax-pre-ltirr 8244 ax-pre-ltwlin 8245 ax-pre-lttrn 8246 ax-pre-apti 8247 ax-pre-ltadd 8248 ax-pre-mulgt0 8249 ax-pre-mulext 8250 ax-arch 8251 ax-caucvg 8252

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-if 3623 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-tr 4211 df-id 4416 df-po 4419 df-iso 4420 df-iord 4489 df-on 4491 df-ilim 4492 df-suc 4494 df-iom 4715 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-isom 5363 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-of 6268 df-1st 6336 df-2nd 6337 df-recs 6538 df-irdg 6603 df-frec 6624 df-1o 6649 df-oadd 6653 df-er 6769 df-map 6886 df-en 6978 df-dom 6979 df-fin 6980 df-sup 7277 df-pnf 8315 df-mnf 8316 df-xr 8317 df-ltxr 8318 df-le 8319 df-sub 8451 df-neg 8452 df-reap 8854 df-ap 8861 df-div 8952 df-inn 9243 df-2 9301 df-3 9302 df-4 9303 df-n0 9502 df-z 9583 df-uz 9860 df-q 9958 df-rp 9993 df-fz 10349 df-fzo 10484 df-seqfrec 10817 df-exp 10908 df-ihash 11147 df-cj 11535 df-re 11536 df-im 11537 df-rsqrt 11691 df-abs 11692 df-clim 11972 df-sumdc 12047 df-ply 15644

This theorem is referenced by: plyadd 15665

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