plyaddlem - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > plyaddlem		GIF version

Theorem plyaddlem 15069

Description: Lemma for plyadd 15071. (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 15053	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
7	1, 6	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
8		0cnd 8036	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℂ)
9	8	snssd 3768	. . . . . . . . 9 ⊢ (𝜑 → {0} ⊆ ℂ)
10	7, 9	unssd 3340	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
11		cnex 8020	. . . . . . . 8 ⊢ ℂ ∈ V
12		ssexg 4173	. . . . . . . 8 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V)
13	10, 11, 12	sylancl 413	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∪ {0}) ∈ V)
14		nn0ex 9272	. . . . . . 7 ⊢ ℕ₀ ∈ V
15		elmapg 6729	. . . . . . 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 5423	. . . 4 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
19		plyadd.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))
20		elmapg 6729	. . . . . . 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 5423	. . . 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 15067	. . 3 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘_𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘))))
29	3	nn0zd 9463	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
30	4	nn0zd 9463	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
31		zdcle 9419	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑀 ≤ 𝑁)
32	29, 30, 31	syl2anc 411	. . . . 5 ⊢ (𝜑 → DECID 𝑀 ≤ 𝑁)
33	4, 3, 32	ifcldcd 3598	. . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ₀)
34		eqid 2196	. . . . . . 7 ⊢ (𝑆 ∪ {0}) = (𝑆 ∪ {0})
35		plyadd.3	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
36	7, 34, 35	un0addcl 9299	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∪ {0}) ∧ 𝑦 ∈ (𝑆 ∪ {0}))) → (𝑥 + 𝑦) ∈ (𝑆 ∪ {0}))
37	14	a1i 9	. . . . . 6 ⊢ (𝜑 → ℕ₀ ∈ V)
38		inidm 3373	. . . . . 6 ⊢ (ℕ₀ ∩ ℕ₀) = ℕ₀
39	36, 17, 22, 37, 37, 38	off 6152	. . . . 5 ⊢ (𝜑 → (𝐴 ∘_𝑓 + 𝐵):ℕ₀⟶(𝑆 ∪ {0}))
40		elfznn0 10206	. . . . 5 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ₀)
41		ffvelcdm 5698	. . . . 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 15061	. . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘_𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘(𝑆 ∪ {0})))
44	28, 43	eqeltrd 2273	. 2 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) ∈ (Poly‘(𝑆 ∪ {0})))
45		plyun0 15056	. 2 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
46	44, 45	eleqtrdi 2289	1 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) ∈ (Poly‘𝑆))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 835 = wceq 1364 ∈ wcel 2167 Vcvv 2763 ∪ cun 3155 ⊆ wss 3157 ifcif 3562 {csn 3623 class class class wbr 4034 ↦ cmpt 4095 “ cima 4667 ⟶wf 5255 ‘cfv 5259 (class class class)co 5925 ∘_𝑓 cof 6137 ↑_𝑚 cmap 6716 ℂcc 7894 0cc0 7896 1c1 7897 + caddc 7899 · cmul 7901 ≤ cle 8079 ℕ₀cn0 9266 ℤcz 9343 ℤ_≥cuz 9618 ...cfz 10100 ↑cexp 10647 Σcsu 11535 Polycply 15048

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-of 6139 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-frec 6458 df-1o 6483 df-oadd 6487 df-er 6601 df-map 6718 df-en 6809 df-dom 6810 df-fin 6811 df-sup 7059 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-n0 9267 df-z 9344 df-uz 9619 df-q 9711 df-rp 9746 df-fz 10101 df-fzo 10235 df-seqfrec 10557 df-exp 10648 df-ihash 10885 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181 df-clim 11461 df-sumdc 11536 df-ply 15050

This theorem is referenced by: plyadd 15071

W3C validator