plyval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > plyval		Structured version Visualization version GIF version

Theorem plyval 26247

Description: Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)

**Assertion**
Ref	Expression
plyval	⊢ (𝑆 ⊆ ℂ → (Poly‘𝑆) = {𝑓 ∣ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})

Distinct variable group: 𝑘,𝑎,𝑛,𝑧,𝑓,𝑆

Proof of Theorem plyval Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnex 11234	. . 3 ⊢ ℂ ∈ V
2	1	elpw2 5340	. 2 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
3		uneq1 4171	. . . . . . 7 ⊢ (𝑥 = 𝑆 → (𝑥 ∪ {0}) = (𝑆 ∪ {0}))
4	3	oveq1d 7446	. . . . . 6 ⊢ (𝑥 = 𝑆 → ((𝑥 ∪ {0}) ↑_m ℕ₀) = ((𝑆 ∪ {0}) ↑_m ℕ₀))
5	4	rexeqdv 3325	. . . . 5 ⊢ (𝑥 = 𝑆 → (∃𝑎 ∈ ((𝑥 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) ↔ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
6	5	rexbidv 3177	. . . 4 ⊢ (𝑥 = 𝑆 → (∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) ↔ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
7	6	abbidv 2806	. . 3 ⊢ (𝑥 = 𝑆 → {𝑓 ∣ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} = {𝑓 ∣ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
8		df-ply 26242	. . 3 ⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
9		nn0ex 12530	. . . 4 ⊢ ℕ₀ ∈ V
10		ovex 7464	. . . 4 ⊢ ((𝑆 ∪ {0}) ↑_m ℕ₀) ∈ V
11	9, 10	ab2rexex 8003	. . 3 ⊢ {𝑓 ∣ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} ∈ V
12	7, 8, 11	fvmpt 7016	. 2 ⊢ (𝑆 ∈ 𝒫 ℂ → (Poly‘𝑆) = {𝑓 ∣ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
13	2, 12	sylbir 235	1 ⊢ (𝑆 ⊆ ℂ → (Poly‘𝑆) = {𝑓 ∣ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {cab 2712 ∃wrex 3068 ∪ cun 3961 ⊆ wss 3963 𝒫 cpw 4605 {csn 4631 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ↑_m cmap 8865 ℂcc 11151 0cc0 11153 · cmul 11158 ℕ₀cn0 12524 ...cfz 13544 ↑cexp 14099 Σcsu 15719 Polycply 26238

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-n0 12525 df-ply 26242

This theorem is referenced by: elply 26249 plyss 26253

W3C validator