Theorem plyss 15841

Description: The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)

Assertion

Ref	Expression
plyss	⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘𝑇))

Proof of Theorem plyss

Dummy variables 𝑎 𝑓 𝑛 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . . . 8 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → 𝑇 ⊆ ℂ)
2		cnex 8303	. . . . . . . 8 ⊢ ℂ ∈ V
3		ssexg 4272	. . . . . . . 8 ⊢ ((𝑇 ⊆ ℂ ∧ ℂ ∈ V) → 𝑇 ∈ V)
4	1, 2, 3	sylancl 417	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → 𝑇 ∈ V)
5		c0ex 8320	. . . . . . . 8 ⊢ 0 ∈ V
6	5	snex 4322	. . . . . . 7 ⊢ {0} ∈ V
7		unexg 4589	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ {0} ∈ V) → (𝑇 ∪ {0}) ∈ V)
8	4, 6, 7	sylancl 417	. . . . . 6 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑇 ∪ {0}) ∈ V)
9		unss1 3398	. . . . . . 7 ⊢ (𝑆 ⊆ 𝑇 → (𝑆 ∪ {0}) ⊆ (𝑇 ∪ {0}))
10	9	adantr 276	. . . . . 6 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑆 ∪ {0}) ⊆ (𝑇 ∪ {0}))
11		mapss 6973	. . . . . 6 ⊢ (((𝑇 ∪ {0}) ∈ V ∧ (𝑆 ∪ {0}) ⊆ (𝑇 ∪ {0})) → ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ⊆ ((𝑇 ∪ {0}) ↑𝑚 ℕ0))
12	8, 10, 11	syl2anc 415	. . . . 5 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ⊆ ((𝑇 ∪ {0}) ↑𝑚 ℕ0))
13		ssrexv 3313	. . . . 5 ⊢ (((𝑆 ∪ {0}) ↑𝑚 ℕ0) ⊆ ((𝑇 ∪ {0}) ↑𝑚 ℕ0) → (∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) → ∃𝑎 ∈ ((𝑇 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
14	12, 13	syl 14	. . . 4 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) → ∃𝑎 ∈ ((𝑇 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
15	14	reximdv 2651	. . 3 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) → ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑇 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
16	15	ss2abdv 3321	. 2 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} ⊆ {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑇 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
17		sstr 3256	. . 3 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → 𝑆 ⊆ ℂ)
18		plyval 15835	. . 3 ⊢ (𝑆 ⊆ ℂ → (Poly‘𝑆) = {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
19	17, 18	syl 14	. 2 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (Poly‘𝑆) = {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
20		plyval 15835	. . 3 ⊢ (𝑇 ⊆ ℂ → (Poly‘𝑇) = {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑇 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
21	20	adantl 277	. 2 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (Poly‘𝑇) = {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑇 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
22	16, 19, 21	3sstr4d 3293	1 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘𝑇))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {cab 2224  ∃wrex 2529  Vcvv 2821   ∪ cun 3218   ⊆ wss 3220  {csn 3709   ↦ cmpt 4192  ‘cfv 5377  (class class class)co 6085   ↑_𝑚 cmap 6922  ℂcc 8177  0cc0 8179   · cmul 8184  ℕ₀cn0 9565  ...cfz 10413  ↑cexp 10977  Σcsu 12121  Polycply 15831

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-i2m1 8284

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-inn 9306  df-n0 9566  df-ply 15833

This theorem is used by:  plyssc  15842