Theorem plyss 15491

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 8161	. . . . . . . 8 ⊢ ℂ ∈ V
3		ssexg 4229	. . . . . . . 8 ⊢ ((𝑇 ⊆ ℂ ∧ ℂ ∈ V) → 𝑇 ∈ V)
4	1, 2, 3	sylancl 413	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → 𝑇 ∈ V)
5		c0ex 8178	. . . . . . . 8 ⊢ 0 ∈ V
6	5	snex 4277	. . . . . . 7 ⊢ {0} ∈ V
7		unexg 4542	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ {0} ∈ V) → (𝑇 ∪ {0}) ∈ V)
8	4, 6, 7	sylancl 413	. . . . . 6 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑇 ∪ {0}) ∈ V)
9		unss1 3375	. . . . . . 7 ⊢ (𝑆 ⊆ 𝑇 → (𝑆 ∪ {0}) ⊆ (𝑇 ∪ {0}))
10	9	adantr 276	. . . . . 6 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑆 ∪ {0}) ⊆ (𝑇 ∪ {0}))
11		mapss 6865	. . . . . 6 ⊢ (((𝑇 ∪ {0}) ∈ V ∧ (𝑆 ∪ {0}) ⊆ (𝑇 ∪ {0})) → ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ⊆ ((𝑇 ∪ {0}) ↑𝑚 ℕ0))
12	8, 10, 11	syl2anc 411	. . . . 5 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ⊆ ((𝑇 ∪ {0}) ↑𝑚 ℕ0))
13		ssrexv 3291	. . . . 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 2632	. . 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 3299	. 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 3234	. . 3 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → 𝑆 ⊆ ℂ)
18		plyval 15485	. . 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 15485	. . 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 3271	1 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘𝑇))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2201  {cab 2216  ∃wrex 2510  Vcvv 2801   ∪ cun 3197   ⊆ wss 3199  {csn 3670   ↦ cmpt 4151  ‘cfv 5328  (class class class)co 6023   ↑_𝑚 cmap 6822  ℂcc 8035  0cc0 8037   · cmul 8042  ℕ₀cn0 9407  ...cfz 10248  ↑cexp 10806  Σcsu 11936  Polycply 15481

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-i2m1 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-map 6824  df-inn 9149  df-n0 9408  df-ply 15483

This theorem is referenced by:  plyssc  15492