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Theorem plyss 26164

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

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . . 8 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → 𝑇 ⊆ ℂ)
2		cnex 11119	. . . . . . . 8 ⊢ ℂ ∈ V
3		ssexg 5264	. . . . . . . 8 ⊢ ((𝑇 ⊆ ℂ ∧ ℂ ∈ V) → 𝑇 ∈ V)
4	1, 2, 3	sylancl 587	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → 𝑇 ∈ V)
5		snex 5381	. . . . . . 7 ⊢ {0} ∈ V
6		unexg 7697	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ {0} ∈ V) → (𝑇 ∪ {0}) ∈ V)
7	4, 5, 6	sylancl 587	. . . . . 6 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑇 ∪ {0}) ∈ V)
8		unss1 4125	. . . . . . 7 ⊢ (𝑆 ⊆ 𝑇 → (𝑆 ∪ {0}) ⊆ (𝑇 ∪ {0}))
9	8	adantr 480	. . . . . 6 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑆 ∪ {0}) ⊆ (𝑇 ∪ {0}))
10		mapss 8837	. . . . . 6 ⊢ (((𝑇 ∪ {0}) ∈ V ∧ (𝑆 ∪ {0}) ⊆ (𝑇 ∪ {0})) → ((𝑆 ∪ {0}) ↑_m ℕ₀) ⊆ ((𝑇 ∪ {0}) ↑_m ℕ₀))
11	7, 9, 10	syl2anc 585	. . . . 5 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → ((𝑆 ∪ {0}) ↑_m ℕ₀) ⊆ ((𝑇 ∪ {0}) ↑_m ℕ₀))
12		ssrexv 3991	. . . . 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)))))
13	11, 12	syl 17	. . . 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)))))
14	13	reximdv 3152	. . 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)))))
15	14	ss2abdv 4005	. 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)))})
16		sstr 3930	. . 3 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → 𝑆 ⊆ ℂ)
17		plyval 26158	. . 3 ⊢ (𝑆 ⊆ ℂ → (Poly‘𝑆) = {𝑓 ∣ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
18	16, 17	syl 17	. 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)))})
19		plyval 26158	. . 3 ⊢ (𝑇 ⊆ ℂ → (Poly‘𝑇) = {𝑓 ∣ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑇 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
20	19	adantl 481	. 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)))})
21	15, 18, 20	3sstr4d 3977	1 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {cab 2714 ∃wrex 3061 Vcvv 3429 ∪ cun 3887 ⊆ wss 3889 {csn 4567 ↦ cmpt 5166 ‘cfv 6498 (class class class)co 7367 ↑_m cmap 8773 ℂcc 11036 0cc0 11038 · cmul 11043 ℕ₀cn0 12437 ...cfz 13461 ↑cexp 14023 Σcsu 15648 Polycply 26149

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-1cn 11096 ax-addcl 11098

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-map 8775 df-nn 12175 df-n0 12438 df-ply 26153

This theorem is referenced by: plyssc 26165 elqaa 26288 aacjcl 26293 aalioulem3 26300 itgoss 43591 cnsrplycl 43595

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