Theorem itgoss 43270

Description: An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Assertion

Ref	Expression
itgoss	⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (IntgOver‘𝑆) ⊆ (IntgOver‘𝑇))

Proof of Theorem itgoss

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plyss 26141	. . . . 5 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘𝑇))
2		ssrexv 4001	. . . . 5 ⊢ ((Poly‘𝑆) ⊆ (Poly‘𝑇) → (∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ (Poly‘𝑇)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
3	1, 2	syl 17	. . . 4 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ (Poly‘𝑇)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
4	3	adantr 480	. . 3 ⊢ (((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) ∧ 𝑎 ∈ ℂ) → (∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ (Poly‘𝑇)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
5	4	ss2rabdv 4025	. 2 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑇)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
6		sstr 3940	. . 3 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → 𝑆 ⊆ ℂ)
7		itgoval 43268	. . 3 ⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
8	6, 7	syl 17	. 2 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
9		itgoval 43268	. . 3 ⊢ (𝑇 ⊆ ℂ → (IntgOver‘𝑇) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑇)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
10	9	adantl 481	. 2 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (IntgOver‘𝑇) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑇)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
11	5, 8, 10	3sstr4d 3987	1 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (IntgOver‘𝑆) ⊆ (IntgOver‘𝑇))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058  {crab 3397   ⊆ wss 3899  ‘cfv 6489  ℂcc 11014  0cc0 11016  1c1 11017  Polycply 26126  coeffccoe 26128  degcdgr 26129  IntgOvercitgo 43264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11072  ax-1cn 11074  ax-addcl 11076

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-map 8761  df-nn 12136  df-n0 12392  df-ply 26130  df-itgo 43266

This theorem is referenced by: (None)