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Theorem itgoss 40508
 Description: An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Assertion
Ref Expression
itgoss ((𝑆𝑇𝑇 ⊆ ℂ) → (IntgOver‘𝑆) ⊆ (IntgOver‘𝑇))

Proof of Theorem itgoss
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyss 24900 . . . . 5 ((𝑆𝑇𝑇 ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘𝑇))
2 ssrexv 3961 . . . . 5 ((Poly‘𝑆) ⊆ (Poly‘𝑇) → (∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ (Poly‘𝑇)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
31, 2syl 17 . . . 4 ((𝑆𝑇𝑇 ⊆ ℂ) → (∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ (Poly‘𝑇)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
43ralrimivw 3114 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → ∀𝑎 ∈ ℂ (∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ (Poly‘𝑇)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
5 ss2rab 3977 . . 3 ({𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑇)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ↔ ∀𝑎 ∈ ℂ (∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ (Poly‘𝑇)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
64, 5sylibr 237 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑇)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
7 sstr 3902 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → 𝑆 ⊆ ℂ)
8 itgoval 40506 . . 3 (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
97, 8syl 17 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
10 itgoval 40506 . . 3 (𝑇 ⊆ ℂ → (IntgOver‘𝑇) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑇)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
1110adantl 485 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → (IntgOver‘𝑇) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑇)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
126, 9, 113sstr4d 3941 1 ((𝑆𝑇𝑇 ⊆ ℂ) → (IntgOver‘𝑆) ⊆ (IntgOver‘𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∀wral 3070  ∃wrex 3071  {crab 3074   ⊆ wss 3860  ‘cfv 6339  ℂcc 10578  0cc0 10580  1c1 10581  Polycply 24885  coeffccoe 24887  degcdgr 24888  IntgOvercitgo 40502 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-1cn 10638  ax-addcl 10640 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-map 8423  df-nn 11680  df-n0 11940  df-ply 24889  df-itgo 40504 This theorem is referenced by: (None)
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