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Theorem itgoss 42457
Description: An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Assertion
Ref Expression
itgoss ((𝑆 βŠ† 𝑇 ∧ 𝑇 βŠ† β„‚) β†’ (IntgOverβ€˜π‘†) βŠ† (IntgOverβ€˜π‘‡))

Proof of Theorem itgoss
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyss 26077 . . . . 5 ((𝑆 βŠ† 𝑇 ∧ 𝑇 βŠ† β„‚) β†’ (Polyβ€˜π‘†) βŠ† (Polyβ€˜π‘‡))
2 ssrexv 4044 . . . . 5 ((Polyβ€˜π‘†) βŠ† (Polyβ€˜π‘‡) β†’ (βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1) β†’ βˆƒπ‘ ∈ (Polyβ€˜π‘‡)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)))
31, 2syl 17 . . . 4 ((𝑆 βŠ† 𝑇 ∧ 𝑇 βŠ† β„‚) β†’ (βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1) β†’ βˆƒπ‘ ∈ (Polyβ€˜π‘‡)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)))
43adantr 480 . . 3 (((𝑆 βŠ† 𝑇 ∧ 𝑇 βŠ† β„‚) ∧ π‘Ž ∈ β„‚) β†’ (βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1) β†’ βˆƒπ‘ ∈ (Polyβ€˜π‘‡)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)))
54ss2rabdv 4066 . 2 ((𝑆 βŠ† 𝑇 ∧ 𝑇 βŠ† β„‚) β†’ {π‘Ž ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)} βŠ† {π‘Ž ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘‡)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
6 sstr 3983 . . 3 ((𝑆 βŠ† 𝑇 ∧ 𝑇 βŠ† β„‚) β†’ 𝑆 βŠ† β„‚)
7 itgoval 42455 . . 3 (𝑆 βŠ† β„‚ β†’ (IntgOverβ€˜π‘†) = {π‘Ž ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
86, 7syl 17 . 2 ((𝑆 βŠ† 𝑇 ∧ 𝑇 βŠ† β„‚) β†’ (IntgOverβ€˜π‘†) = {π‘Ž ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
9 itgoval 42455 . . 3 (𝑇 βŠ† β„‚ β†’ (IntgOverβ€˜π‘‡) = {π‘Ž ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘‡)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
109adantl 481 . 2 ((𝑆 βŠ† 𝑇 ∧ 𝑇 βŠ† β„‚) β†’ (IntgOverβ€˜π‘‡) = {π‘Ž ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘‡)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
115, 8, 103sstr4d 4022 1 ((𝑆 βŠ† 𝑇 ∧ 𝑇 βŠ† β„‚) β†’ (IntgOverβ€˜π‘†) βŠ† (IntgOverβ€˜π‘‡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3062  {crab 3424   βŠ† wss 3941  β€˜cfv 6534  β„‚cc 11105  0cc0 11107  1c1 11108  Polycply 26062  coeffccoe 26064  degcdgr 26065  IntgOvercitgo 42451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-1cn 11165  ax-addcl 11167
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-map 8819  df-nn 12212  df-n0 12472  df-ply 26066  df-itgo 42453
This theorem is referenced by: (None)
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