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Theorem itgoss 42587
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 26146 . . . . 5 ((𝑆 βŠ† 𝑇 ∧ 𝑇 βŠ† β„‚) β†’ (Polyβ€˜π‘†) βŠ† (Polyβ€˜π‘‡))
2 ssrexv 4049 . . . . 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 4071 . 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 3988 . . 3 ((𝑆 βŠ† 𝑇 ∧ 𝑇 βŠ† β„‚) β†’ 𝑆 βŠ† β„‚)
7 itgoval 42585 . . 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 42585 . . 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 4027 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 1534   ∈ wcel 2099  βˆƒwrex 3067  {crab 3429   βŠ† wss 3947  β€˜cfv 6548  β„‚cc 11137  0cc0 11139  1c1 11140  Polycply 26131  coeffccoe 26133  degcdgr 26134  IntgOvercitgo 42581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-1cn 11197  ax-addcl 11199
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-map 8847  df-nn 12244  df-n0 12504  df-ply 26135  df-itgo 42583
This theorem is referenced by: (None)
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