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Mirrors > Home > MPE Home > Th. List > ply1val | Structured version Visualization version GIF version |
Description: The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
ply1val.1 | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1val.2 | ⊢ 𝑆 = (PwSer1‘𝑅) |
Ref | Expression |
---|---|
ply1val | ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1val.1 | . 2 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | fveq2 6825 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = (PwSer1‘𝑅)) | |
3 | ply1val.2 | . . . . . 6 ⊢ 𝑆 = (PwSer1‘𝑅) | |
4 | 2, 3 | eqtr4di 2794 | . . . . 5 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = 𝑆) |
5 | oveq2 7345 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1o mPoly 𝑟) = (1o mPoly 𝑅)) | |
6 | 5 | fveq2d 6829 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(1o mPoly 𝑟)) = (Base‘(1o mPoly 𝑅))) |
7 | 4, 6 | oveq12d 7355 | . . . 4 ⊢ (𝑟 = 𝑅 → ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))) = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))) |
8 | df-ply1 21459 | . . . 4 ⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟)))) | |
9 | ovex 7370 | . . . 4 ⊢ (𝑆 ↾s (Base‘(1o mPoly 𝑅))) ∈ V | |
10 | 7, 8, 9 | fvmpt 6931 | . . 3 ⊢ (𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))) |
11 | fvprc 6817 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
12 | ress0 17050 | . . . . 5 ⊢ (∅ ↾s (Base‘(1o mPoly 𝑅))) = ∅ | |
13 | 11, 12 | eqtr4di 2794 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (∅ ↾s (Base‘(1o mPoly 𝑅)))) |
14 | fvprc 6817 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅) | |
15 | 3, 14 | eqtrid 2788 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅) |
16 | 15 | oveq1d 7352 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑆 ↾s (Base‘(1o mPoly 𝑅))) = (∅ ↾s (Base‘(1o mPoly 𝑅)))) |
17 | 13, 16 | eqtr4d 2779 | . . 3 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))) |
18 | 10, 17 | pm2.61i 182 | . 2 ⊢ (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))) |
19 | 1, 18 | eqtri 2764 | 1 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2105 Vcvv 3441 ∅c0 4269 ‘cfv 6479 (class class class)co 7337 1oc1o 8360 Basecbs 17009 ↾s cress 17038 mPoly cmpl 21215 PwSer1cps1 21452 Poly1cpl1 21454 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-1cn 11030 ax-addcl 11032 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-nn 12075 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-ply1 21459 |
This theorem is referenced by: ply1bas 21472 ply1crng 21475 ply1assa 21476 ply1bascl 21480 ply1plusg 21502 ply1vsca 21503 ply1mulr 21504 ply1ring 21525 ply1lmod 21529 ply1sca 21530 |
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