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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 6884 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = (PwSer1‘𝑅)) | |
3 | ply1val.2 | . . . . . 6 ⊢ 𝑆 = (PwSer1‘𝑅) | |
4 | 2, 3 | eqtr4di 2784 | . . . . 5 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = 𝑆) |
5 | oveq2 7412 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1o mPoly 𝑟) = (1o mPoly 𝑅)) | |
6 | 5 | fveq2d 6888 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(1o mPoly 𝑟)) = (Base‘(1o mPoly 𝑅))) |
7 | 4, 6 | oveq12d 7422 | . . . 4 ⊢ (𝑟 = 𝑅 → ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))) = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))) |
8 | df-ply1 22051 | . . . 4 ⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟)))) | |
9 | ovex 7437 | . . . 4 ⊢ (𝑆 ↾s (Base‘(1o mPoly 𝑅))) ∈ V | |
10 | 7, 8, 9 | fvmpt 6991 | . . 3 ⊢ (𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))) |
11 | fvprc 6876 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
12 | ress0 17194 | . . . . 5 ⊢ (∅ ↾s (Base‘(1o mPoly 𝑅))) = ∅ | |
13 | 11, 12 | eqtr4di 2784 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (∅ ↾s (Base‘(1o mPoly 𝑅)))) |
14 | fvprc 6876 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅) | |
15 | 3, 14 | eqtrid 2778 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅) |
16 | 15 | oveq1d 7419 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑆 ↾s (Base‘(1o mPoly 𝑅))) = (∅ ↾s (Base‘(1o mPoly 𝑅)))) |
17 | 13, 16 | eqtr4d 2769 | . . 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 2754 | 1 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∅c0 4317 ‘cfv 6536 (class class class)co 7404 1oc1o 8457 Basecbs 17150 ↾s cress 17179 mPoly cmpl 21795 PwSer1cps1 22044 Poly1cpl1 22046 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-1cn 11167 ax-addcl 11169 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-nn 12214 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-ply1 22051 |
This theorem is referenced by: ply1bas 22064 ply1crng 22067 ply1assa 22068 ply1bascl 22072 ply1plusg 22092 ply1vsca 22093 ply1mulr 22094 ply1ring 22116 ply1lmod 22120 ply1sca 22121 |
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