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Mirrors > Home > MPE Home > Th. List > fvcoe1 | Structured version Visualization version GIF version |
Description: Value of a multivariate coefficient in terms of the coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
coe1fval.a | ⊢ 𝐴 = (coe1‘𝐹) |
Ref | Expression |
---|---|
fvcoe1 | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1o)) → (𝐹‘𝑋) = (𝐴‘(𝑋‘∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 7839 | . . . . 5 ⊢ 1o = {∅} | |
2 | nn0ex 11625 | . . . . 5 ⊢ ℕ0 ∈ V | |
3 | 0ex 5014 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 1, 2, 3 | mapsnconst 8170 | . . . 4 ⊢ (𝑋 ∈ (ℕ0 ↑𝑚 1o) → 𝑋 = (1o × {(𝑋‘∅)})) |
5 | 4 | adantl 475 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1o)) → 𝑋 = (1o × {(𝑋‘∅)})) |
6 | 5 | fveq2d 6437 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1o)) → (𝐹‘𝑋) = (𝐹‘(1o × {(𝑋‘∅)}))) |
7 | elmapi 8144 | . . . 4 ⊢ (𝑋 ∈ (ℕ0 ↑𝑚 1o) → 𝑋:1o⟶ℕ0) | |
8 | 0lt1o 7851 | . . . 4 ⊢ ∅ ∈ 1o | |
9 | ffvelrn 6606 | . . . 4 ⊢ ((𝑋:1o⟶ℕ0 ∧ ∅ ∈ 1o) → (𝑋‘∅) ∈ ℕ0) | |
10 | 7, 8, 9 | sylancl 582 | . . 3 ⊢ (𝑋 ∈ (ℕ0 ↑𝑚 1o) → (𝑋‘∅) ∈ ℕ0) |
11 | coe1fval.a | . . . 4 ⊢ 𝐴 = (coe1‘𝐹) | |
12 | 11 | coe1fv 19936 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ (𝑋‘∅) ∈ ℕ0) → (𝐴‘(𝑋‘∅)) = (𝐹‘(1o × {(𝑋‘∅)}))) |
13 | 10, 12 | sylan2 588 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1o)) → (𝐴‘(𝑋‘∅)) = (𝐹‘(1o × {(𝑋‘∅)}))) |
14 | 6, 13 | eqtr4d 2864 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1o)) → (𝐹‘𝑋) = (𝐴‘(𝑋‘∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∅c0 4144 {csn 4397 × cxp 5340 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 1oc1o 7819 ↑𝑚 cmap 8122 ℕ0cn0 11618 coe1cco1 19908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-1cn 10310 ax-addcl 10312 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-map 8124 df-nn 11351 df-n0 11619 df-coe1 19913 |
This theorem is referenced by: coe1mul2 19999 ply1coe 20026 deg1ldg 24251 deg1leb 24254 |
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