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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 8110 | . . . . 5 ⊢ 1o = {∅} | |
2 | nn0ex 11897 | . . . . 5 ⊢ ℕ0 ∈ V | |
3 | 0ex 5203 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 1, 2, 3 | mapsnconst 8450 | . . . 4 ⊢ (𝑋 ∈ (ℕ0 ↑m 1o) → 𝑋 = (1o × {(𝑋‘∅)})) |
5 | 4 | adantl 484 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑m 1o)) → 𝑋 = (1o × {(𝑋‘∅)})) |
6 | 5 | fveq2d 6668 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑m 1o)) → (𝐹‘𝑋) = (𝐹‘(1o × {(𝑋‘∅)}))) |
7 | elmapi 8422 | . . . 4 ⊢ (𝑋 ∈ (ℕ0 ↑m 1o) → 𝑋:1o⟶ℕ0) | |
8 | 0lt1o 8123 | . . . 4 ⊢ ∅ ∈ 1o | |
9 | ffvelrn 6843 | . . . 4 ⊢ ((𝑋:1o⟶ℕ0 ∧ ∅ ∈ 1o) → (𝑋‘∅) ∈ ℕ0) | |
10 | 7, 8, 9 | sylancl 588 | . . 3 ⊢ (𝑋 ∈ (ℕ0 ↑m 1o) → (𝑋‘∅) ∈ ℕ0) |
11 | coe1fval.a | . . . 4 ⊢ 𝐴 = (coe1‘𝐹) | |
12 | 11 | coe1fv 20368 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ (𝑋‘∅) ∈ ℕ0) → (𝐴‘(𝑋‘∅)) = (𝐹‘(1o × {(𝑋‘∅)}))) |
13 | 10, 12 | sylan2 594 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑m 1o)) → (𝐴‘(𝑋‘∅)) = (𝐹‘(1o × {(𝑋‘∅)}))) |
14 | 6, 13 | eqtr4d 2859 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑m 1o)) → (𝐹‘𝑋) = (𝐴‘(𝑋‘∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∅c0 4290 {csn 4560 × cxp 5547 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 1oc1o 8089 ↑m cmap 8400 ℕ0cn0 11891 coe1cco1 20340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-1cn 10589 ax-addcl 10591 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-map 8402 df-nn 11633 df-n0 11892 df-coe1 20345 |
This theorem is referenced by: coe1mul2 20431 ply1coe 20458 deg1ldg 24680 deg1leb 24683 |
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