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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 7572 | . . . . 5 ⊢ 1𝑜 = {∅} | |
2 | nn0ex 11298 | . . . . 5 ⊢ ℕ0 ∈ V | |
3 | 0ex 4790 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 1, 2, 3 | mapsnconst 7903 | . . . 4 ⊢ (𝑋 ∈ (ℕ0 ↑𝑚 1𝑜) → 𝑋 = (1𝑜 × {(𝑋‘∅)})) |
5 | 4 | adantl 482 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → 𝑋 = (1𝑜 × {(𝑋‘∅)})) |
6 | 5 | fveq2d 6195 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝐹‘𝑋) = (𝐹‘(1𝑜 × {(𝑋‘∅)}))) |
7 | elmapi 7879 | . . . 4 ⊢ (𝑋 ∈ (ℕ0 ↑𝑚 1𝑜) → 𝑋:1𝑜⟶ℕ0) | |
8 | 0lt1o 7584 | . . . 4 ⊢ ∅ ∈ 1𝑜 | |
9 | ffvelrn 6357 | . . . 4 ⊢ ((𝑋:1𝑜⟶ℕ0 ∧ ∅ ∈ 1𝑜) → (𝑋‘∅) ∈ ℕ0) | |
10 | 7, 8, 9 | sylancl 694 | . . 3 ⊢ (𝑋 ∈ (ℕ0 ↑𝑚 1𝑜) → (𝑋‘∅) ∈ ℕ0) |
11 | coe1fval.a | . . . 4 ⊢ 𝐴 = (coe1‘𝐹) | |
12 | 11 | coe1fv 19576 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ (𝑋‘∅) ∈ ℕ0) → (𝐴‘(𝑋‘∅)) = (𝐹‘(1𝑜 × {(𝑋‘∅)}))) |
13 | 10, 12 | sylan2 491 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝐴‘(𝑋‘∅)) = (𝐹‘(1𝑜 × {(𝑋‘∅)}))) |
14 | 6, 13 | eqtr4d 2659 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝐹‘𝑋) = (𝐴‘(𝑋‘∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∅c0 3915 {csn 4177 × cxp 5112 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1𝑜c1o 7553 ↑𝑚 cmap 7857 ℕ0cn0 11292 coe1cco1 19548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-map 7859 df-nn 11021 df-n0 11293 df-coe1 19553 |
This theorem is referenced by: coe1mul2 19639 ply1coe 19666 deg1ldg 23852 deg1leb 23855 |
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