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Mirrors > Home > MPE Home > Th. List > coe11 | Structured version Visualization version GIF version |
Description: The coefficient function is one-to-one, so if the coefficients are equal then the functions are equal and vice-versa. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
coefv0.1 | ⊢ 𝐴 = (coeff‘𝐹) |
coeadd.2 | ⊢ 𝐵 = (coeff‘𝐺) |
Ref | Expression |
---|---|
coe11 | ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 𝐺 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6906 | . . 3 ⊢ (𝐹 = 𝐺 → (coeff‘𝐹) = (coeff‘𝐺)) | |
2 | coefv0.1 | . . 3 ⊢ 𝐴 = (coeff‘𝐹) | |
3 | coeadd.2 | . . 3 ⊢ 𝐵 = (coeff‘𝐺) | |
4 | 1, 2, 3 | 3eqtr4g 2799 | . 2 ⊢ (𝐹 = 𝐺 → 𝐴 = 𝐵) |
5 | simp3 1137 | . . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
6 | 5 | cnveqd 5888 | . . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → ◡𝐴 = ◡𝐵) |
7 | 6 | imaeq1d 6078 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (◡𝐴 “ (ℂ ∖ {0})) = (◡𝐵 “ (ℂ ∖ {0}))) |
8 | 7 | supeq1d 9483 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ) = sup((◡𝐵 “ (ℂ ∖ {0})), ℕ0, < )) |
9 | 2 | dgrval 26281 | . . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )) |
10 | 9 | 3ad2ant1 1132 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )) |
11 | 3 | dgrval 26281 | . . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) = sup((◡𝐵 “ (ℂ ∖ {0})), ℕ0, < )) |
12 | 11 | 3ad2ant2 1133 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (deg‘𝐺) = sup((◡𝐵 “ (ℂ ∖ {0})), ℕ0, < )) |
13 | 8, 10, 12 | 3eqtr4d 2784 | . . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (deg‘𝐹) = (deg‘𝐺)) |
14 | 13 | oveq2d 7446 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (0...(deg‘𝐹)) = (0...(deg‘𝐺))) |
15 | simpl3 1192 | . . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → 𝐴 = 𝐵) | |
16 | 15 | fveq1d 6908 | . . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → (𝐴‘𝑘) = (𝐵‘𝑘)) |
17 | 16 | oveq1d 7445 | . . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = ((𝐵‘𝑘) · (𝑧↑𝑘))) |
18 | 14, 17 | sumeq12dv 15738 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → Σ𝑘 ∈ (0...(deg‘𝐹))((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(deg‘𝐺))((𝐵‘𝑘) · (𝑧↑𝑘))) |
19 | 18 | mpteq2dv 5249 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐺))((𝐵‘𝑘) · (𝑧↑𝑘)))) |
20 | eqid 2734 | . . . . . 6 ⊢ (deg‘𝐹) = (deg‘𝐹) | |
21 | 2, 20 | coeid 26291 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))((𝐴‘𝑘) · (𝑧↑𝑘)))) |
22 | 21 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))((𝐴‘𝑘) · (𝑧↑𝑘)))) |
23 | eqid 2734 | . . . . . 6 ⊢ (deg‘𝐺) = (deg‘𝐺) | |
24 | 3, 23 | coeid 26291 | . . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐺))((𝐵‘𝑘) · (𝑧↑𝑘)))) |
25 | 24 | 3ad2ant2 1133 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐺))((𝐵‘𝑘) · (𝑧↑𝑘)))) |
26 | 19, 22, 25 | 3eqtr4d 2784 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐹 = 𝐺) |
27 | 26 | 3expia 1120 | . 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 = 𝐵 → 𝐹 = 𝐺)) |
28 | 4, 27 | impbid2 226 | 1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 𝐺 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∖ cdif 3959 {csn 4630 ↦ cmpt 5230 ◡ccnv 5687 “ cima 5691 ‘cfv 6562 (class class class)co 7430 supcsup 9477 ℂcc 11150 0cc0 11152 · cmul 11157 < clt 11292 ℕ0cn0 12523 ...cfz 13543 ↑cexp 14098 Σcsu 15718 Polycply 26237 coeffccoe 26239 degcdgr 26240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-rlim 15521 df-sum 15719 df-0p 25718 df-ply 26241 df-coe 26243 df-dgr 26244 |
This theorem is referenced by: (None) |
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