Theorem coe11 26486

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.)

Hypotheses

Ref	Expression
coefv0.1	⊢ 𝐴 = (coeff‘𝐹)
coeadd.2	⊢ 𝐵 = (coeff‘𝐺)

Assertion

Ref	Expression
coe11	⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 𝐺 ↔ 𝐴 = 𝐵))

Proof of Theorem coe11

Dummy variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6882	. . 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 2822	. 2 ⊢ (𝐹 = 𝐺 → 𝐴 = 𝐵)
5		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
6	5	cnveqd 5859	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → ◡𝐴 = ◡𝐵)
7	6	imaeq1d 6059	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (◡𝐴 “ (ℂ ∖ {0})) = (◡𝐵 “ (ℂ ∖ {0})))
8	7	supeq1d 9420	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ) = sup((◡𝐵 “ (ℂ ∖ {0})), ℕ0, < ))
9	2	dgrval 26461	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
10	9	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
11	3	dgrval 26461	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) = sup((◡𝐵 “ (ℂ ∖ {0})), ℕ0, < ))
12	11	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (deg‘𝐺) = sup((◡𝐵 “ (ℂ ∖ {0})), ℕ0, < ))
13	8, 10, 12	3eqtr4d 2807	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (deg‘𝐹) = (deg‘𝐺))
14	13	oveq2d 7433	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (0...(deg‘𝐹)) = (0...(deg‘𝐺)))
15		simpl3 1212	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → 𝐴 = 𝐵)
16	15	fveq1d 6884	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → (𝐴‘𝑘) = (𝐵‘𝑘))
17	16	oveq1d 7432	. . . . . 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 15796	. . . . 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 5203	. . . 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 2762	. . . . . 6 ⊢ (deg‘𝐹) = (deg‘𝐹)
21	2, 20	coeid 26471	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))((𝐴‘𝑘) · (𝑧↑𝑘))))
22	21	3ad2ant1 1151	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))((𝐴‘𝑘) · (𝑧↑𝑘))))
23		eqid 2762	. . . . . 6 ⊢ (deg‘𝐺) = (deg‘𝐺)
24	3, 23	coeid 26471	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐺))((𝐵‘𝑘) · (𝑧↑𝑘))))
25	24	3ad2ant2 1152	. . . 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 2807	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐹 = 𝐺)
27	26	3expia 1139	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 = 𝐵 → 𝐹 = 𝐺))
28	4, 27	impbid2 229	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 𝐺 ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∖ cdif 3899  {csn 4587   ↦ cmpt 5190  ^◡ccnv 5658   “ cima 5662  ‘cfv 6537  (class class class)co 7417  supcsup 9414  ℂcc 11126  0cc0 11128   · cmul 11133   < clt 11271  ℕ₀cn0 12532  ...cfz 13565  ↑cexp 14129  Σcsu 15777  Polycply 26416  coeffccoe 26418  degcdgr 26419

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-inf2 9624  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-pm 8833  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-sup 9416  df-inf 9417  df-oi 9486  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-n0 12533  df-z 12620  df-uz 12892  df-rp 13047  df-fz 13566  df-fzo 13714  df-fl 13857  df-seq 14070  df-exp 14130  df-hash 14399  df-cj 15190  df-re 15191  df-im 15192  df-sqrt 15326  df-abs 15327  df-clim 15579  df-rlim 15580  df-sum 15778  df-0p 25904  df-ply 26420  df-coe 26422  df-dgr 26423

This theorem is used by: (None)