Theorem coe11 26293

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 6863	. . 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 2821	. 2 ⊢ (𝐹 = 𝐺 → 𝐴 = 𝐵)
5		simp3 1150	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
6	5	cnveqd 5845	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → ◡𝐴 = ◡𝐵)
7	6	imaeq1d 6045	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (◡𝐴 “ (ℂ ∖ {0})) = (◡𝐵 “ (ℂ ∖ {0})))
8	7	supeq1d 9389	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ) = sup((◡𝐵 “ (ℂ ∖ {0})), ℕ0, < ))
9	2	dgrval 26268	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
10	9	3ad2ant1 1145	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
11	3	dgrval 26268	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) = sup((◡𝐵 “ (ℂ ∖ {0})), ℕ0, < ))
12	11	3ad2ant2 1146	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (deg‘𝐺) = sup((◡𝐵 “ (ℂ ∖ {0})), ℕ0, < ))
13	8, 10, 12	3eqtr4d 2806	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (deg‘𝐹) = (deg‘𝐺))
14	13	oveq2d 7408	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → (0...(deg‘𝐹)) = (0...(deg‘𝐺)))
15		simpl3 1206	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → 𝐴 = 𝐵)
16	15	fveq1d 6865	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → (𝐴‘𝑘) = (𝐵‘𝑘))
17	16	oveq1d 7407	. . . . . 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 15716	. . . . 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 5193	. . . 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 2761	. . . . . 6 ⊢ (deg‘𝐹) = (deg‘𝐹)
21	2, 20	coeid 26278	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))((𝐴‘𝑘) · (𝑧↑𝑘))))
22	21	3ad2ant1 1145	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))((𝐴‘𝑘) · (𝑧↑𝑘))))
23		eqid 2761	. . . . . 6 ⊢ (deg‘𝐺) = (deg‘𝐺)
24	3, 23	coeid 26278	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐺))((𝐵‘𝑘) · (𝑧↑𝑘))))
25	24	3ad2ant2 1146	. . . 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 2806	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐴 = 𝐵) → 𝐹 = 𝐺)
27	26	3expia 1133	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 = 𝐵 → 𝐹 = 𝐺))
28	4, 27	impbid2 228	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 𝐺 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ∖ cdif 3901  {csn 4581   ↦ cmpt 5180  ^◡ccnv 5644   “ cima 5648  ‘cfv 6517  (class class class)co 7392  supcsup 9383  ℂcc 11068  0cc0 11070   · cmul 11075   < clt 11213  ℕ₀cn0 12478  ...cfz 13509  ↑cexp 14071  Σcsu 15696  Polycply 26224  coeffccoe 26226  degcdgr 26227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-inf2 9593  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-of 7656  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-map 8805  df-pm 8806  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-sup 9385  df-inf 9386  df-oi 9455  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-2 12277  df-3 12278  df-n0 12479  df-z 12566  df-uz 12837  df-rp 12991  df-fz 13510  df-fzo 13657  df-fl 13799  df-seq 14012  df-exp 14072  df-hash 14341  df-cj 15109  df-re 15110  df-im 15111  df-sqrt 15245  df-abs 15246  df-clim 15498  df-rlim 15499  df-sum 15697  df-0p 25712  df-ply 26228  df-coe 26230  df-dgr 26231

This theorem is referenced by: (None)